Lihat sumber Daftar identitas trigonometri

{{Trigonometri}}

[[Trigonometri]] merupakan salah satu cabang matematika yang mempelajari sudut dalam segitiga siku-siku (yang dijelaskan secara geometri). Identitas trigonometri merupakan salah satu fungsi trigonometri dimana rumus tersebut memiliki hasil yang sama bila diuji suatu nilai [[Variabel (matematika)|variabel]]. Identitas berikut ini sangatlah penting dan berguna dalam komputasi yang elusif.

Daftar ini menjelaskan dasar-dasar fungsi, invers fungsi, beserta nilai sudut istimewa pada fungsi trigonometri. Dan juga mengenai jumlah dan perkalian sudut. Mengenai daftar identitas fungsi invers juga dimasukkan ke dalam halaman ini. Terdapat bukti-bukti mengenai rumus-rumus di bawah. Meski begitu, halaman ini hanya menjelaskan bukti singkat pada rumus dan adapula yang tidak. Untuk melihat bukti, lihat [[Bukti identitas trigonometri]]. Berikut adalah '''daftar identitas trigonometri'''.

== Fungsi dasar trigonometri ==
{{Main|Fungsi trigonometri}}
[[Berkas:TrigonometryTriangle.svg|jmpl|330x330px|Segitiga siku-siku <math>ABC</math> dimana <math>AC = b</math> dan <math>BC = a</math> adalah [[Kaki (geometri)|sisi segitiga]] dan <math>AB=c</math> adalah [[hipotenusa]].]]
Salah satu fungsi trigonometri paling umum, semenjak kita duduk di bangku sekolah menengah atas adalah fungsi trigonometri seperti [[Sinus (trigonometri)|sinus]], [[kosinus]], [[tangen]], [[sekan]], [[kosekan]], dan [[kotangen]]. Secara geometri, keenam fungsi trigonometri tersebut dapat didefinisikan melalui sudut pada segitiga. Misalkan <math>ABC</math> adalah [[segitiga siku-siku]], <math>a</math> dan <math>b</math> adalah sisi-sisi segitiga beserta <math>c</math> adalah [[hipotenusa]] atau sisi miring segitiga. Misalkan <math>A</math> adalah sudut yang diketahui. Maka,

: <math>\sin A = \frac{a}{c} </math>

: <math>\cos A = \frac{b}{c} </math>

: <math>\tan A = \frac{a}{b} = \frac{\sin A}{\cos A} </math>

: <math>\cot A = \frac{1}{\tan A} = \frac{\cos A}{\sin A} = \frac{b}{a} </math>

: <math>\sec A = \frac{1}{\cos A} = \frac{c}{b} </math>.

: <math>\csc A = \frac{1}{\sin A} = \frac{c}{a} </math>.

Keenam fungsi trigonometri di atas memiliki grafik, dengan ranah dan kisaran pada setiap dari mereka adalah berbeda, terutama periodenya. Berikut adalah daftar fungsi trigonometri yang ditabelkan, dengan periode, ranah, kisaran, beserta visualisasi grafik fungsi.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!Fungsi
!Periode
!Ranah
!Kisaran
!Grafik
|-
![[Sinus (trigonometri)|sinus]]
|<math>2\pi</math>
|<math>(-\infty,\infty)</math>
|<math>[-1,1]</math>
|[[Berkas:Sine one period.svg|400x400px]]
|-
![[kosinus]]
|<math>2\pi</math>
|<math>(-\infty,\infty)</math>
|<math>[-1,1]</math>
|[[Berkas:Cosine one period.svg|400x400px]]
|-
![[tangen]]
|<math>\pi</math>
|<math>x \neq n\pi</math>
|<math>(-\infty,\infty)</math>
|[[Berkas:Tangent-plot.svg|400x400px]]
|-
![[sekan]]
|<math>2\pi</math>
|<math>x \neq \frac{\pi}{2} +n\pi</math>
|<math>(-\infty,-1] \cup [1,\infty)</math>
|[[Berkas:Secant.svg|400x400px]]
|-
![[kosekan]]
|<math>2\pi</math>
|<math>x \neq \frac{\pi}{2} +n\pi</math>
|<math>(-\infty,-1] \cup [1,\infty)</math>
|[[Berkas:Cosecant.svg|400x400px]]
|-
![[kotangen]]
|<math>\pi</math>
|<math>x \neq n\pi</math>
|<math>(-\infty,\infty)</math>
|[[Berkas:Cotangent.svg|400x400px]]
|}

=== Nilai sudut istimewa ===
Berikut adalah nilai sudut istimewa pada keenam fungsi trigonometri:
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
| style="border: solid 1px darkgray;" |<div style="height:300px; overflow:auto; background:#F9F9F9;">{{nilai sudut istimewa}}</div>
|}

