title: Asuka (wrestler)

text: Kana made her debut for Yoshihiro Tajiri's new Smash promotion on June 25, 2010, losing to Syuri in the main event of "Smash.4". After the match, Kana left the arena smiling and later challenged Syuri to a rematch, which was made official on June 30. In the rematch on July 24 at "Smash.5" Kana, now a full blown heel, defeated Syuri via submission in just three minutes. As part of the attempt to get her over as the top villain, Kana wrote a manifesto for the "Weekly Pro-Wrestling" magazine, which was published on August 4 and where she attacked the entire "Joshi Puroresu" community, claiming it was full of wrestlers with no personality and bad attitudes and who wrestled in an unrealistic manner. The article led to a confrontation between Kana and wrestlers from the JWP Joshi Puroresu promotion on August 30 at "Smash.7", but never led to a direct match. Kana then began feuding with Tajiri, which led to an intergender match on December 24 at "Happening Eve", where Tajiri was victorious. Following the loss, Kana reemerged on January 29, 2011, as the leader of the Triple Tails stable with the Shirai sisters, dubbing herself "Sekai no Kana" (literally "World Famous Kana"). Kana then continued working against the opposite sex, teaming with the Shirais to defeat Tajiri, Ken Ohka and Yoshiaki Yago on February 25, but losing to Funaki on March 31.


title: Asuka (wrestler)

text: In April, Kana entered a feud with former WWE wrestler Serena, stemming from a comment she made, calling Kana a talented "amateur" wrestler, while she herself had worked for the top promotion in the world. In response, Kana vowed to make Serena respect her. In the first encounter on April 30 at "Smash.16", Serena teamed with Makoto and Syuri to defeat the Triple Tails in a six-woman tag team match, when Serena pinned Kana with her finishing maneuver, the spear. On May 3 at "Smash.17", Serena defeated Kana with a spear in the first singles match between the two. At the following event on June 9, Smash started a tournament to crown the inaugural Smash Diva Champion, which saw Kana defeat Lin "Bitch" Byron in her first round match. On August 11 at "Smash.20", Kana defeated Syuri to set up a rematch with Serena in the finals of the tournament. On September 8, Kana defeated Serena to become the inaugural Smash Diva Champion. On November 24 at "Smash.23", Kana lost the Smash Diva Championship to Tomoka Nakagawa in her first defense. On January 19, 2012, at "We Are Smash", Kana defeated Nakagawa in a rematch to regain the title. Kana's second reign ended also in her first defense, when she lost the title to Syuri on February 19 at "Smash.25", after which the two longtime rivals hugged and made peace with each other. On March 14, Smash held "Smash.Final", the promotion's final event before ceasing operations, during which Kana teamed with Último Dragón to defeat Mentallo and Syuri in a tag team match.


title: Asuka (wrestler)

text: On August 3, 2011, it was reported that Kana would be making her debut in the United States in early October, working for all-female promotion Shimmer Women Athletes on 1 and 2 and for Chikara on 7 and 8. In her Shimmer debut on "Volume 41", Kana was shown in early segments ignoring Sara Del Rey and Cheerleader Melissa's welcoming, establishing herself as a villain in the process. Later, she defeated Mia Yim in her first match for the promotion. Later that same day on "Volume 42", Kana defeated Del Rey via submission. The following day on "Volume 43", Kana was defeated by Cheerleader Melissa in a match to determine the number one contender to the Shimmer Championship. Later that day, Kana defeated LuFisto on "Volume 44" in her final match for the weekend, after which she shook LuFisto's hand. The following weekend, Kana made her debut for Chikara, defeating Jessie McKay in her first match on October 7 in Burlington, North Carolina. The following day, Kana was defeated by Sara Del Rey in the main event of a show in Kingsport, Tennessee. Kana returned to Shimmer Women Athletes on March 17 on "Volume 45"; now a babyface, Kana teamed with LuFisto in a losing effort against Hailey Hatred and Kalamity. Later that same day on "Volume 46", Kana was defeated in a singles match by Mercedes Martinez. The following day on "Volume 47", Kana and LuFisto defeated Christina Von Eerie and MsChif in a tag team match. Kana then defeated Kellie Skater on "Volume 48", in her final match of the weekend.


title: Asuka (wrestler)

text: Kana returned to Shimmer on October 27, 2012, when she and LuFisto unsuccessfully challenged The Canadian NINJAs (Nicole Matthews and Portia Perez) for the Shimmer Tag Team Championship as part of "Volume 49". Later that same night on "Volume 50", Kana was defeated by Ayako Hamada in a singles match. The following night on "Volume 51", Kana defeated Athena in a singles match. Later that night on "Volume 51", Kana teamed with LuFisto in a tag team match (her final match of the weekend), where they defeated Made in Sin (Allysin Kay and Taylor Made). On April 6, 2013, at the Volume 53 internet pay-per-view, Kana and LuFisto unsuccessfully challenged the Canadian NINJAs for the Shimmer Tag Team Championship in a four-way match, which also included the teams of Global Green Gangsters (Kellie Skater and Tomoka Nakagawa) and Made in Sin. Kana returned to singles action a week later on April 13 at the tapings of "Volume 54", where she defeated Jessie McKay. Later that same day on "Volume 55", Kana was defeated by Madison Eagles in another singles match. The following day, Kana first defeated Ayako Hamada, Mercedes Martinez and Yuu Yamagata in a four-way match on "Volume 56", before finishing her weekend with a win over Kalamity in a singles match on "Volume 57". Upon her return to Shimmer on October 19 at "Volume 58", Kana and LuFisto were given another shot at the Shimmer Tag Team Championship, but were defeated by the defending champions, the Global Green Gangsters. Kana returned to Shimmer on October 18, 2014, unsuccessfully challenging Cheerleader Melissa for the Shimmer Championship at "Volume 67".


title: Asuka (wrestler)

text: On April 5, 2012, Tajiri named Kana as part of the roster of Wrestling New Classic (WNC), the follow-up promotion to Smash. For the first time since 2006, when she was affiliated with AtoZ, Kana signed a contract to make WNC her new home promotion, ending her freelancing days, though the deal allowed her to continue working for Pro Wrestling Wave and producing her own independent events. On April 26 at WNC's first event, "Before the Dawn", Kana teamed with Mio Shirai in a tag team match, where they defeated Makoto and Syuri, with Kana pinning Makoto for the win. Post-match, Kana verbally assaulted Makoto, which led to Shirai turning on her and siding with Makoto. At WNC's second event, "Starting Over" on May 24, Kana submitted Makoto in a tag team match, where she teamed with Syuri and Makoto with Riho, a replacement for an injured Mio Shirai. Two days later at "Go! Go! West: Osaka", Kana submitted Makoto again, this time in a singles match. On June 17, Kana produced her first Kana Pro event in collaboration with WNC, where she and Kenichi Yamamoto were defeated in the main event by Daisuke Ikeda and Syuri. The rivalry between Kana and Makoto continued at WNC's following event on June 22, where Makoto ended her losing streak by pinning Kana in a tag team match, where Kana teamed with Syuri and Makoto with Mio Shirai. On July 15, Kana defeated Shirai in a singles match, after which she had another heated exchange with Makoto, who had been sidelined from in-ring action with a fractured wrist.


