# Trigonometriskās funkcijas[labot | labot pirmkodu]

Trigonometrisko funkciju grafiki: sinuss, kosinuss, tangenss, kotangenss, sekanss, kosekanss
**Trigonometriska funkcija** ir jebkura no funkcijām sin *x*, cos *x*, tg *x*, ctg *x*, sec *x* un cosec *x*, kur arguments *x* ir leņķis. Raksturīga šo funkciju īpašība ir to periodiskums.

Ne katra periodiska funkcija, kuras arguments ir leņķis, ir trigonometriska funkcija. Piemēram, funkcija WIKINIUM\_MATH\_PH\_0 nav trigonometriska funkcija.

## Trigonometrisko funkciju uzskaitījums

Vienības aplis ar kosinusa un sinusa vērtībām

| Funkcija | Apzīmējums | Apraksts | Sakarības (izmantojot radiānus) |
| --- | --- | --- | --- |
| **Sinuss** | sin |  WIKINIUM\_MATH\_PH\_2  |  WIKINIUM\_MATH\_PH\_4  |
| **Kosinuss** | cos |  WIKINIUM\_MATH\_PH\_6  |  WIKINIUM\_MATH\_PH\_8  |
| **Tangenss** | tg |  WIKINIUM\_MATH\_PH\_10  |  WIKINIUM\_MATH\_PH\_12  |
| **Kotangenss** | ctg |  WIKINIUM\_MATH\_PH\_14  |  WIKINIUM\_MATH\_PH\_16  |
| **Sekanss** | sec |  WIKINIUM\_MATH\_PH\_18  |  WIKINIUM\_MATH\_PH\_20  |
| **Kosekanss** | cosec (vai csc) |  WIKINIUM\_MATH\_PH\_22  |  WIKINIUM\_MATH\_PH\_24  |

## Trigonometrisko funkciju īpašības

### Galvenās vienādības

Tā kā sinuss un kosinuss ir attiecīgi punkta ordināta un abscisa, kas atbilst leņķa α riņķim, tad, atbilstoši Pitagora teorēmai

 WIKINIUM\_MATH\_PH\_26
Dalot šīs vienādības abas puses ar sinusa kvadrātu vai kosinusa kvadrātu, iegūstam:

 WIKINIUM\_MATH\_PH\_28
 WIKINIUM\_MATH\_PH\_30
### Nepārtrauktība

Sinuss un kosinuss ir nepārtrauktas funkcijas, bet tangensam, kotangensam, sekansam un kosekansam ir pārtraukuma punkti WIKINIUM\_MATH\_PH\_32 kotangenss un kosekanss — WIKINIUM\_MATH\_PH\_34

### Paritāte

Kosinuss un sekanss ir funkcijas, kurām ir simetrija attiecībā uz funkcijas zīmes maiņu. Pārējām četrām funkcijām tādas īpašības nav, t.i.:

 WIKINIUM\_MATH\_PH\_36
 WIKINIUM\_MATH\_PH\_38
 WIKINIUM\_MATH\_PH\_40
 WIKINIUM\_MATH\_PH\_42
 WIKINIUM\_MATH\_PH\_44
 WIKINIUM\_MATH\_PH\_46
### Periodiskums

Funkcijas WIKINIUM\_MATH\_PH\_48 , WIKINIUM\_MATH\_PH\_50 , WIKINIUM\_MATH\_PH\_52 un WIKINIUM\_MATH\_PH\_54 ir periodiskas funkcijas ar periodu WIKINIUM\_MATH\_PH\_56 . Savukārt, funkcijas WIKINIUM\_MATH\_PH\_58 un WIKINIUM\_MATH\_PH\_60 ir periodiskas ar periodu WIKINIUM\_MATH\_PH\_62

### Saskaitīšanas formulas

Summas trigonometriskās funkcijas nozīme un divu leņķu starpība:

