Sine, Cosine, Tangent Three Functions, but same idea. Right Triangle Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle . Before getting stuck into the functions, it helps to give a name to each side of a right triangle: "Opposite" is opposite to the angle θ "Adjacent" is adjacent (next to) to the angle θ "Hypotenuse" is the long one Adjacent is always next to the angle And Opposite is opposite the angle Sine, Cosine and Tangent Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle: For a given angle θ each ratio stays the same no matter how big or small the triangle is To calculate them: Divide the length of one side by another side Example: What is the sine of 35°? Using this triangle (lengths are only to one decimal place): sin(35°) d = 18.88 to 2 decimal places.

The depth "d" is 18.88 m Exercise Try this paper-based exercise where you can calculate the sine function for all angles from 0° to 360°, and then graph the result. It will help you to understand these relatively simple functions. You can also see Graphs of Sine, Cosine and Tangent . Less Common Functions To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used. They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan: Secant Function: What Is Trigonometry? What Is Trigonometry? We learn from The Words of Mathematics : trigonometry (noun), trigonometric (adjective): the first part of the word is from Greek trigon "triangle". The second part of trigonometry is from Greek metron "a measure." The Indo-European root is probably me- "to measure." Trigonometry is literally the measuring (of angles and sides) of triangles. Historically speaking, the triangular approach to trigonometry is ancient, wheres the circular approach now taught in our schools is relatively recent.

So originally trigonometry was understood to define relations between elements of a triangle. In a triangle, there are six basic elements: 3 sides and 3 angles. Not any three line segments may serve as the sides of a triangle. They do iff they satisfy the triangle inequality, or rather three triangle inequalities. Not any three angles may be the angles of a triangle. In Euclidean geometry, the three angles of a triangle add up to a straight angle . These requirements impose limitations on the manner in which the relations between the elements are defined. In modern trigonometry these relations are extended to arbitrary angles. This can be done, for example, by observing the projections of a rotating radius of a circle and a tangent at the end of the radius . If the sides a, b, c of a triangle lie opposite angles α, β, γ then a + b > c is oneof the inequalities that the sides obey, and α + β + γ = 180° is the identity that holds in Euclidean geometry. We also know that, if γ is right, then the Pythagorean theorem holds: a² + b² = c². (Its converse holds too.) Trigonometric relations involve trigonometric functions .

There is an awful amount of trigonometric identities. The most basic one is the Pythagorean theorem expressed in terms of sine and cosine: sin² α + cos² α = 1. Then there are double argument formulas: sin 2α = 2 sin α cos α cos 2α = cos² α - sin² α tan 2α = 2 tan α / (1 - tan² α) cot 2α = (cot² α - 1) / 2cot α. sin (α + β) = sin α cos β + cos α sin β cos (α + β) = cos α cos β - sin α sin β sin (α - β) = sin α cos β - cos α sin β cos (α - β) = cos α cos β + sin α sin β. And, of course, no list of trigonometric relations could be complete unless the Laws of Cosines and Sines are mentioned. Trigonometry is a methodology for finding some unknown elements of a triangle (or other geometric shapes) provided the data includes a sufficient amount of linear and angular measurements to define a shape uniquely.

For example, two sides a and b of a triangle and the angle they include define the triangle uniquely. The third side c can then be found from the Law of Cosines while the angles α and β are determined from the Law of Sines . The latter can be used to find the circumradius. The area of the triangle can be found from S = (ab sin γ)/2 and knowing that we can determine the inradius from S = (a + b + c)r/2, HS-mathematics - Trigonometry Trigonometry Pioneered By: Andrew Sciacchitano, Jocele Shelton, Jacob Montgomery, Jordan McClure Revised By: Molly Carter, Jennifer Moon, Jim Nordman, & Amanda VanderMeulen Revised By: Trevor Goodman, Will Braschler Table of Contents COMMON CORE STANDARDS FOR TRIGONOMETRY: WHAT IS TRIGONOMETRY? According to Wolfram MathWorld, trigonometry is "the study of angles and of the angular relationships of planar and three-dimensional figures." For more in-depth information, see "What is Trigonometry?" courtesy of Dave's Short Trig Course.