title: Asuka (wrestler)

text: Later that same event, Kana's uneasy alliance with Syuri came to an end, when Syuri turned on her, after losing to Ayako Hamada, and joined a new villainous stable formed by Akira and StarBuck. The first match between Kana and Syuri since the latter's turn took place at a WNC event on August 2; Syuri, Akira and StarBuck defeated Kana, Hajime Ohara and Tajiri. On August 30, Kana, Mikey Whipwreck and Tajiri were defeated in a Barbed Wire Board Deathmatch by Syuri, Akira and StarBuck. Kana, Whipwreck and Tajiri were also defeated in a rematch the following day in Osaka, before coming back to win the second rematch on September 1 in Toyohashi, ending the villainous trio's win streak in the process. In the main event of a WNC event on September 17, Kana earned a big win over Syuri, defeating her via submission in a tag team match, where she and Tajiri faced Syuri and Akira. During a press conference the following day, Kana and Makoto announced that they were coming together to produce an event at Shinjuku Face on November 9, celebrating their common birthday, respectively. On September 20, Kana was defeated by Akira in an intergender match at Korakuen Hall. On November 9, Kana and Makoto produced the first MakoKana Pro event. Early in the event, the two teamed to defeat the Ice Ribbon team of Hamuko Hoshi and Hikaru Shida. Later, in the main event, Kana teamed with Sendai Girls' Pro Wrestling's Kagetsu, wrestling Syuri and JWP's Arisa Nakajima to a thirty-minute time limit draw. The match was billed as the first confrontation between Kana and Nakajima, who both started their careers in AtoZ. On November 26, Kana entered the WNC Women's Championship tournament, defeating Makoto in her first round match. Two days later, Kana was eliminated in the semi-finals of the tournament by Nagisa Nozaki, following interference from her Kabushiki gaisha DQN stablemate Jiro Kuroshio. Following the match, Kana quit WNC and once again became a freelancer, claiming that she had not been paid for months and had not been able to get in contact with the promotion's president Tsutomu Takashima. WNC officially released Kana from her contract on December 4.


title: Asuka (wrestler)

text: On August 30, 2014, Kana began working for Reina Joshi Puroresu as a storyline consultant. After Kana used her power to strip Ariya and Makoto of the Reina World Tag Team Championship, she and Arisa Nakajima became the new champions on November 20 by defeating Lin Byron and Syuri in a tournament final, when Syuri was betrayed by her own partner, paid off by Kana. On December 26, Kana defeated Syuri to win the Reina World Women's Championship. Following the win, Kana reformed Reina as the promotion's new "general producer" (GP). In early 2015, Kana launched her own stable, originally named Kana-gun (Kana Army), but later renamed Piero-gun (Clown Army) due to all members adopting Kana's clown makeup; members of the stable included native wrestlers Makoto and Rina Yamashita, foreigners Alex Lee, Cat Power and La Comandante and male wrestlers Hercules Senga and Yuko Miyamoto. On January 12, 2015, Kana and Nakajima made their first successful defense of the Reina World Tag Team Championship against Goya Kong and Muñeca de Plata. On February 25 at "KanaProMania: Advance", Kana and Arisa Nakajima lost the Reina World Tag Team Championship to Hikaru Shida and Syuri. On March 25, Kana lost her position as the Reina GP to Shida after she and her stablemates Cat Power and Yuko Miyamoto were defeated in a six-person tag team match by Shida, Shiro Koshinaka and Zeus. In May, Kana made her first successful defenses of the Reina World Women's Championship against visiting Mexican wrestlers, defeating Gitana on May 17 and Reyna Isis on May 29. On June 13, Kana made her third successful title defense against Yako Fujigasaki. On August 31, ahead of her impending departure from the promotion, Kana relinquished the Reina World Women's Championship. She wrestled her final match for the promotion on September 4, when she and Syuri defeated Konami and Makoto in a tag team match.


title: Asuka (wrestler)

text: On August 22, 2015, Urai attended WWE's event in Brooklyn, New York, Rumours about her having signed with WWE were soon confirmed at a press conference on September 8 in Tokyo; this made her the first female Japanese wrestler to sign with the company since Bull Nakano in 1994. She joined WWE's developmental territory NXT, and adopted the ring name Asuka. Her debut appearance, which aired on September 23, ended in a confrontation with Dana Brooke and Emma. In her debut match on October 7 at , Asuka defeated Dana. Asuka continued racking victories over various wrestlers including main roster wrestler Cameron and continued her feud with Dana Brooke and Emma. At , Asuka defeated Emma. Both wrestlers' performance was highly praised by critics such as Dave Meltzer, who called the match "fantastic".


title: Asuka (wrestler)

text: On the January 13, 2016, episode of "NXT", Asuka participated in a battle royal to determine the number one contender for Bayley's NXT Women's Championship; she was eliminated second-to-last by Eva Marie. On the February 10 episode, battle royal winner Carmella lost her title match against Bayley. Asuka saved Carmella from a post-match attack by Eva Marie and Nia Jax and afterwards had a staredown with Bayley, indicating that she wanted a shot at her title. On April 1, Asuka defeated Bayley by technical submission to win the NXT Women's Championship at and successfully defended the title against Nia Jax on June 8 at . In summer 2016, Vince McMahon contemplated moving Asuka to the main roster during the 2016 draft, but NXT head Triple H dissuaded him, calling Asuka "the one person NXT could not afford to lose" and the brand's "anchor".


title: Asuka (wrestler)

text: Asuka remained in NXT and rekindled her feud with Bayley, whom she again defeated on August 20 at . After her successful title defense against Mickie James on November 19 at , Asuka displayed the first signs of villainous behavior when she refused to shake her opponent's hand after the match. On the January 11, 2017, episode of "NXT" when Asuka was overpowered by The Iconic Duo (Billie Kay and Peyton Royce), Nikki Cross came to her rescue but then herself attacked Asuka. Asuka demanded to defend her title against all three and won the subsequent fatal four-way match on January 28 at . In early February, Asuka became the longest-reigning singles champion in NXT history, male or female. In March, Asuka wrestled on several house shows with WWE's main roster as a replacement for an injured Naomi, including a show at Madison Square Garden on March 12. Around this time, WWE began promoting Asuka's "undefeated streak [...] since her arrival" in NXT. Asuka had never lost a match by submitting or being pinned, though she had come up short in the aforementioned battle royal in January 2016 and tag team match and one fatal four-way match on NXT house shows in March and May 2016, respectively, when other wrestlers were pinned.


title: Asuka (wrestler)

text: In her successful title defense against Ember Moon on April 1 at , Asuka again displayed a villainous demeanor, shoving the referee into her opponent to avoid Moon's finishing maneuver (the Eclipse) en route to winning the match. Asuka also attacked Ember Moon, Nikki Cross and Ruby Riott, who were the final competitors in a battle royal to determine her challenger for , officially cementing herself as a villainess. Subsequently, Asuka had to defend her title against Nikki Cross and Ruby Riott in a triple threat match (Ember Moon missed out on the match due a shoulder injury), but she managed to retain her title at the event on June 20. In June, Asuka decisively defeated Nikki Cross in a Last Woman Standing match to retain her championship. During that time, she surpassed Goldberg's winning streak and Rockin' Robin's record-setting 502-day WWF Women's Championship reign.


title: Asuka (wrestler)

text: On August 19 at , Asuka defeated the returning Ember Moon to retain her title reign. However, during the match she broke her right collarbone which forced her out of action for six to eight weeks, thus Asuka had to relinquish the NXT Women's Championship at the August 24 "NXT" tapings which aired on September 6, ending her reign at 510 days. During her farewell segment, NXT General Manager William Regal called her "one of the greatest champions in WWE history" and also mentioned negotiations to move Asuka to WWE's main roster.


title: Asuka (wrestler)

text: After multiple vignettes in the upcoming weeks, it was revealed on the September 11, 2017, episode of "Raw", Asuka was assigned to the Raw brand. A month later, on October 22, at , Asuka defeated her old rival Emma in her in-ring debut. On November 19 at Survivor Series, Asuka represented Team Raw in a Survivor Series match against Team SmackDown and was the sole survivor, lastly eliminating Tamina and Natalya. Throughout the next few months, Asuka continued her streak by defeating the likes of Dana Brooke (whom she defeated in three seconds, setting a new WWE record for the fastest submission win in a women's match), Alicia Fox and Raw Women's Champion Alexa Bliss in a non-title match.