 WIKINIUM\_MATH\_PH\_64
 WIKINIUM\_MATH\_PH\_66
 WIKINIUM\_MATH\_PH\_68
 WIKINIUM\_MATH\_PH\_70
Līdzīgas formulas trim leņķiem:

 WIKINIUM\_MATH\_PH\_72
 WIKINIUM\_MATH\_PH\_74
### Formulas leņķu daudzkārtņiem

Divkārša leņķa formulas:

 WIKINIUM\_MATH\_PH\_76
 WIKINIUM\_MATH\_PH\_78
 WIKINIUM\_MATH\_PH\_80
 WIKINIUM\_MATH\_PH\_82
Trīskārša leņķa formulas:

 WIKINIUM\_MATH\_PH\_84
 WIKINIUM\_MATH\_PH\_86
 WIKINIUM\_MATH\_PH\_88
 WIKINIUM\_MATH\_PH\_90
Citas leņķu daudzkārtņu formulas:

 WIKINIUM\_MATH\_PH\_92
 WIKINIUM\_MATH\_PH\_94
 WIKINIUM\_MATH\_PH\_96
 WIKINIUM\_MATH\_PH\_98
 WIKINIUM\_MATH\_PH\_100
 WIKINIUM\_MATH\_PH\_102
 WIKINIUM\_MATH\_PH\_104
 WIKINIUM\_MATH\_PH\_106
Pusleņķa formulas:

 WIKINIUM\_MATH\_PH\_108
 WIKINIUM\_MATH\_PH\_110
 WIKINIUM\_MATH\_PH\_112
 WIKINIUM\_MATH\_PH\_114
 WIKINIUM\_MATH\_PH\_116
 WIKINIUM\_MATH\_PH\_118
### Reizināšana

Formulas divu leņķu reizināšanai:

 WIKINIUM\_MATH\_PH\_120
 WIKINIUM\_MATH\_PH\_122
 WIKINIUM\_MATH\_PH\_124
 WIKINIUM\_MATH\_PH\_126
 WIKINIUM\_MATH\_PH\_128
 WIKINIUM\_MATH\_PH\_130
Līdzīgas formulas triju leņķu sinusu un kosinusu reizināšanai:

 WIKINIUM\_MATH\_PH\_132
 WIKINIUM\_MATH\_PH\_134
 WIKINIUM\_MATH\_PH\_136
 WIKINIUM\_MATH\_PH\_138
Attiecīgās formulas triju leņķu tangensiem un kotangensiem var iegūt, izdalot augstāk minēto vienādojumu labās puses ar kreisajām.

### Pakāpes

|  |  |
| --- | --- |
|  WIKINIUM\_MATH\_PH\_140  |  WIKINIUM\_MATH\_PH\_142  |
|  WIKINIUM\_MATH\_PH\_144  |  WIKINIUM\_MATH\_PH\_146  |
|  WIKINIUM\_MATH\_PH\_148  |  WIKINIUM\_MATH\_PH\_150  |
|  WIKINIUM\_MATH\_PH\_152  |  WIKINIUM\_MATH\_PH\_154  |
|  WIKINIUM\_MATH\_PH\_156  |  WIKINIUM\_MATH\_PH\_158  |
|  WIKINIUM\_MATH\_PH\_160  |  WIKINIUM\_MATH\_PH\_162  |

### Summas

 WIKINIUM\_MATH\_PH\_164
 WIKINIUM\_MATH\_PH\_166
 WIKINIUM\_MATH\_PH\_168
 WIKINIUM\_MATH\_PH\_170
 WIKINIUM\_MATH\_PH\_172
Funkcijām ar argumentu WIKINIUM\_MATH\_PH\_174 ir vienādojums:

 WIKINIUM\_MATH\_PH\_176
kur leņķi WIKINIUM\_MATH\_PH\_178 atrod pēc formulas:

 WIKINIUM\_MATH\_PH\_180
### Tangensa vienādības

Jebkuru trigonometrisko funkciju var izteikt kā pusleņķa tangensu.