THE HISTORY OF TRIGONOMETRY: Hipparchus of Rhodes The first trigonometry problem may have been solved by Hipparchus of Rhodes (ca. 190-120). As a result of this, he is generally considered to be the father of trigonometry. His problem dealt with why the seasons are different lengths. Hipparchus used the observed lengths of each of the seasons to determine the arc length traveled by the sun in its orbit during each season. He then found the lengths of the chords that connect the sun’s position at the ends of the seasons. These lengths enabled him to determine how far the earth was from the center of the sun’s orbit. A diagram of this can be found below. Image Source: In addition, Hipparchus was able to create a table to approximate the chord lengths of a circle. More can be found about Hipparchus of Rhodes HERE . Finding Chord Lengths Euclid may have been the first person to show how to calculate other chord lengths when he determined the lengths of the sides of regular inscribed pentagons and decagons. A diagram of this can be found below.

Image Source: Using the foundation that Euclid started, Ptolemy of Alexandria (90–168 CE) constructed a table of chord lengths for a circle of radius 60 in half-degree increments. Ptolemy’s table is equivalent to a table of sines in quarter-degree increments. Interestingly, his calculations were carried out to seven-digit accuracy. Indian astronomers in the third fourth or fifth century CE started the shift to the half-cord or sine. The use of sine was then adopted by Arabic mathematicians, later translated into the Latin word Sinus, literally meaning half-cord. From the Latin root is where we get the English Sine! The Unit Circle The development of the unit circle can be credited to Leonhard Euler (1707-83). It was Euler who decided that for the purposes of calculus, the radius should be fixed at 1. He realized that measuring the arc length and the line length in the same units would be beneficial; in addition, he concluded that if the radius of the circle is 1, then the circumference is 2 Pi. From this, it seems as though he was using radians. However, radians weren't used for almost a hundred years after Euler's death.

Triangle Trigonometry Triangle trigonometry started with the simple problem of finding the length of a shadow cast by an object at certain angles of the sun. An example of this can be found below. Image Source: Finding the answer to this question were mathematicians such as Ptolemy and Al-Khwarizmi of Baghdad (ca. 790–840). Interestingly, Al-Khwarizmi's name is the root for the word "algorithm" and whose book, al-jabr, is the root for the word algebra. The early triangle trigonometry problems led to the formation of trigonometric identities. The name for tangent comes from the fact the length being solved for is tangent to the circle. The secant, arising from the Latin secantem, meaning “cutting,” is the length of the radial line segment cut off by the tangent. The cosine, cotangent, and cosecant are the corresponding line segments for the complementary angle. Using trigonometry to solve for right triangles became prominent with the posthumous publishing of Johann Müller's De Triangulis Omnimodis (or, On Triangles of Every Kind).

Müller, Georg Rheticus, and Bartholomew Pitiscus also used trigonometry and similar triangles to solve for an unknown side of any given right triangle for which one of the acute angles and one other side are given. For more on the history of trigonometry: History of trig BASIC IDEAS OF TRIGONOMETRY: The Unit Circle Trigonometric functions are defined using the unit circle. The Unit Circle is centered at (0,0) and has a radius of 1 unit. Let Θ (theta) be an angle measured counterclockwise from the x-axis along an arc of the circle. Then cos Θ is the horizontal coorinate of the arc endpoint, and sin Θis the vertical coordinate, as shown below. Degrees A circle is divided into 360 parts called degrees. There are many theories about the use of degrees as a measure of the angles around a circle. Some say that the use of the number 360 corresponds to the amount of days in a year rounded down to the nearest 10. (for more theories click here ).

While the background of degrees is debatable the fact that they exist and can be used to measure angles in many different forms has provided a great opportunities. Radians We also can measure the angles around a circle in units called radians. Unlike degrees, radians have a more mathematical background.They are a pure measure based on the radius of a circle. To Demonstrate we will start with the unit circle, which has a radius of 1 unit. To show 1 radian, we will leave the radius connected to the outside edge of the circle and rotate it until it is standing straight up, perpendicular to the original radius line. Next we will lay the line down along the curve of the circle. The angle this creates is 1 radian. If we continue to add the same arc length all of the way around the circle, we will find that the circle is about 6.2831 radians. We can easily compare this to the circumference of the circle, which for the unit circle is 2π or 6.2831.

We can see an illustration of this below: Converting Radians to Degrees or Degrees to Radians Knowing that the entire way around the circle is 2π radians, we also know that half way around the circle is π radians. We can compare this to the fact that half way around the circle is 180° and get π radians = 180°. We can rewrite this as 1 radian = 180°/π degrees and 1 degree = π/180° radians. Below is a graph of all of the radian values corresponding with their matching degree values. Right Triangles Right triangles are often used in trigonometry, as seen above. A right triangle has three sides, hypotenuse, adjacent to a given angle Θ and opposite to given angle Θ as shown below. The angle can be found using the definitions of the trigonometric functions. There are many mnemonics used to help students remember these, such as the most commonly used "SOHCAHTOA" where sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.