title: Asuka (wrestler)

text: On January 28, 2018, at the Royal Rumble, Asuka won the first women's Royal Rumble match, entering at number 25 and eliminating her former rival Ember Moon and lastly Nikki Bella to earn herself a women's championship match at WrestleMania 34. Throughout January and February, Asuka started a feud with Nia Jax, who attacked her during and before her matches and wanted her title opportunity at Wrestlemania. Asuka defeated Nia Jax on February 25 at Elimination Chamber and again on the March 5 episode of "Raw". In parallel, she also took a part of the Mixed Match Challenge tournament, with The Miz as her partner in support of the charity Rescue Dogs Rock. With wins over Carmella and Big E, Sasha Banks and Finn Bálor, Alexa Bliss and Braun Strowman and finally Charlotte Flair and Bobby Roode, they won the tournament. After Flair retained the SmackDown Women's Championship against Ruby Riott on March 11 at Fastlane, Asuka challenged Charlotte for the SmackDown Women's Championship at WrestleMania 34. At the event on April 8, Asuka submitted to Charlotte, ending her WWE undefeated streak at 914 days, or roughly 2 and a half years.


title: Asuka (wrestler)

text: On April 17, Asuka was drafted to SmackDown as part of the 2018 WWE Superstar Shake-up, making her first appearance by saving Charlotte Flair and Becky Lynch from an attack by The IIconics (a duo made of Billie Kay and Peyton Royce) and then—SmackDown Women's Champion Carmella. On the May 15 episode of "SmackDown", SmackDown general manager Paige interrupted Carmella, to announce that she would be defending the SmackDown Women's Championship against Asuka at Money in the Bank. At the event on June 17, Asuka lost to Carmella after an interference from the returning James Ellsworth, suffering her first official pinfall loss in WWE. Paige then granted Asuka a rematch that took place on July 15 at Extreme Rules, which Asuka lost after another distraction from Ellsworth, who was hanging above the ring in a shark cage, which Carmella used to win the match. In September, Asuka helped Naomi, who after defeating Peyton Royce was attacked by The IIconics. This led to a tag team match between the two teams on October 6 at WWE Super Show-Down, where Asuka and Naomi were defeated. Also during that time, it was announced that Asuka and The Miz would take part in the upcoming season two of Mixed Match Challenge. They lasted until the semi-finals, where they were eliminated by eventual winners R-Truth and Carmella. On November 18, Asuka took part in the Survivor Series as part of Team Smackdown, where she was the last woman eliminated from Team SmackDown.


title: Asuka (wrestler)

text: Towards the end of November, Asuka won a battle royal to enter her way into the first-ever women's Tables, ladders, and chairs match against both Becky Lynch and Charlotte Flair for the SmackDown Women's Championship. On December 16 at the , in what was also the first ever women's championship match to main event a dual–branded event, Asuka won the SmackDown Women's Championship for the first time in her career after Ronda Rousey interfered and pushed both Lynch and Flair off a ladder. Throughout her reign, Asuka was able to fend off different challengers such as Naomi on the "SmackDown" following TLC, Becky Lynch at the Royal Rumble and Mandy Rose at Fastlane. On the March 26 episode of "SmackDown", Asuka dropped the title to Charlotte Flair in an impromptu match, ending her reign at 100 days. However, Dave Meltzer from "The Wrestling Observer" revealed that Asuka was supposed to defend the SmackDown Women's Championship at WrestleMania 35 against the winner of a fatal four-way match between Carmella, Mandy Rose, Naomi and Sonya Deville that was cancelled in the last minute.


title: Asuka (wrestler)

text: On the April 16, 2019, episode of "SmackDown", Paige announced that she would be managing a newly formed women's tag team, consisting of Asuka and the debuting Kairi Sane from NXT. The team of Asuka and Kairi Sane, later dubbed as "The Kabuki Warriors", was immediately put in a feud with The IIconics (Billie Kay and Peyton Royce) over the WWE Women's Tag Team Championship. After weeks of avoiding them, The IIconics lost to The Kabuki Warriors during WWE's tour in Tokyo, which earned The Kabuki Warriors a title match. The title match took place on the July 16 episode of "SmackDown", where The IIconics retained after they got themselves counted–out. In August, The Kabuki Warriors failed to capture the titles from then–champions Alexa Bliss and Nikki Cross.


title: Asuka (wrestler)

text: After a short hiatus, The Kabuki Warriors returned in September and defeated Mandy Rose and Sonya Deville. On October 6 at the Hell in a Cell pay–per–view, The Kabuki Warriors finally won the Women's Tag Team Championship from Bliss and Cross, after Asuka used a green mist on Cross. The next night on "Raw", The Kabuki Warriors completed their transition into heels, as they cut a promo on both Becky Lynch and Charlotte Flair and went onto defeat them in a non–title match, once again using green mist as leverage. As part of the 2019 Draft in mid-October, Asuka and Sane were both drafted to the Raw brand. The duo further solidified their heel turn after Asuka spit green mist on their manager Paige, ending their association with Paige. She renewed her rivalry with Becky Lynch after Lynch challenged Asuka to a Raw Women's Championship match at the Royal Rumble in a rematch from the previous year's event. At the Royal Rumble, Asuka failed to win the title from Lynch. On the February 3, 2020, episode of "Raw", Asuka demanded a rematch for the title, in which Lynch accepted, but a week later on "Raw", Asuka again failed to win the title.


title: Asuka (wrestler)

text: On the first night of WrestleMania 36, The Kabuki Warriors lost their WWE Women's Tag Team Championship to Alexa Bliss and Nikki Cross, ending their reign at 181 days, the longest reign for the titles so far. On the April 13 episode of "Raw", Asuka qualified for the Money in the Bank ladder match after defeating Ruby Riott. At the eponymous event, Asuka won the Women's Money in the Bank match.


title: Asuka (wrestler)

text: On the "Raw" immediately following Money in the Bank, Becky Lynch revealed that she had vacated the Raw Women's Championship due to pregnancy, and that the ladder match was in fact for the vacated Raw Women's Championship rather than a championship match contract, making Asuka the new champion. She also became the second Women's Grand Slam winner after Bayley and third Women's Triple Crown winner after Bayley and Alexa Bliss. She embraced Becky Lynch after the announcement and chanted her name, thus turning face in the process. On the following "Raw", The Kabuki Warriors officiated their face turn, beginning a feud with the returning Nia Jax. At Backlash, Asuka retained the title against Jax after both were counted out. A rematch was scheduled the following night on "Raw", where Asuka again retained the title. Asuka presumably lost the Raw Women's Championship to Sasha Banks at The Horror Show at Extreme Rules after she indirectly blinded the referee with the green mist and Bayley removed the referee's shirt, put it on herself and counted the pinfall, which was unofficial. The next night on "Raw", Stephanie McMahon ruled that the result was reversed to a no-contest, meaning that Asuka retained the championship, despite Banks having taken possession of the actual belt. However, the following week, Asuka officially lost the title via count out in their rematch, where the belt could change hands by pinfall, submission, count out, disqualification or interference. Sasha Banks won the title when Asuka chose to get herself counted out to save Sane from an attack by Bayley. This also marked Sane's final appearance in WWE, as she left the company to be with her new husband in Japan, therefore disbanding the Kabuki Warriors.