 WIKINIUM\_MATH\_PH\_182

 WIKINIUM\_MATH\_PH\_184

 WIKINIUM\_MATH\_PH\_186

 WIKINIUM\_MATH\_PH\_188

 WIKINIUM\_MATH\_PH\_190

 WIKINIUM\_MATH\_PH\_192

## Trigonometrisko funkciju galveno vērtību tabula

|  WIKINIUM\_MATH\_PH\_194  | 0° (0 rad) | 30° (π/6) | 45° (π/4) | 60° (π/3) | 90° (π/2) | 180° (π) | 270° (3π/2) | 360° (2π) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  WIKINIUM\_MATH\_PH\_196  |  WIKINIUM\_MATH\_PH\_198  |  WIKINIUM\_MATH\_PH\_200  |  WIKINIUM\_MATH\_PH\_202  |  WIKINIUM\_MATH\_PH\_204  |  WIKINIUM\_MATH\_PH\_206  |  WIKINIUM\_MATH\_PH\_208  |  WIKINIUM\_MATH\_PH\_210  |  WIKINIUM\_MATH\_PH\_212  |
|  WIKINIUM\_MATH\_PH\_214  |  WIKINIUM\_MATH\_PH\_216  |  WIKINIUM\_MATH\_PH\_218  |  WIKINIUM\_MATH\_PH\_220  |  WIKINIUM\_MATH\_PH\_222  |  WIKINIUM\_MATH\_PH\_224  |  WIKINIUM\_MATH\_PH\_226  |  WIKINIUM\_MATH\_PH\_228  |  WIKINIUM\_MATH\_PH\_230  |
|  WIKINIUM\_MATH\_PH\_232  |  WIKINIUM\_MATH\_PH\_234  |  WIKINIUM\_MATH\_PH\_236  |  WIKINIUM\_MATH\_PH\_238  |  WIKINIUM\_MATH\_PH\_240  |  WIKINIUM\_MATH\_PH\_242  |  WIKINIUM\_MATH\_PH\_244  |  WIKINIUM\_MATH\_PH\_246  |  WIKINIUM\_MATH\_PH\_248  |
|  WIKINIUM\_MATH\_PH\_250  |  WIKINIUM\_MATH\_PH\_252  |  WIKINIUM\_MATH\_PH\_254  |  WIKINIUM\_MATH\_PH\_256  |  WIKINIUM\_MATH\_PH\_258  |  WIKINIUM\_MATH\_PH\_260  |  WIKINIUM\_MATH\_PH\_262  |  WIKINIUM\_MATH\_PH\_264  |  WIKINIUM\_MATH\_PH\_266  |
|  WIKINIUM\_MATH\_PH\_268  |  WIKINIUM\_MATH\_PH\_270  |  WIKINIUM\_MATH\_PH\_272  |  WIKINIUM\_MATH\_PH\_274  |  WIKINIUM\_MATH\_PH\_276  |  WIKINIUM\_MATH\_PH\_278  |  WIKINIUM\_MATH\_PH\_280  |  WIKINIUM\_MATH\_PH\_282  |  WIKINIUM\_MATH\_PH\_284  |
|  WIKINIUM\_MATH\_PH\_286  |  WIKINIUM\_MATH\_PH\_288  |  WIKINIUM\_MATH\_PH\_290  |  WIKINIUM\_MATH\_PH\_292  |  WIKINIUM\_MATH\_PH\_294  |  WIKINIUM\_MATH\_PH\_296  |  WIKINIUM\_MATH\_PH\_298  |  WIKINIUM\_MATH\_PH\_300  |  WIKINIUM\_MATH\_PH\_302  |