Another mnemonic used is "Tommy On A Ship Of His Caught A Herring." The trigonometric functions are cosine (cos x), sine (sin x), tangent (tan x), cosecant (csc x), secant (sec x), and cotangent (cot x). Below are some of the trigonometric functions. Geogebra Example (an example of what the values of sine, cosine, tangent, cosecant, secant, and cotangent relate to on a triangle on the unit circle can be found on this webpage ) (an example of how the values of sine, cosine, tangent, cosecant, secant, and cotangent change as theta changes can be found on the webpage TouchTrigonometry.org ) When finding the lengths of the sides of a right triangle, a simple formula that most students know is the Pythagorean theorem. However, most students do not understand how the formula actually works. Below is an illustration of how this holds: This illustration demonstrates that the lengths of the two smaller sides, a and b, individually squared create two squares in which the area is equal to that of the the square created by the hypotenuse, or c, squared. -links given from HERE .

TRIGONOMETRIC IDENTITIES: A mathematical identity (as defined by the Merriam-Webster's Dictionary - is an equation that is satisfied by all values of the symbol. Here, we can think of the word symbol as the exact same thing as a variable. For example, though trivial, x = x is an identity because for every value you can substitute for 'x', the equation will always be true. In trigonometry, identities are very important and help us tremendously. They can allow us to put a function in terms of one trigonometric function into terms of another. The most basic, and likely most common, identity is the Pythagorean identity, which states: (from: ) While these three equations appear different, they are equal to one another. Divide the first equation by cos^2 and you get the second equation. Divide the first equation by sin^2 and you get the third equation. Other useful identities can be found at TrigIDs . These identities are put into groups relevant to what they represent. For example, the half angle identities represent what happens when the variable of a trigonometric funciton is divided by two.

The most intriguing and useful of these identities is probably the cos(2x) identity (seen in the 'double angle identities' group in the list of identities). What makes this so interesting is that it takes a function of degree one - (cos(2x)) - and converts it into a function of degree two. The same can be thought of in reverse as well. Trigonometric identities are very useful when changing the representation of a trigonometric function. For example, if you have the square root of (1 - cosx)/2, then you can simply write this as sin(x/2). Applications of this and similar processes can be seen using all of the other trigonometric identities. Putting together the things that we now know about trigonometric functions and the Unit Circle, we can construct a very well-defined graphical image of the Unit Circle itself, shown below, which is considered by many to be the base from which all of trigonometry stems. The Unit Circle with points labeled in degrees, radians, and cartesian coordinates. Trigonometric Identities Proofs Adding Angles Trigonometry can explain not only the properties of angles but also the properties of multiple angles.

Let's look at an angle defined as the addition of angle A and angle B. We can draw this by drawing a right triangle with angle A touching a right triangle with angle B as shown below. Since we define the hypotenuse of the second triangle to have length 1, we know that the legs of that triangle are cos(B) and sin(B). Since our triangles share a side, we know that the hypotenuse of the bottom triangle is also cos(B). This tells us that the bottom triangle has legs of length cos(B)cos(A) and cos(B)sin(A). Now if we look at the highlighted triangle, we know that the hypotenuse is equal to sin(B). This tells us that the highlighted triangle has legs of length sin(B)cos(A) and sin(B)sin(A). The sin(A+B) is going to equal the sum of the vertical components of the bottom triangle and the highlighted triangle. This means that sin(A+B) = sin(A)cos(B) + cos(A)sin(B) We also know that the cos(A+B) is going to equal the horizontal component of the bottom triangle minus the horizontal component of the highlighted triangle.

This means that cos(A+B) = cos(A)cos(B) - sin(A)sin(B) Since the tangent is defined as sin/cos, we know that tan(A+B) = sin(A+B) / cos(A+B) We can substitute our known values for the sin(A+B) and the cos(A+B) from above to find Since the sin(B) = cos(B)tan(B), we can factor out the cos(B) term from the top and the bottom to get the tan(A+B) is equal to Note again that sin(A) = cos(A)tan(A). We can therefore factor out the cos(A) term from the top and bottom to get We have some common terms in both the top and bottom. We can eliminate the cos(A) and cos(B) terms to find Therefore we know that the tan(A+B) = [tan(A) + tan(B)] \ [1 - tan(A)tan(B)] Pythagorean Identity First recall that the equation to the unit circle centered at the origin is x2+y2=1. Let P be a point on the unit circle with an angle Ɵ from the x-axis.