title: Asuka (wrestler)

text: At Summerslam, Asuka competed against both Bayley for the SmackDown Women's Championship, and later Banks for the Raw Women's Championship; she lost the first match, but later defeated Banks to win back the Raw Women's Championship, marking her first time winning any WWE title more than once. At Clash of Champions, Asuka defeated Zelina Vega to retain her championship and, later in the night, answered Bayley's open challenge for the Smackdown Women's Championship, which she won via disqualification. The following night on "Raw", Asuka faced Zelina Vega for the Raw Women's Championship and again successfully retained. On the October 5 episode of "Raw", Asuka teamed with Dana Brooke and Mandy Rose to defeat Zelina Vega, Natalya and Lana. On the October 19 episode of "Raw," Asuka successfully retained her title against Lana. Asuka then faced Sasha Banks at Survivor Series in a losing effort. At , Asuka and Charlotte Flair defeat Jax and Baszler to win the tag team titles, making Asuka a double champion. They lost the titles back to Jax and Baszler at the Royal Rumble. She also lost the Raw Women's title at WrestleMania 37 against Rhea Ripley, ending her reign at 231 days. On the "Raw" after WrestleMania, Asuka faced Ripley for the title in a rematch which resulted in a no-contest due to Charlotte Flair attacking both of them. On the May 3 episode of "Raw," another rematch between Ripley and Asuka for the Raw Women's Championship was scheduled for WrestleMania Backlash. However, later that night, WWE official Sonya Deville added Flair to the championship match to make it a triple threat match. At the event, Asuka was unsuccessful to win the title and was pinned by Ripley.


title: Asuka (wrestler)

text: On the June 21 episode of "Raw", Asuka and Naomi qualified for the women's Money in the Bank ladder match by defeating the team of Eva Marie and Doudrop (formerly Piper Niven). At Money in the Bank, she was unsuccessful in winning the match. It was her last match in 2021 before becoming inactive due to an injury. During the 2021 WWE Draft, Asuka was not drafted and was declared a free agent. After 9 months of inactivity due to her injury recovery, Asuka made her return on the April 25, 2022, episode of "Raw" confronting Becky Lynch. On the May 9 episode of "Raw", WWE official Adam Pearce gave Asuka a championship contender's match against Raw Women's Champion Bianca Belair. Lynch became upset and later attacked both of them during their match. The following week, Asuka defeated Lynch using green mist and earned a title match for the Raw Women's Championship at Hell in a Cell. Lynch defeated Asuka in a rematch on the next episode of "Raw" and she was added to the title match. At the event, Belair retained her title. On the June 20 episode of "Raw", Asuka defeated Lynch to qualify for the Women's Money in the Bank ladder match. At Money in the Bank, she was unsuccessful in winning the match. On the October 31 episode of "Raw", Asuka and Alexa Bliss defeated women's tag team champions Dakota Kai and Iyo Sky in a title match and won the championship. In a rematch at Crown Jewel, Asuka and Bliss lost the titles back to Kai and Sky due to an interference from Nikki Cross.


title: Asuka (wrestler)

text: In December 2015, Dave Meltzer of the "Wrestling Observer Newsletter" stated that Urai "may be the best worker in WWE, man or woman". In 2017, Urai became the first Japanese wrestler to top "Pro Wrestling Illustrated"'s annual "PWI Female 50" list. She is the first person in WWE history to win NXT Women's Championship, Raw Women's Championship, SmackDown Women's Championship, Women's Tag Team Championship, Royal Rumble match, and Money in the Bank match. In 2020, she competed in more WWE matches than all other superstars. In 2021, WWE named her as one of the top 5 of the greatest female superstars of all time.


title: Asuka (wrestler)

text: Urai has released three gravure DVDs, titled , and . She and Mio Shirai have also released a gravure DVD together, titled "Sadistic Tails" (2012). In September 2019, Urai launched her own gaming and cooking channel on YouTube, "KanaChanTV". She had teased the channel with several posts on her Twitter account before formally announcing the channel. "KanaChanTV" features playthrough videos with her own bilingual English/Japanese commentary as well as lifestyle content.


title: Asuka (wrestler)

text: Urai is a graduate of the Osaka University of Arts Junior College. She has written for "Xbox Magazine" and designed graphics for Nintendo DS and various mobile applications; she also owns a hair salon in Yokohama called Another Heaven. Urai is a mother, as mentioned by Becky Lynch and Naomi.


title: Asuka (wrestler)

text: In an interview with "ET Canada" in 2021, Urai detailed her own experiences with racism and shared her thoughts on the rise of anti-Asian sentiment during the COVID-19 pandemic.


title: Cartesian coordinate system

text: A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its "origin", at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.


title: Cartesian coordinate system

text: One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, "n" Cartesian coordinates (an element of real "n"-space) specify the point in an "n"-dimensional Euclidean space for any dimension "n". These coordinates are equal, up to sign, to distances from the point to "n" mutually perpendicular hyperplanes.


title: Cartesian coordinate system

text: The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: "Cartesius") revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates "x" and "y" satisfy the equation .


title: Cartesian coordinate system

text: Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.


title: Cartesian coordinate system

text: The adjective "Cartesian" refers to the French mathematician and philosopher René Descartes, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.


title: Cartesian coordinate system

text: Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' "La Géométrie" was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.


title: Cartesian coordinate system

text: The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces.


title: Cartesian coordinate system

text: Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.


title: Cartesian coordinate system

text: Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—involves choosing a point "O" of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by "O" is the positive and which is negative; we then say that the line "is oriented" (or "points") from the negative half towards the positive half. Then each point "P" of the line can be specified by its distance from "O", taken with a + or − sign depending on which half-line contains "P".


title: Cartesian coordinate system

text: A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum such as the real numbers.


title: Cartesian coordinate system

text: A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For any point "P", a line is drawn through "P" perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, are the "Cartesian coordinates" of "P". The reverse construction allows one to determine the point "P" given its coordinates.


title: Cartesian coordinate system

text: The first and second coordinates are called the "abscissa" and the "ordinate" of "P", respectively; and the point where the axes meet is called the "origin" of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in . Thus the origin has coordinates , and the points on the positive half-axes, one unit away from the origin, have coordinates and .


title: Cartesian coordinate system

text: In mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. (However, in some computer graphics contexts, the ordinate axis may be oriented downwards.) The origin is often labeled "O", and the two coordinates are often denoted by the letters "X" and "Y", or "x" and "y". The axes may then be referred to as the "X"-axis and "Y"-axis. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.


title: Cartesian coordinate system

text: A Euclidean plane with a chosen Cartesian coordinate system is called a . In a Cartesian plane one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at and ), the unit hyperbola, and so on.


title: Cartesian coordinate system

text: The two axes divide the plane into four right angles, called "quadrants". The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the "first quadrant".


title: Cartesian coordinate system

text: If the coordinates of a point are , then its distances from the "X"-axis and from the "Y"-axis are and , respectively; where denotes the absolute value of a number.


title: Cartesian coordinate system

text: A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the "axes") that go through a common point (the "origin"), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point "P" of space, one considers a hyperplane through "P" perpendicular to each coordinate axis, and interprets the point where that hyperplane cuts the axis as a number. The Cartesian coordinates of "P" are those three numbers, in the chosen order. The reverse construction determines the point "P" given its three coordinates.


title: Cartesian coordinate system

text: Alternatively, each coordinate of a point "P" can be taken as the distance from "P" to the hyperplane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.


title: Cartesian coordinate system

text: Each pair of axes defines a "coordinate hyperplane". These hyperplanes divide space into eight "octants". The octants are:


title: Cartesian coordinate system

text: The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in or . Thus, the origin has coordinates , and the unit points on the three axes are , , and .


title: Cartesian coordinate system

text: There are no standard names for the coordinates in the three axes (however, the terms "abscissa", "ordinate" and "applicate" are sometimes used). The coordinates are often denoted by the letters "X", "Y", and "Z", or "x", "y", and "z". The axes may then be referred to as the "X"-axis, "Y"-axis, and "Z"-axis, respectively. Then the coordinate hyperplanes can be referred to as the "XY"-plane, "YZ"-plane, and "XZ"-plane.