### Trigonometrisko funkciju vērtības nestandarta leņķiem

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
|  WIKINIUM\_MATH\_PH\_304  |  WIKINIUM\_MATH\_PH\_306  |  WIKINIUM\_MATH\_PH\_308  |  WIKINIUM\_MATH\_PH\_310  |  WIKINIUM\_MATH\_PH\_312  |  WIKINIUM\_MATH\_PH\_314  |  WIKINIUM\_MATH\_PH\_316  |  WIKINIUM\_MATH\_PH\_318  |
|  WIKINIUM\_MATH\_PH\_320  |  WIKINIUM\_MATH\_PH\_322  |  WIKINIUM\_MATH\_PH\_324  |  WIKINIUM\_MATH\_PH\_326  |  WIKINIUM\_MATH\_PH\_328  |  WIKINIUM\_MATH\_PH\_330  |  WIKINIUM\_MATH\_PH\_332  |  WIKINIUM\_MATH\_PH\_334  |
|  WIKINIUM\_MATH\_PH\_336  |  WIKINIUM\_MATH\_PH\_338  |  WIKINIUM\_MATH\_PH\_340  |  WIKINIUM\_MATH\_PH\_342  |  WIKINIUM\_MATH\_PH\_344  |  WIKINIUM\_MATH\_PH\_346  |  WIKINIUM\_MATH\_PH\_348  |  WIKINIUM\_MATH\_PH\_350  |
|  WIKINIUM\_MATH\_PH\_352  |  WIKINIUM\_MATH\_PH\_354  |  WIKINIUM\_MATH\_PH\_356  |  WIKINIUM\_MATH\_PH\_358  |  WIKINIUM\_MATH\_PH\_360  |  WIKINIUM\_MATH\_PH\_362  |  WIKINIUM\_MATH\_PH\_364  |  WIKINIUM\_MATH\_PH\_366  |
|  WIKINIUM\_MATH\_PH\_368  |  WIKINIUM\_MATH\_PH\_370  |  WIKINIUM\_MATH\_PH\_372  |  WIKINIUM\_MATH\_PH\_374  |  WIKINIUM\_MATH\_PH\_376  |  WIKINIUM\_MATH\_PH\_378  |  WIKINIUM\_MATH\_PH\_380  |  WIKINIUM\_MATH\_PH\_382  |

 WIKINIUM\_MATH\_PH\_384

 WIKINIUM\_MATH\_PH\_386 , kur WIKINIUM\_MATH\_PH\_388 .