Image Source: The x and y coordinates of P are given as x=cos Ɵ and y=sin Ɵ. By substituting these values of x and y into our equation for the unit circle we can see that (A worksheet about the Pythagorean Identity can be found HERE .) The Law of Sines & The Law of Cosines: Pythagorean The Functions Applied to Non-Right Triangles We use the Law of Sines and Cosines to solve oblique triangles, which are triangles that do not have a right angle. If you are given information about a triangle and asked to solve for the unknowns you can use these two formulas. Law of Sines You use two equivalent ratios at a time based of the information that you have been given and what you are being asked to find. Law of Cosines But which one do you use? Below shows the method you would use to solve the unknown values, depending on what you are given. 1. If you are given two angles and any side of a triangle, also known as AAS or ASA, you will use the Law of Sines to solve. 2.

If you are given two sides and an angle opposite one of the given sides, also know as SSA, you will you the Law of Sines to solve. *This is also know as the ambiguous case, since the situation could produce one of three possible solutions, depending on the specific values of the two sides and the opposite angle. You would start by solving for one of the unknown angles and if |sin A| > 1, then no solution is possible. If |sin A|< 1, then there are two angles to consider, A and A', which is equal to (180° - A). We have at least one solution with A, but we need to check A' by adding it to the angle that was given. If it is greater than 180°, then there is only the one solution, but if it is less than 180°, there is a second solution. 3. If you given all three sides, or SSS, you will use the Law of Cosines to solve. 4. If you are given two sides and the included angle, or SAS, you will use the Law of Cosines to solve.

Pythagorean Proofs of the Law of Sines and Cosines The Law of Sines asserts that a certain ratio holds true in any triangle, namely, that the ratio of the sine of an angle to the side opposite that angle is a constant. The proof is as follows. Consider the triangle ABC. Now we will drop a perpendicular from one of the vertices of the triangle and label the length of the perpendicular h. Let's consider the angle A. Now, sinA is defined as the side opposite angle A divided by the hypotenuse of ACD, or h/b. So using algebra, bsinAh. Now also consider angle B. similarly, sinBh/a. So then asinB = h. Now we have two equal expressions. asinB = bsinA (sinA)/a = (sinB)/b Since this will work with any arbitrary perpendicular, we can say with certainty that the ratio will hold true for all sides and all angles of our triangle, therefore (sinA)/a = (sinB)/b = (sinC)/c The Law of Cosines is a little bit more complex of a claim for triangles. Consider the triangle ABC again, redrawn below.

Looking at the triangle ACD and using the Pythagorean Theorem, we can see that (x^2) + (h^2) = (b^2) and we know from our Trigonometric Identities that cosA = x/bbcosA = x We will next examine triangle BCD and using the Pythagorean Theorem, we can see that ((c - x)^2)+(h^2) = (a^2)(c^2) - 2cx + (x^2) + (h^2) = (a^2)(c^2) - 2c(bcosA) + (b^2) = (a^2)(b^2) + (c^2) - 2bccosA = (a^2) The last line of the equation we have derived is known as the Law of Cosines, and since this property will hold true for any perpendicular, we can conclude that (a^2) = (b^2) + (c^2) - 2bccosA(b^2) = (a^2) + (c^2) - 2accosB(c^2) = (a^2) + (

b^2) - 2abcosC Making Connections with a Model: Great for Visual Learners There are many connections that can be made with this picture.

To get your students started you may show them that sin^2(x) + cos^2(x) = 1. You can see in this picture that all these can be connected and help the students start to formulate their own connections. Example Problem: APPLICATIONS OF TRIGONOMETRY: Architecture In architecture, trigonometry plays a massive role, especially in the compilation of building plans. Architects have to calculate exact angles of intersection for the components of their structure to ensure stability and safety. Some instances of trigonometric use in architecture include: arches, domes, support beams, and suspension bridges. More information on how trigonometry is used in facades can be found HERE . Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles. Architecture remains one of the most important sectors of our society. As architects plan the design of buildings trigonometry is used to insure its longevity. In addition, trigonometry can be used for aesthetic purposes, as seen architecturally below. Image Source: Astronomy Astronomy has been studied by civilizations in all regions of the world from ancient times onward.