title: Cartesian coordinate system

text: In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called "height" or "altitude". The orientation is usually chosen so that the 90 degree angle from the first axis to the second axis looks counter-clockwise when seen from the point ; a convention that is commonly called "the right hand rule".


title: Cartesian coordinate system

text: Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; that is, with the Cartesian product formula_2, where formula_3 is the set of all real numbers. In the same way, the points in any Euclidean space of dimension "n" be identified with the tuples (lists) of "n" real numbers; that is, with the Cartesian product formula_4.


title: Cartesian coordinate system

text: The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In such an oblique coordinate system the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see affine plane).


title: Cartesian coordinate system

text: The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in or . The origin is often labelled with the capital letter "O". In analytic geometry, unknown or generic coordinates are often denoted by the letters ("x", "y") in the plane, and ("x", "y", "z") in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as the coordinates of points in many geometric problems), and letters near the beginning for given quantities.


title: Cartesian coordinate system

text: These conventional names are often used in other domains, such as physics and engineering, although other letters may be used. For example, in a graph showing how a pressure varies with time, the graph coordinates may be denoted "p" and "t". Each axis is usually named after the coordinate which is measured along it; so one says the "x-axis", the "y-axis", the "t-axis", etc.


title: Cartesian coordinate system

text: Another common convention for coordinate naming is to use subscripts, as ("x", "x", ..., "x") for the "n" coordinates in an "n"-dimensional space, especially when "n" is greater than 3 or unspecified. Some authors prefer the numbering ("x", "x", ..., "x"). These notations are especially advantageous in computer programming: by storing the coordinates of a point as an array, instead of a record, the subscript can serve to index the coordinates.


title: Cartesian coordinate system

text: In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top. Young children learning the Cartesian system, commonly learn the order to read the values before cementing the "x"-, "y"-, and "z"-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the "x"-axis then up vertically along the "y"-axis).


title: Cartesian coordinate system

text: Computer graphics and image processing, however, often use a coordinate system with the "y"-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers.


title: Cartesian coordinate system

text: For three-dimensional systems, a convention is to portray the "xy"-plane horizontally, with the "z"-axis added to represent height (positive up). Furthermore, there is a convention to orient the "x"-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the "x"- and "y"-axis horizontally and vertically, respectively, then the "z"-axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the "z"-axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.


title: Cartesian coordinate system

text: For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for "x" and "y", respectively. When they are, the "z"-coordinate is sometimes called the applicate. The words "abscissa", "ordinate" and "applicate" are sometimes used to refer to coordinate axes rather than the coordinate values.


title: Cartesian coordinate system

text: The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called "quadrants", each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the coordinates both have positive signs), II (where the abscissa is negative − and the ordinate is positive +), III (where both the abscissa and the ordinate are −), and IV (abscissa +, ordinate −). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("north-east") quadrant.


title: Cartesian coordinate system

text: Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs; for example, or . The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant, and a similar naming system applies.


title: Cartesian coordinate system

text: The Euclidean distance between two points of the plane with Cartesian coordinates formula_5 and formula_6 is


title: Cartesian coordinate system

text: This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points formula_8 and formula_9 is


title: Cartesian coordinate system

text: The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations, rotations, reflections and glide reflections.


title: Cartesian coordinate system

text: Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are , after the translation they will be


title: Cartesian coordinate system

text: To rotate a figure counterclockwise around the origin by some angle formula_12 is equivalent to replacing every point with coordinates ("x","y") by the point with coordinates ("x<nowiki>'</nowiki>","y<nowiki>'</nowiki>"), where


title: Cartesian coordinate system

text: If are the Cartesian coordinates of a point, then are the coordinates of its reflection across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle formula_12 with the x-axis, is equivalent to replacing every point with coordinates by the point with coordinates , where


title: Cartesian coordinate system

text: A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection).


title: Cartesian coordinate system

text: All affine transformations of the plane can be described in a uniform way by using matrices. For this purpose the coordinates formula_18 of a point are commonly represented as the column matrix formula_19 The result formula_20 of applying an affine transformation to a point formula_18 is given by the formula


title: Cartesian coordinate system

text: Among the affine transformations, the Euclidean transformations are characterized by the fact that the matrix formula_26 is orthogonal; that is, its columns are orthogonal vectors of Euclidean norm one, or, explicitly,


title: Cartesian coordinate system

text: This is equivalent to saying that times its transpose is the identity matrix. If these conditions do not hold, the formula describes a more general affine transformation.


title: Cartesian coordinate system

text: The transformation is a translation if and only if is the identity matrix. The transformation is a rotation around some point if and only if is a rotation matrix, meaning that it is orthogonal and


title: Cartesian coordinate system

text: Assuming that translations are not used (that is, formula_31) transformations can be composed by simply multiplying the associated transformation matrices. In the general case, it is useful to use the augmented matrix of the transformation; that is, to rewrite the transformation formula


title: Cartesian coordinate system

text: With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.


title: Cartesian coordinate system

text: Affine transformations of the Euclidean plane are transformations that map lines to lines, but may change distances and angles. As said in the preceding section, they can be represented with augmented matrices:


title: Cartesian coordinate system

text: The Euclidean transformations are the affine transformations such that the 2×2 matrix of the formula_35 is orthogonal.


title: Cartesian coordinate system

text: The augmented matrix that represents the composition of two affine transformations is obtained by multiplying their augmented matrices.


title: Cartesian coordinate system

text: An example of an affine transformation which is not Euclidean is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number "m". If are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates


title: Cartesian coordinate system

text: A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:


title: Cartesian coordinate system

text: Fixing or choosing the "x"-axis determines the "y"-axis up to direction. Namely, the "y"-axis is necessarily the perpendicular to the "x"-axis through the point marked 0 on the "x"-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called "handedness") of the Cartesian plane.


title: Cartesian coordinate system

text: The usual way of orienting the plane, with the positive "x"-axis pointing right and the positive "y"-axis pointing up (and the "x"-axis being the "first" and the "y"-axis the "second" axis), is considered the "positive" or "standard" orientation, also called the "right-handed" orientation.


title: Cartesian coordinate system

text: A commonly used mnemonic for defining the positive orientation is the "right-hand rule". Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the "x"-axis to the "y"-axis, in a positively oriented coordinate system.


title: Cartesian coordinate system

text: The other way of orienting the plane is following the "left hand rule", placing the left hand on the plane with the thumb pointing up.


title: Cartesian coordinate system

text: When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis.


title: Cartesian coordinate system

text: Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any one axis will reverse the orientation, but switching both will leave the orientation unchanged.


title: Cartesian coordinate system

text: Once the "x"- and "y"-axes are specified, they determine the line along which the "z"-axis should lie, but there are two possible orientation for this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'. The standard orientation, where the "xy"-plane is horizontal and the "z"-axis points up (and the "x"- and the "y"-axis form a positively oriented two-dimensional coordinate system in the "xy"-plane if observed from "above" the "xy"-plane) is called right-handed or positive.


title: Cartesian coordinate system

text: The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative orientation of the "x"-, "y"-, and "z"-axes in a "right-handed" system. The thumb indicates the "x"-axis, the index finger the "y"-axis and the middle finger the "z"-axis. Conversely, if the same is done with the left hand, a left-handed system results.


title: Cartesian coordinate system

text: Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point "towards" the observer, whereas the "middle"-axis is meant to point "away" from the observer. The red circle is "parallel" to the horizontal "xy"-plane and indicates rotation from the "x"-axis to the "y"-axis (in both cases). Hence the red arrow passes "in front of" the "z"-axis.