 WIKINIUM\_MATH\_PH\_390 , kur WIKINIUM\_MATH\_PH\_392

## Trigonometrisko funkciju skaitlisko vērtību tabula

| *θ* grādos | *θ* radiānos | sin *θ* | cos *θ* | tan *θ* |
| --- | --- | --- | --- | --- |
| 0 | 0 | 0.0 | 1.0 | 0.0
  |
| 1 | 0.017453293 | 0.01745240 | 0.9998477 | 0.017455065
  |
| 2 | 0.034906585 | 0.034899497 | 0.99939083 | 0.034920769
  |
| 3 | 0.052359878 | 0.052335956 | 0.99862953 | 0.052407779
  |
| 4 | 0.06981317 | 0.069756474 | 0.99756405 | 0.069926812
  |
| 5 | 0.087266463 | 0.087155743 | 0.9961947 | 0.087488664
  |
| 6 | 0.10471976 | 0.10452846 | 0.9945219 | 0.10510424
  |
| 7 | 0.12217305 | 0.12186934 | 0.99254615 | 0.12278456
  |
| 8 | 0.13962634 | 0.1391731 | 0.99026807 | 0.14054083
  |
| 9 | 0.15707963 | 0.15643447 | 0.98768834 | 0.15838444
  |
| 10 | 0.17453293 | 0.17364818 | 0.98480775 | 0.17632698
  |
| 11 | 0.19198622 | 0.190809 | 0.98162718 | 0.19438031
  |
| 12 | 0.20943951 | 0.20791169 | 0.9781476 | 0.21255656
  |
| 13 | 0.2268928 | 0.22495105 | 0.97437006 | 0.23086819
  |
| 14 | 0.2443461 | 0.2419219 | 0.97029573 | 0.249328
  |
| 15 | 0.26179939 | 0.25881905 | 0.96592583 | 0.26794919
  |
| 16 | 0.27925268 | 0.27563736 | 0.9612617 | 0.28674539
  |
| 17 | 0.29670597 | 0.2923717 | 0.95630476 | 0.30573068
  |
| 18 | 0.31415927 | 0.30901699 | 0.95105652 | 0.3249197
  |
| 19 | 0.33161256 | 0.32556815 | 0.94551858 | 0.34432761
  |
| 20 | 0.34906585 | 0.34202014 | 0.93969262 | 0.36397023
  |
| 21 | 0.36651914 | 0.35836795 | 0.93358043 | 0.38386404
  |
| 22 | 0.38397244 | 0.37460659 | 0.92718385 | 0.40402623
  |
| 23 | 0.40142573 | 0.39073113 | 0.92050485 | 0.42447482
  |
| 24 | 0.41887902 | 0.40673664 | 0.91354546 | 0.44522869
  |
| 25 | 0.43633231 | 0.42261826 | 0.90630779 | 0.46630766
  |
| 26 | 0.45378561 | 0.43837115 | 0.89879405 | 0.48773259
  |
| 27 | 0.4712389 | 0.4539905 | 0.89100652 | 0.50952545
  |
| 28 | 0.48869219 | 0.46947156 | 0.88294759 | 0.53170943
  |
| 29 | 0.50614548 | 0.48480962 | 0.87461971 | 0.55430905
  |
| 30 | 0.52359878 | 0.5 | 0.8660254 | 0.57735027
  |
| 31 | 0.54105207 | 0.51503807 | 0.8571673 | 0.60086062
  |
| 32 | 0.55850536 | 0.52991926 | 0.8480481 | 0.62486935
  |
| 33 | 0.57595865 | 0.54463904 | 0.83867057 | 0.64940759
  |
| 34 | 0.59341195 | 0.5591929 | 0.82903757 | 0.67450852
  |
| 35 | 0.61086524 | 0.57357644 | 0.81915204 | 0.70020754
  |
| 36 | 0.62831853 | 0.58778525 | 0.80901699 | 0.72654253
  |
| 37 | 0.64577182 | 0.60181502 | 0.79863551 | 0.75355405
  |
| 38 | 0.66322512 | 0.61566148 | 0.78801075 | 0.78128563
  |
| 39 | 0.68067841 | 0.62932039 | 0.77714596 | 0.80978403
  |
| 40 | 0.6981317 | 0.64278761 | 0.76604444 | 0.83909963
  |
| 41 | 0.71558499 | 0.65605903 | 0.75470958 | 0.86928674
  |
| 42 | 0.73303829 | 0.66913061 | 0.74314483 | 0.90040404
  |
| 43 | 0.75049158 | 0.68199836 | 0.7313537 | 0.93251509
  |
| 44 | 0.76794487 | 0.69465837 | 0.7193398 | 0.96568877
  |
| 45 | 0.78539816 | 0.70710678 | 0.70710678 | 1.0
  |