In our modern age, trigonometry helps calculate distances between stars and learn more about the universe. Menelaus' Theorem helps astronomers gather information by providing a back drop in spherical triangle calculation. Geology Trigonometry is used in geology to estimate the true dip of bedding angles. An example of this can be found below. Calculating the true dip allows geologist to determine slope stability. Image Source: ~sadura/strref/str14.jpg Navigation Trigonometry is used in navigation to find distances of the shore from a point in the sea. It is also used in oceanography in calculating the height of the tides in the oceans. One day, it can be used in space travel, specifically finding the distance between celestial bodies in space. In addition, people and places can be found using triangulation. Triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline. A diagram of this can be found below. Photo Source: Physics Trigonometry plays a huge role in all varieties of physics. For example, radio, microwave, and electromagnetic waves are all measured and graphed with trigonometric functions such as sine and cosine.

Physics concerns itself greatly with three-dimensional space. Mathematics, specifically trigonometry, is critical to being able to determine values and help identify the nature of something in the space it exists in. Trigonometry is used so much in physics that it is as prevalent as simple addition and subtraction for the average person. A worksheet using trigonometry and physics can be found HERE . More For jobs that use trigonometry, see XP Math - Jobs That Use Right Triangle Trigonometry . In addition, each individual career reveals math content areas necessary for employment. COMMON CORE STANDARDS FOR TRIGONOMETRY: Common Core State Standards: Trigonometric Functions Standards (The links below certain standards provide activities or resources related to those standards) Extend the Domain of Trigonometric Functions Using the Unit Circle Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. National Council of Teachers of Mathematics Expectations: In grades 9–12 all students should Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them; Use trigonometric relationships to determine lengths and angle measures.

State Standards (As set by the CCSS institution): Define trigonometric ratios and solve problems involving right triangles Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Explain and use the relationship between the sine and cosine of complementary angles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply trigonometry to general triangles Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Prove the Laws of Sines and Cosines and use them to solve problems. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Note: No reference needed because of non-commercial use which is allowable under the CCSS institution terms of use Sine, Cosine and Tangent in Four Quadrants Sine, Cosine and Tangent in Four Quadrants Sine, Cosine and Tangent The three main functions in trigonometry are Sine, Cosine and Tangent .

They are easy to calculate: Divide the length of one side of a right angled triangle by another side ... but we must know which sides! For an angle θ, the functions are calculated this way: Sine Function: Example: What is the sine of 35°? Using this triangle (lengths are only to one decimal place): sin(35°) = Opposite / Hypotenuse = 2.8/4.9 = 0.57... Cartesian Coordinates Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is: The point (12,5) is 12 units along, and 5 units up. Four Quadrants When we include negative values, the x and y axes divide the space up into 4 pieces: Quadrants I, II, III and IV (They are numbered in a counter-clockwise direction) In Quadrant I both x and y are positive, in Quadrant II x is negative (y is still positive), in Quadrant III both x and y are negative, and in Quadrant IV x is positive again, and y is negative.

Like this: θ − 180º And if any angle is less than 0º, then add 360º.

We can now solve equations for angles between 0º and 360º (using Inverse Sine Cosine and Tangent ) Example: Solve sin θ = 0.5 We get the first solution from the calculator = sin-1(0.5) = 30º (it is in Quadrant I) The other solution is 180º − 30º = 150º (Quadrant II) Example: Solve tan θ = −1.3 We get the first solution from the calculator = tan-1(−1.3) = −52.4º This is less than 0º, so we add 360º: −52.4º + 360º = 307.6º (Quadrant IV) The other solution is 307.6º − 180º = 127.6º (Quadrant II) Example: Solve cos θ = −0.85 We get the first solution from the calculator = cos-1(−0.85) = 148.2º (Quadrant II) The other solution is 

360º − 148.2º = 211.8º (Quadrant III) trigonometry | Britannica.com Trigonometry analysis Trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations.

There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure . For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A, or sin A; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures. Trigonometry developed from a need to compute angles and distances in such fields as astronomy , map making, surveying, and artillery range finding. Problems involving angles and distances in one plane are covered in plane trigonometry . Applications to similar problems in more than one plane of three-dimensional space are considered in spherical trigonometry .