title: Cartesian coordinate system

text: Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the "x"-axis as pointing "towards" the observer and thus seeing a concave corner.


title: Cartesian coordinate system

text: A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as formula_39. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:


title: Cartesian coordinate system

text: where formula_41 and formula_42 are unit vectors in the direction of the "x"-axis and "y"-axis respectively, generally referred to as the "standard basis" (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates formula_43 can be written as:


title: Cartesian coordinate system

text: There is no "natural" interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates with the complex number . Here, "i" is the imaginary unit and is identified with the point with coordinates , so it is "not" the unit vector in the direction of the "x"-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three-dimensional cartesian space a similar identification can be made with a subset of the quaternions.


title: Cartesian coordinate system

text: Cartesian coordinates are an abstraction that have a multitude of possible applications in the real world. However, three constructive steps are involved in superimposing coordinates on a problem application.


title: Cartesian coordinate system

text: Consider as an example superimposing 3D Cartesian coordinates over all points on the Earth (that is, geospatial 3D). Kilometers are a good choice of units, since the original definition of the kilometer was geospatial, with equaling the surface distance from the equator to the North Pole. Based on symmetry, the gravitational center of the Earth suggests a natural placement of the origin (which can be sensed via satellite orbits). The axis of Earth's rotation provides a natural orientation for the "X", "Y", and "Z" axes, strongly associated with "up vs. down", so positive "Z" can adopt the direction from the geocenter to the North Pole. A location on the equator is needed to define the "X"-axis, and the prime meridian stands out as a reference orientation, so the "X"-axis takes the orientation from the geocenter out to longitude, latitude. Note that with three dimensions, and two perpendicular axes orientations pinned down for "X" and "Z", the "Y"-axis is determined by the first two choices. In order to obey the right-hand rule, the "Y"-axis must point out from the geocenter to longitude, latitude. From a longitude of , a latitude , and Earth radius of 40,000/2"π" km, and transforming from spherical to Cartesian coordinates, one can estimate the geocentric coordinates of the Empire State Building, . GPS navigation relies on such geocentric coordinates.


title: Cartesian coordinate system

text: In engineering projects, agreement on the definition of coordinates is a crucial foundation. One cannot assume that coordinates come predefined for a novel application, so knowledge of how to erect a coordinate system where there previously was no such coordinate system is essential to applying René Descartes' thinking.


title: Cartesian coordinate system

text: While spatial applications employ identical units along all axes, in business and scientific applications, each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables.


title: Cartesian coordinate system

text: The graph of a function or relation is the set of all points satisfying that function or relation. For a function of one variable, "f", the set of all points , where is the graph of the function "f". For a function "g" of two variables, the set of all points , where is the graph of the function "g". A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.


title: Firestarter (2022 film)

text: Firestarter is a 2022 American science fiction horror film directed by Keith Thomas, from a screenplay by Scott Teems, based on the novel of the same name by Stephen King and serving as a remake of the 1984 film adaptation. The film stars Zac Efron, Ryan Kiera Armstrong, Sydney Lemmon, Kurtwood Smith, John Beasley, Michael Greyeyes, and Gloria Reuben. It is produced by Jason Blum and Akiva Goldsman under their Blumhouse Productions and Weed Road Pictures banners, respectively, alongside BoulderLight Pictures and Night Platform.


title: Firestarter (2022 film)

text: "Firestarter" was released in the United States on May 13, 2022, by Universal Pictures simultaneously theatrically and through streaming on Peacock. The film received negative reviews from critics, with many deeming it to be inferior to the original 1984 film, and grossed $15 million worldwide against a $12 million budget.


title: Firestarter (2022 film)

text: In a flashback, baby Charlene "Charlie" McGee sits in her crib, spontaneously setting the room ablaze with her pyrokinesis power and sending her father Andrew "Andy" McGee into a panic. In another flashback, a young Andy and his girlfriend Victoria "Vicky" Tomlinson talk to a doctor in a clinical trial, who explains to them that they will be injected with the experimental chemical drug Lot-6, which secretly gives them supernatural powers: Andy gains telepathy, and Vicky obtains telekinesis.


title: Firestarter (2022 film)

text: In the present day, Charlie is sitting at the kitchen table after having a nightmare. Her parents join her and Charlie explains that she has been repressing something bad, her powers becoming more unstable. She unintentionally causes a ruckus at her school after exploding a bathroom stall due to anger at being bullied. Andy is shown using his power, "the push", to influence a client to stop smoking, although the strain causes his eyes to bleed.


title: Firestarter (2022 film)

text: Meanwhile, in a secret facility, Captain Jane Hollister, leader of the Department of Scientific Intelligence (DSI), is monitoring thermal signatures caused by Charlie's outbursts. She visits Doctor Joseph Wanless, creator of Lot-6 and the resulting superhumans, who implores Hollister to terminate Charlie before her powers become uncontrollable. Hollister enlists fellow superhuman John Rainbird to assist her. Rainbird visits the McGee home, confronting Vicky who attempts to counterattack with her repressed telekinetic powers. Rainbird overpowers and kills her, holding Charlie at knifepoint as she and Andy enter the home. Charlie's powers overwhelm him and she sends a concussive burst of flames throughout the house. Andy and Charlie escape into their truck.


title: Firestarter (2022 film)

text: On the road, they encounter a man named Irv. After using the push to convince Irv to take them to Boston, they hitch a ride with him, stopping off at his house. After Charlie accidentally stumbles upon his paralyzed wife, Irv flies into a rage before conceding that he occasionally overreacts. Irv sits up all night watching a news report of the incident at the McGee home, which is being framed as murder by Andy. Irv and Andy argue before Andy explains to Irv that he is just trying to protect his daughter. Charlie tells Irv, after speaking telepathically with his wife, that she forgives him for the accident that left her paralyzed, causing him to relent and attempt to protect Andy and Charlie when the police appear due to his prior emergency call. Rainbird appears in the bushes, kills the policemen, and then shoots Irv in the knee before black trucks roll in to pick up Charlie and Andy. Andy uses his push one last time to trick Rainbird so Charlie can escape to a forest. Charlie spends time honing her fire powers before stealing a bike and clothes to follow her father's telepathic message to her from his cell at DSI.


title: Firestarter (2022 film)

text: Charlie finds DSI and takes an agent's pass card from him. He pleads for her not to kill him and tells her he doesn't have a gun. She kills him after he draws the gun to attack her. She follows a large staircase down to the restricted area where her father is being kept. She reaches her father's glass-fronted cell, from inside which Captain Hollister tells her not to try and burn her, lest she burn her father in the process. Andy tells Charlie that Rainbird, not he, telepathically called for her. Seeing no other way out, he apologizes to her and then mentally pushes her to burn the entire place down, starting with Hollister and himself. The now-rogue Charlie sets both on fire, mentally unlocks all the security doors, and walks through the facility killing everyone. Rainbird is released when his holding cell is unlocked. Charlie is captured by men in fire suits, unable to harm them. The men are about to subdue her when Rainbird shoots them from behind. He surrenders to Charlie and kneels for her judgment. Charlie starts to kill him but sees herself in the mirror and, realizing that the DSI is also controlling him, she spares him, before finally burning the rest of the building down. Lastly, Charlie is seen walking onto a beach with Rainbird following behind. Knowing that she and Rainbird are now alone in the world, Charlie allows Rainbird to carry her, and they walk off together into the night.


title: Firestarter (2022 film)

text: On April 27, 2017, Universal Pictures and Blumhouse Productions announced a remake of "Firestarter" (1984), with Akiva Goldsman set to direct and produce alongside Jason Blum. On June 28, 2018, Fatih Akin replaced Goldsman as director, with Scott Teems set to write the script. On December 16, 2019, Keith Thomas replaced Akin as director. In September 2020, Zac Efron joined the cast. In February 2021, Michael Greyeyes joined the cast. In June 2021, Ryan Kiera Armstrong, Gloria Reuben, and Sydney Lemmon were added to the cast. Principal photography began on May 25, 2021, in Toronto and Hamilton, Ontario, and wrapped on July 16.