| 46 | 0.80285146 | 0.7193398 | 0.69465837 | 1.03553031
  |
| 47 | 0.82030475 | 0.7313537 | 0.68199836 | 1.07236871
  |
| 48 | 0.83775804 | 0.74314483 | 0.66913061 | 1.11061251
  |
| 49 | 0.85521133 | 0.75470958 | 0.65605903 | 1.15036841
  |
| 50 | 0.87266463 | 0.76604444 | 0.64278761 | 1.19175359
  |
| 51 | 0.89011792 | 0.77714596 | 0.62932039 | 1.23489716
  |
| 52 | 0.90757121 | 0.78801075 | 0.61566148 | 1.27994163
  |
| 53 | 0.9250245 | 0.79863551 | 0.60181502 | 1.32704482
  |
| 54 | 0.9424778 | 0.80901699 | 0.58778525 | 1.37638192
  |
| 55 | 0.95993109 | 0.81915204 | 0.57357644 | 1.42814801
  |
| 56 | 0.97738438 | 0.82903757 | 0.5591929 | 1.48256097
  |
| 57 | 0.99483767 | 0.83867057 | 0.54463904 | 1.53986496
  |
| 58 | 1.01229097 | 0.8480481 | 0.52991926 | 1.60033453
  |
| 59 | 1.02974426 | 0.8571673 | 0.51503807 | 1.66427948
  |
| 60 | 1.04719755 | 0.8660254 | 0.5 | 1.73205081
  |
| 61 | 1.06465084 | 0.87461971 | 0.48480962 | 1.80404776
  |
| 62 | 1.08210414 | 0.88294759 | 0.46947156 | 1.88072647
  |
| 63 | 1.09955743 | 0.89100652 | 0.4539905 | 1.96261051
  |
| 64 | 1.11701072 | 0.89879405 | 0.43837115 | 2.05030384
  |
| 65 | 1.13446401 | 0.90630779 | 0.42261826 | 2.14450692
  |
| 66 | 1.15191731 | 0.91354546 | 0.40673664 | 2.24603677
  |
| 67 | 1.1693706 | 0.92050485 | 0.39073113 | 2.35585237
  |
| 68 | 1.18682389 | 0.92718385 | 0.37460659 | 2.47508685
  |
| 69 | 1.20427718 | 0.93358043 | 0.35836795 | 2.60508906
  |
| 70 | 1.22173048 | 0.93969262 | 0.34202014 | 2.74747742
  |
| 71 | 1.23918377 | 0.94551858 | 0.32556815 | 2.90421088
  |
| 72 | 1.25663706 | 0.95105652 | 0.30901699 | 3.07768354
  |
| 73 | 1.27409035 | 0.95630476 | 0.2923717 | 3.27085262
  |
| 74 | 1.29154365 | 0.9612617 | 0.27563736 | 3.48741444
  |
| 75 | 1.30899694 | 0.96592583 | 0.25881905 | 3.73205081
  |
| 76 | 1.32645023 | 0.97029573 | 0.2419219 | 4.01078093
  |
| 77 | 1.34390352 | 0.97437006 | 0.22495105 | 4.33147587
  |
| 78 | 1.36135682 | 0.9781476 | 0.20791169 | 4.70463011
  |
| 79 | 1.37881011 | 0.98162718 | 0.190809 | 5.14455402
  |
| 80 | 1.3962634 | 0.98480775 | 0.17364818 | 5.67128182
  |
| 81 | 1.41371669 | 0.98768834 | 0.15643447 | 6.31375151
  |
| 82 | 1.43116999 | 0.99026807 | 0.1391731 | 7.11536972
  |
| 83 | 1.44862328 | 0.99254615 | 0.12186934 | 8.14434643
  |
| 84 | 1.46607657 | 0.9945219 | 0.10452846 | 9.51436445
  |
| 85 | 1.48352986 | 0.9961947 | 0.087155743 | 11.4300523
  |
| 86 | 1.50098316 | 0.99756405 | 0.069756474 | 14.3006663
  |
| 87 | 1.51843645 | 0.99862953 | 0.052335956 | 19.0811367
  |
| 88 | 1.53588974 | 0.99939083 | 0.034899497 | 28.6362533
  |
| 89 | 1.55334303 | 0.9998477 | 0.01745240 | 57.2899616
  |
| 90 | 1.57079633 | 1.0 | 0.0 | ∞ |

## Skatīt arī

* Sinuss, kosinuss, tangenss, kotangenss, sekanss, kosekanss
* Trigonometrisko funkciju integrēšana
* Inversās trigonometriskās funkcijas
* Periodiska funkcija
* Leņķis
* Taisnleņķa trijstūris

## Ārējās saites

* Eric W. Weisstein, Trigonometric Functions, MathWorld.