History of trigonometry Classical trigonometry The word trigonometry comes from the Greek words trigonon (“triangle”) and metron (“to measure”). Until about the 16th century, trigonometry was chiefly concerned with computing the numerical values of the missing parts of a triangle (or any shape that can be dissected into triangles) when the values of other parts were given. For example, if the lengths of two sides of a triangle and the measure of the enclosed angle are known, the third side and the two remaining angles can be calculated. Such calculations distinguish trigonometry from geometry , which mainly investigates qualitative relations. Of course, this distinction is not always absolute: the Pythagorean theorem , for example, is a statement about the lengths of the three sides in a right triangle and is thus quantitative in nature. Still, in its original form, trigonometry was by and large an offspring of geometry; it was not until the 16th century that the two became separate branches of mathematics. Ancient Egypt and the Mediterranean world Similar Topics Principles of trigonometry Trigonometric functions A somewhat more general concept of angle is required for trigonometry than for geometry.

An angle A with vertex at V, the initial side of which is VP and the terminal side of which is VQ, is indicated in the figure by the solid circular arc. This angle is generated by the continuous counterclockwise rotation of a line segment about the point V from the position VP to the position VQ. A second angle A′ with the same initial and terminal sides, indicated in the figure by the broken circular arc, is generated by the clockwise rotation of the line segment from the position VP to the position VQ. Angles are considered positive when generated by counterclockwise rotations, negative when generated by clockwise rotations. The positive angle A and the negative angle A′ in the figure are generated by less than one complete rotation of the line segment about the point V. All other positive and negative angles with the same initial and terminal sides are obtained by rotating the line segment one or more complete turns before coming to rest at VQ. Numerical values can be assigned to angles by selecting a unit of measure. The most common units are the degree and the radian . There are 360° in a complete revolution, with each degree further divided into 60′ (minutes) and each minute divided into 60″ (seconds).

In theoretical work, the radian is the most convenient unit. It is the angle at the centre of a circle that intercepts an arc equal in length to the radius; simply put, there are 2π radians in one complete revolution. From these definitions, it follows that 1° = π/180 radians. Equal angles are angles with the same measure; i.e., they have the same sign and the same number of degrees. Any angle −A has the same number of degrees as A but is of opposite sign. Its measure, therefore, is the negative of the measure of A. If two angles, A and B, have the initial sides VP and VQ and the terminal sides VQ and VR, respectively, then the angle A + B has the initial and terminal sides VP and VR (see the figure ). The angle A + B is called the sum of the angles A and B, and its relation to A and B when A is positive and B is positive or negative is illustrated in the figure. The sum A + B is the angle the measure of which is the algebraic sum of the measures of A and B. The difference A − B is the sum of A and −B.

Thus, all angles coterminal with angle A (i.e., with the same initial and terminal sides as angle A) are given by A ± 360n, in which 360n is an angle of n complete revolutions. The angles (180 − A) and (90 − A) are the supplement and complement of angle A, respectively. Trigonometric functions of an angle To define trigonometric functions for any angle A, the angle is placed in position (see the figure ) on a rectangular coordinate system with the vertex of A at the origin and the initial side of A along the positive x-axis; r (positive) is the distance from V to any point Q on the terminal side of A, and (x, y) are the rectangular coordinates of Q. The six functions of A are then defined by six ratios exactly as in the earlier case for the triangle given in the introduction (see the figure ). Because division by zero is not allowed, the tangent and secant are not defined for angles the terminal side of which falls on the y-axis, and the cotangent and cosecant are undefined for angles the terminal side of which falls on the x-axis.

When the Pythagorean equality x2 + y2 = r2 is divided in turn by r2, x2, and y2, the three squared relations relating cosine and sine, tangent and secant, cotangent and cosecant are obtained. If the point Q on the terminal side of angle A in standard position has coordinates (x, y), this point will have coordinates (x, −y) when on the terminal side of −A in standard position. From this fact and the definitions are obtained further identities for negative angles. These relations may also be stated briefly by saying that cosine and secant are even functions (symmetrical about the y-axis), while the other four are odd functions (symmetrical about the origin). It is evident that a trigonometric function has the same value for all coterminal angles. When n is an integer, therefore, sin (A ± 360n) = sin A; there are similar relations for the other five functions. These results may be expressed by saying that the trigonometric functions are periodic and have a period of 360° or 180°.