title: Firestarter (2022 film)

text: The score was composed by John Carpenter, Cody Carpenter, and Daniel Davies, who previously collaborated with Blumhouse on "Halloween" (2018) and "Halloween Kills" (2021). The score was digitally released on May 13, 2022, by Back Lot Music and on LP/CD and Cassette in October by Sacred Bones Records.


title: Firestarter (2022 film)

text: "Firestarter" was released in wide theaters by Universal Pictures and on Peacock in the United States on May 13, 2022. Over its first week of streaming, the film was watched in two million households.


title: Firestarter (2022 film)

text: The film was released on Video on demand on June 12, 2022, and then on DVD on June 28, 2022, 45 days after its theatrical release.


title: Firestarter (2022 film)

text: "Firestarter" has grossed $9.6million in the United States and Canada and $5.4million in other territories, for a worldwide total of $15million.


title: Firestarter (2022 film)

text: In the United States and Canada, the film was released alongside "Family Camp", and was projected to gross $5–7 million from 3,412 theaters in its opening weekend. It made $1.54 million on its first day, including $375,000 from Thursday night previews. It went on to debut to $3.8 million, finishing fourth at the box office. The film made $2 million in its second weekend. It dropped out of the box office top ten in its third weekend with $285,270, and returned in its fifth weekend once put in a drive-in theater double feature with "Jurassic World Dominion", earning $833,340, only to fall once again in its sixth with $182,365.


title: Firestarter (2022 film)

text: Outside the US and Canada, the film earned $2.1 million from 40 markets in its opening weekend. It made $833,000 from 46 markets in its second weekend.


title: Firestarter (2022 film)

text: Audiences polled by CinemaScore gave the film an average grade of "C–" on an A+ to F scale, while PostTrak reported 50% of audience members gave it a positive score, with 27% saying they would definitely recommend it.


title: Firestarter (2022 film)

text: In May 2022, Thomas stated that there are ongoing discussions to possibly expand the film into a franchise, acknowledging that this may be in the form of sequel, prequel, or spin-off media.


title: Trigonometric functions

text: In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.


title: Trigonometric functions

text: The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.


title: Trigonometric functions

text: The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.


title: Trigonometric functions

text: Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression formula_1 would typically be interpreted to mean formula_2 so parentheses are required to express formula_3


title: Trigonometric functions

text: A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example formula_4 and formula_5 denote formula_6 not formula_7 This differs from the (historically later) general functional notation in which formula_8


title: Trigonometric functions

text: However, the exponent formula_9 is commonly used to denote the inverse function, not the reciprocal. For example formula_10 and formula_11 denote the inverse trigonometric function alternatively written formula_12 The equation formula_13 implies formula_14 not formula_15 In this case, the superscript "could" be considered as denoting a composed or iterated function, but negative superscripts other than formula_9 are not in common use.


title: Trigonometric functions

text: If the acute angle is given, then any right triangles that have an angle of are similar to each other. This means that the ratio of any two side lengths depends only on . Thus these six ratios define six functions of , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, "opposite" represents the side opposite the given angle , and "adjacent" represents the side between the angle and the right angle.


title: Trigonometric functions

text: In a right-angled triangle, the sum of the two acute angles is a right angle, that is, or . Therefore formula_17 and formula_18 represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.


title: Trigonometric functions

text: In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in conventional units of degrees in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics).


title: Trigonometric functions

text: However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function via power series or as solutions to differential equations given particular initial values ("see below"), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions "if" "the argument is regarded as an angle given in radians". Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.


title: Trigonometric functions

text: When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 (≈ 6.28) rad. For real number "x", the notations sin "x", cos "x", etc. refer to the value of the trigonometric functions evaluated at an angle of "x" rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin "x°", cos "x°", etc.). Using this standard notation, the argument "x" for the trigonometric functions satisfies the relationship "x" = (180"x"/)°, so that, for example, sin = sin 180° when we take "x" = . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = /180 ≈ 0.0175.


title: Trigonometric functions

text: The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between and formula_19 radians the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.


title: Trigonometric functions

text: Let formula_20 be the ray obtained by rotating by an angle the positive half of the -axis (counterclockwise rotation for formula_21 and clockwise rotation for formula_22). This ray intersects the unit circle at the point formula_23 The ray formula_24 extended to a line if necessary, intersects the line of equation formula_25 at point formula_26 and the line of equation formula_27 at point formula_28 The tangent line to the unit circle at the point , is perpendicular to formula_24 and intersects the - and -axes at points formula_30 and formula_31 The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of in the following manner.


title: Trigonometric functions

text: The trigonometric functions and are defined, respectively, as the "x"- and "y"-coordinate values of point . That is,


title: Trigonometric functions

text: In the range formula_34, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius as hypotenuse. And since the equation formula_35 holds for all points formula_36 on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.


title: Trigonometric functions

text: By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is


title: Trigonometric functions

text: Since a rotation of an angle of formula_43 does not change the position or size of a shape, the points , , , , and are the same for two angles whose difference is an integer multiple of formula_44. Thus trigonometric functions are periodic functions with period formula_44. That is, the equalities


title: Trigonometric functions

text: hold for any angle and any integer . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that formula_44 is the smallest value for which they are periodic (i.e., formula_44 is the fundamental period of these functions). However, after a rotation by an angle formula_50, the points and already return to their original position, so that the tangent function and the cotangent function have a fundamental period of formula_50. That is, the equalities


title: Trigonometric functions

text: Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.


title: Trigonometric functions

text: Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.


title: Trigonometric functions

text: The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.


title: Trigonometric functions

text: The modern trend in mathematics is to build geometry from calculus rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus.


title: Trigonometric functions

text: Trigonometric functions are differentiable and analytic at every point where they are defined; that is, everywhere for the sine and the cosine, and, for the tangent, everywhere except at for every integer .


title: Trigonometric functions

text: The trigonometric function are periodic functions, and their primitive period is for the sine and the cosine, and for the tangent, which is increasing in each open interval . At each end point of these intervals, the tangent function has a vertical asymptote.


title: Trigonometric functions

text: In calculus, there are two equivalent definitions of trigonometric functions, either using power series or differential equations. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.


title: Trigonometric functions

text: Differentiating again, formula_60 and formula_61, so both sine and cosine are solutions of the ordinary differential equation


title: Trigonometric functions

text: Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions


title: Trigonometric functions

text: The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.


title: Trigonometric functions

text: Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form formula_66 for the tangent and the secant, or formula_67 for the cotangent and the cosecant, where is an arbitrary integer.


title: Trigonometric functions

text: Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.


title: Trigonometric functions

text: There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:


title: Trigonometric functions

text: "Proof": Let formula_84 and formula_85 One has formula_86 for . The quotient rule implies thus that formula_87. Therefore, formula_88 is a constant function, which equals , as formula_89 This proves the formula.


title: Trigonometric functions

text: Solving this linear system in sine and cosine, one can express them in terms of the exponential function:


title: Trigonometric functions

text: Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity formula_93 for simplifying the result.


title: Trigonometric functions

text: For example, the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula


title: Trigonometric functions

text: The sine and cosine of a complex number formula_96 can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:


title: Trigonometric functions

text: By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of formula_98 becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.


title: Trigonometric functions

text: Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval , see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.


title: Trigonometric functions

text: The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is:


title: Trigonometric functions

text: All trigonometric functions are periodic functions of period . This is the smallest period, except for the tangent and the cotangent, which have as smallest period. This means that, for every integer , one has


title: Trigonometric functions

text: The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is


title: Trigonometric functions

text: The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.