When Q on the terminal side of A in standard position has coordinates (x, y), it has coordinates (−y, x) and (y, −x) on the terminal side of A + 90 and A − 90 in standard position, respectively. Consequently, six formulas follow which display that a function of the complement of A is equal to the corresponding cofunction of A. Of fundamental importance for the study of trigonometry are the addition formulas, functions of the sum or difference of two angles. From the addition formulas are derived the double-angle and half-angle formulas. Numerous identities of lesser importance can be derived from the above basic identities. Tables of natural functions To be of practical use, the values of the trigonometric functions must be readily available for any given angle. Various trigonometric identities show that the values of the functions for all angles can readily be found from the values for angles from 0° to 45°. For this reason, it is sufficient to list in a table the values of sine, cosine, and tangent for all angles from 0° to 45° that are integral multiples of some convenient unit (commonly 1′).

Before computers rendered them obsolete in the late 20th century, such trigonometry tables were helpful to astronomers, surveyors, and engineers. For angles that are not integral multiples of the unit, the values of the functions may be interpolated. Because the values of the functions are in general irrational numbers, they are entered in the table as decimals, rounded off at some convenient place. For most purposes, four or five decimal places are sufficient, and tables of this accuracy are common. Simple geometrical facts alone, however, suffice to determine the values of the trigonometric functions for the angles 0°, 30°, 45°, 60°, and 90°. Plane trigonometry In many applications of trigonometry the essential problem is the solution of triangles . If enough sides and angles are known, the remaining sides and angles as well as the area can be calculated, and the triangle is then said to be solved. Triangles can be solved by the law of sines and the law of cosines.

To secure symmetry in the writing of these laws, the angles of the triangle are lettered A, B, and C and the lengths of the sides opposite the angles are lettered a, b, and c, respectively. An example of this standardization is shown in the figure . The law of sines is expressed as an equality involving three sine functions while the law of cosines is an identification of the cosine with an algebraic expression formed from the lengths of sides opposite the corresponding angles. To solve a triangle, all the known values are substituted into equations expressing the laws of sines and cosines, and the equations are solved for the unknown quantities. For example, the law of sines is employed when two angles and a side are known or when two sides and an angle opposite one are known. Similarly, the law of cosines is appropriate when two sides and an included angle are known or three sides are known. Texts on trigonometry derive other formulas for solving triangles and for checking the solution. Older textbooks frequently included formulas especially suited to logarithmic calculation. Newer textbooks, however, frequently include simple computer instructions for use with a symbolic mathematical program such as Mathematica™ or Maple™.

Spherical trigonometry Spherical trigonometry involves the study of spherical triangles , which are formed by the intersection of three great circle arcs on the surface of a sphere. Spherical triangles were subject to intense study from antiquity because of their usefulness in navigation , cartography , and astronomy. (See the section Passage to Europe .) The angles of a spherical triangle are defined by the angle of intersection of the corresponding tangent lines to each vertex. The sum of the angles of a spherical triangle is always greater than the sum of the angles in a planar triangle (π radians, equivalent to two right angles). The amount by which each spherical triangle exceeds two right angles (in radians) is known as its spherical excess. The area of a spherical triangle is given by the product of its spherical excess E and the square of the radius r of the sphere it resides on—in symbols, Er2. By connecting the vertices of a spherical triangle with the centre O of the sphere that it resides on, a special “angle” known as a trihedral angle is formed.

The central angles (also known as dihedral angles) between each pair of line segments OA, OB, and OC are labeled α, β, and γ to correspond to the sides (arcs) of the spherical triangle labeled a, b, and c, respectively. Because a trigonometric function of a central angle and its corresponding arc have the same value, spherical trigonometry formulas are given in terms of the spherical angles A, B, and C and, interchangeably, in terms of the arcs a, b, and c and the dihedral angles α, β, and γ. Furthermore, most formulas from plane trigonometry have an analogous representation in spherical trigonometry. For example, there is a spherical law of sines and a spherical law of cosines. As was described for a plane triangle, the known values involving a spherical triangle are substituted in the analogous spherical trigonometry formulas, such as the laws of sines and cosines, and the resulting equations are then solved for the unknown quantities. Many other relations exist between the sides and angles of a spherical triangle.

Worth mentioning are Napier’s analogies (derivable from the spherical trigonometry half-angle or half-side formulas), which are particularly well suited for use with logarithmic tables. Analytic trigonometry Analytic trigonometry combines the use of a coordinate system, such as the Cartesian coordinate system used in analytic geometry, with algebraic manipulation of the various trigonometry functions to obtain formulas useful for scientific and engineering applications. Trigonometric functions of a real variable x are defined by means of the trigonometric functions of an angle. For example, sin x in which x is a real number is defined to have the value of the sine of the angle containing x radians. Similar definitions are made for the other five trigonometric functions of the real variable x. These functions satisfy the previously noted trigonometric relations with A, B, 90°, and 360° replaced by x, y, π/2 radians, and 2π radians, respectively. The minimum period of tan x and cot x is π, and of the other four functions it is 2π. In the calculus it is shown that sin x and cos x are sums of power series .

These series may be used to compute the sine and cosine of any angle. For example, to compute the sine of 10°, it is necessary to find the value of sin π/18 because 10° is the angle containing π/18 radians. When π/18 is substituted in the series for sin x, it is found that the first two terms give 0.17365, which is correct to five decimal places for the sine of 10°. By taking enough terms of the series, any number of decimal places can be correctly obtained. Tables of the functions may be used to sketch the graphs of the functions, as shown in the figure . Each trigonometric function has an inverse function , that is, a function that “undoes” the original function. For example, the inverse function for the sine function is written arc sin or sin−1, thus sin−1(sin x) = sin (sin−1 x) = x. The other trigonometric inverse functions are defined similarly.

Coordinates and transformation of coordinates Polar coordinates For problems involving directions from a fixed origin (or pole) O, it is often convenient to specify a point P by its polar coordinates (r, θ), in which r is the distance OP and θ is the angle that the direction of r makes with a given initial line. The initial line may be identified with the x-axis of rectangular Cartesian coordinates, as shown in the figure . The point (r, θ) is the same as (r, θ + 2nπ) for any integer n. It is sometimes desirable to allow r to be negative, so that (r, θ) is the same as (−r, θ + π). Given the Cartesian equation for a curve, the polar equation for the same curve can be obtained in terms of the radius r and the angle θ by substituting r cos θ and r sin θ for x and y, respectively. For example, the circle x2 + y2 = a2 has the polar equation (r cos θ)2 + (r sin θ)2 = a2, which reduces to r = a.

(The positive value of r is sufficient, if θ takes all values from −π to π or from 0 to 2π). Thus the polar equation of a circle simply expresses the fact that the curve is independent of θ and has constant radius. In a similar manner, the line y = x tan ϕ has the polar equation sin θ = cos θ tan ϕ, which reduces to θ = ϕ. (The other solution, θ = ϕ + π, can be discarded if r is allowed to take negative values.) Transformation of coordinates A transformation of coordinates in a plane is a change from one coordinate system to another. Thus, a point in the plane will have two sets of coordinates giving its position with respect to the two coordinate systems used, and a transformation will express the relationship between the coordinate systems. For example, the transformation between polar and Cartesian coordinates discussed in the preceding section is given by x = r cos θ and y = r sin θ. Similarly, it is possible to accomplish transformations between rectangular and oblique coordinates. In a translation of Cartesian coordinate axes, a transformation is made between two sets of axes that are parallel to each other but have their origins at different positions.

If a point P has coordinates (x, y) in one system, its coordinates in the second system are given by (x − h, y − k) where (h, k) is the origin of the second system in terms of the first coordinate system. Thus, the transformation of P between the first system (x, y) and the second system (x′, y′) is given by the equations x = x′ + h and y = y′ + k. The common use of translations of axes is to simplify the equations of curves. For example, the equation 2x2 + y2 − 12x −2y + 17 = 0 can be simplified with the translations x′ = x − 3 and y′ = y − 1 to an equation involving only squares of the variables and a constant term: (x′)2 + (y′)2/2 = 1. In other words, the curve represents an ellipse with its centre at the point (3, 1) in the original coordinate system.

A rotation of coordinate axes is one in which a pair of axes giving the coordinates of a point (x, y) rotate through an angle ϕ to give a new pair of axes in which the point has coordinates (x′, y′), as shown in the figure . The transformation equations for such a rotation are given by x = x′ cos ϕ − y′ sin ϕ and y = x′ sin ϕ + y′ cos ϕ. The application of these formulas with ϕ = 45° to the difference of squares, x2 − y2 = a2, leads to the equation x′y′ = c (where c is a constant that depends on the value of a). This equation gives the form of the rectangular hyperbola when its asymptotes (the lines that a curve approaches without ever quite meeting) are used as the coordinate axes.

Question: In addition to sine and cosine, what is the third main function of trigonometry?
Answer: Tangent