title: Trigonometric functions

text: When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.


title: Trigonometric functions

text: By setting formula_109 all trigonometric functions of formula_110 can be expressed as rational fractions of formula_111:


title: Trigonometric functions

text: this is the tangent half-angle substitution, which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.


title: Trigonometric functions

text: The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number  is a constant of integration.


title: Trigonometric functions

text: Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:


title: Trigonometric functions

text: The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.


title: Trigonometric functions

text: The notations , , etc. are often used for and , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "arcsecond".


title: Trigonometric functions

text: Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms.


title: Trigonometric functions

text: In this section , , denote the three (interior) angles of a triangle, and , , denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.


title: Trigonometric functions

text: The law of sines states that for an arbitrary triangle with sides , , and and angles opposite those sides , and :


title: Trigonometric functions

text: It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in "triangulation", a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.


title: Trigonometric functions

text: The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:


title: Trigonometric functions

text: In this formula the angle at is opposite to the side . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem.


title: Trigonometric functions

text: The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.


title: Trigonometric functions

text: If "s" is the triangle's semiperimeter, ("a" + "b" + "c")/2, and "r" is the radius of the triangle's incircle, then "rs" is the triangle's area. Therefore Heron's formula implies that:


title: Trigonometric functions

text: The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.


title: Trigonometric functions

text: Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.


title: Trigonometric functions

text: Under rather general conditions, a periodic function can be expressed as a sum of sine waves or cosine waves in a Fourier series. Denoting the sine or cosine basis functions by , the expansion of the periodic function takes the form:


title: Trigonometric functions

text: In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.


title: Trigonometric functions

text: While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) can be traced back to the "jyā" and "koti-jyā" functions used in Gupta period Indian astronomy ("Aryabhatiya", "Surya Siddhanta"), via translation from Sanskrit to Arabic and then from Arabic to Latin. (See Aryabhata's sine table.)


title: Trigonometric functions

text: All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant. Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, and produced tables of tangents and cotangents. Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. The trigonometric functions were later studied by mathematicians including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.


title: Trigonometric functions

text: Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. (See Madhava series and Madhava's sine table.)


title: Trigonometric functions

text: The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.


title: Trigonometric functions

text: The terms "tangent" and "secant" were first introduced by the Danish mathematician Thomas Fincke in his book "Geometria rotundi" (1583).


title: Trigonometric functions

text: The 17th century French mathematician Albert Girard made the first published use of the abbreviations "sin", "cos", and "tan" in his book "Trigonométrie".


title: Trigonometric functions

text: In a paper published in 1682, Gottfried Leibniz proved that is not an algebraic function of . Though introduced as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his "Introduction to the Analysis of the Infinite" (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. He presented "Euler's formula", as well as near-modern abbreviations ("sin.", "cos.", "tang.", "cot.", "sec.", and "cosec.").


title: Trigonometric functions

text: A few functions were common historically, but are now seldom used, such as the chord, the versine (which appeared in the earliest tables), the coversine, the haversine, the exsecant and the excosecant. The list of trigonometric identities shows more relations between these functions.


title: Trigonometric functions

text: The word derives from Latin "sinus", meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word "jaib", meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin.


title: Trigonometric functions

text: The choice was based on a misreading of the Arabic written form "j-y-b" (), which itself originated as a transliteration from Sanskrit ', which along with its synonym ' (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek "string".


title: Trigonometric functions

text: The word "tangent" comes from Latin "tangens" meaning "touching", since the line "touches" the circle of unit radius, whereas "secant" stems from Latin "secans"—"cutting"—since the line "cuts" the circle.


title: Trigonometric functions

text: The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's "Canon triangulorum" (1620), which defines the "cosinus" as an abbreviation for the "sinus complementi" (sine of the complementary angle) and proceeds to define the "cotangens" similarly.


title: Hypoxia (medical)

text: Hypoxia is a condition in which the body or a region of the body is deprived of adequate oxygen supply at the tissue level. Hypoxia may be classified as either "generalized", affecting the whole body, or "local", affecting a region of the body. Although hypoxia is often a pathological condition, variations in arterial oxygen concentrations can be part of the normal physiology, for example, during strenuous physical exercise..


title: Hypoxia (medical)

text: Hypoxia differs from hypoxemia and anoxemia, in that hypoxia refers to a state in which oxygen present in a tissue or the whole body is insufficient, whereas hypoxemia and anoxemia refer specifically to states that have low or no oxygen in the blood. Hypoxia in which there is complete absence of oxygen supply is referred to as anoxia.


title: Hypoxia (medical)

text: Hypoxia can be due to external causes, when the breathing gas is hypoxic, or internal causes, such as reduced effectiveness of gas transfer in the lungs, reduced capacity of the blood to carry oxygen, compromised general or local perfusion, or inability of the affected tissues to extract oxygen from, or metabolically process, an adequate supply of oxygen from an adequately oxygenated blood supply.


title: Hypoxia (medical)

text: Generalized hypoxia occurs in healthy people when they ascend to high altitude, where it causes altitude sickness leading to potentially fatal complications: high altitude pulmonary edema (HAPE) and high altitude cerebral edema (HACE). Hypoxia also occurs in healthy individuals when breathing inappropriate mixtures of gases with a low oxygen content, e.g., while diving underwater, especially when using malfunctioning closed-circuit rebreather systems that control the amount of oxygen in the supplied air. Mild, non-damaging intermittent hypoxia is used intentionally during altitude training to develop an athletic performance adaptation at both the systemic and cellular level.


title: Hypoxia (medical)

text: Hypoxia is a common complication of preterm birth in newborn infants. Because the lungs develop late in pregnancy, premature infants frequently possess underdeveloped lungs. To improve blood oxygenation, infants at risk of hypoxia may be placed inside incubators that provide warmth, humidity, and supplemental oxygen. More serious cases are treated with continuous positive airway pressure (CPAP).


title: Hypoxia (medical)

text: Hypoxia exists when there is a reduced amount of oxygen in the tissues of the body. Hypoxemia refers to a reduction in arterial oxygenation below the normal range, regardless of whether gas exchange is impaired in the lung, arterial oxygen content (C – which represents the amount of oxygen delivered to the tissues) is adequate, or tissue hypoxia exists. The classification categories are not always mutually exclusive, and hypoxia can be a consequence of a wide variety of causes.


title: Hypoxia (medical)

text: Intermittent hypoxic training induces mild generalized hypoxia for short periods as a training method to improve sporting performance. This is not considered a medical condition. Acute cerebral hypoxia leading to blackout can occur during freediving. This is a consequence of prolonged voluntary apnea underwater, and generally occurs in trained athletes in good health and good physical condition.


title: Hypoxia (medical)

text: The term "generalized hypoxia" may refer to hypoxia affecting the whole body, or may be used as a synonym for hypoxic hypoxia, which occurs when there is insufficient oxygen in the breathing gas to oxygenate the blood to a level that will adequately support normal metabolic processes, and which will inherently affect all perfused tissues.


title: Hypoxia (medical)

text: In the case of altitude sickness, where hypoxia develops gradually, the symptoms include fatigue, numbness / tingling of extremities, nausea, and cerebral hypoxia. These symptoms are often difficult to identify, but early detection of symptoms can be critical.


title: Hypoxia (medical)

text: In severe hypoxia, or hypoxia of very rapid onset, ataxia, confusion, disorientation, hallucinations, behavioral change, severe headaches, reduced level of consciousness, papilloedema, breathlessness, pallor, tachycardia, and pulmonary hypertension eventually leading to the late signs cyanosis, slow heart rate, cor pulmonale, and low blood pressure followed by heart failure eventually leading to shock and death.