== Fungsi invers trigonometri ==
Fungsi invers trigonometri merupakan fungsi yang merupakan kebalikan dari fungsi dasar trigonometri. Lazimnya, fungsi invers trigonometri biasanya dinotasikan dengan prefiks '''arc'''-.<ref>{{Cite book|last=Hall|first=Arthur Graham|last2=Frink|first2=Fred Goodrich|date=[c1909]|url=http://archive.org/details/planetrigonometr00hallrich|title=Plane trigonometry|publisher=New York : Henry Holt|others=University of California Libraries}} <q>[…] <code>α = arcsin ''m''</code>: It is frequently read "arc-sine ''m''" or "anti-sine ''m''," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, <code>α = sin<sup>-1</sup>''m''</code>, is still found in English and American texts. The notation <code>α = inv sin ''m''</code> is perhaps better still on account of its general applicability. […]</q></ref> Terkadang, fungsi invers trigonometri juga dituliskan melalui notasi eksponen <math>^{-1}</math>.{{Refn|Misalnya, invers fungsi trigonometri sinus, dinotasikan <math> \arcsin(\dots) </math> atau <math> \sin ^{-1}(\dots) </math>.
.|group=nb}}

Berikut adalah fungsi invers trigonometri, dengan ranah dan kisarannya, antara lain:
{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions}}
[[Komposisi fungsi]] trigonometri dengan invers fungsinya sendiri akan sama dengan menuliskan suatu variabel. Dengan kata lain (tinjau <math>f</math> adalah fungsi),

: <math>f(f^{-1}(x)) = x</math> jika dan hanya jika <math>f^{-1}(f(x)) = x</math>

Hal yang serupa untuk fungsi trigonometri, berikut adalah fungsi yang memetakan fungsi inversnya sendiri:
{{div col|colwidth=22em}}
* <math>\sin(\arcsin(x)) = x</math>
* <math>\cos(\arccos(x)) = x</math>
* <math>\tan(\arctan(x)) = x</math>
* <math>\sec(\arcsec(x)) = x</math>
* <math>\csc(\arccsc(x)) = x</math>
* <math>\cot(\arccot(x)) = x</math>
{{div col end}}

=== Komposisi fungsi trigonometri dengan fungsi invers trigonometri lain ===
Komposisi fungsi invers untuk lebih lanjut dapat dilihat pada tabel di bawah ini.<ref>{{harvnb|Abramowitz|Stegun|1972|loc=hlm. 73, 4.3.45}}</ref><math display="block">
\begin{align}
  \sin(\arcsin x) &=x
& \cos(\arcsin x) &=\sqrt{1-x^2}
& \tan(\arcsin x) &=\frac{x}{\sqrt{1 - x^2}}
\\
  \sin(\arccos x) &=\sqrt{1-x^2}
& \cos(\arccos x) &=x
& \tan(\arccos x) &=\frac{\sqrt{1 - x^2}}{x}
\\
  \sin(\arctan x) &=\frac{x}{\sqrt{1+x^2}}
& \cos(\arctan x) &=\frac{1}{\sqrt{1+x^2}}
& \tan(\arctan x) &=x
\\
  \sin(\arccsc x) &=\frac{1}{x}
& \cos(\arccsc x) &=\frac{\sqrt{x^2 - 1}}{x}
& \tan(\arccsc x) &=\frac{1}{\sqrt{x^2 - 1}}
\\
  \sin(\arcsec x) &=\frac{\sqrt{x^2 - 1}}{x}
& \cos(\arcsec x) &=\frac{1}{x}
& \tan(\arcsec x) &=\sqrt{x^2 - 1}
\\
  \sin(\arccot x) &=\frac{1}{\sqrt{1+x^2}}
& \cos(\arccot x) &=\frac{x}{\sqrt{1+x^2}}
& \tan(\arccot x) &=\frac{1}{x}
\\
\end{align}
</math>

=== Penyelesaian terhadap persamaan trigonometri ===
Berikut adalah penyelesaian persamaan trigonometri, dengan nilai <math>\theta</math> dan <math>x</math>.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!Persamaan
! rowspan="7" |{{math|[[Logical equality|<math>\iff</math>]]}}
! colspan="6" |Penyelesaian
! rowspan="7" |untuk suatu <math>k \in \Z</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\sin \theta = x</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>(-1)^k</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\arcsin (x)</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\cos \theta = x</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>\pm\,</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\arccos (x)</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |<math>2</math>
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\tan \theta = x</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |
| style="border-style: solid none solid none; text-align: left;" |<math>\arctan (x)</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\csc \theta = x</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>(-1)^k</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\arccsc (x)</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\sec \theta = x</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>\pm\,</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\arcsec (x)</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |<math>2</math>
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\cot \theta = x</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |
| style="border-style: solid none solid none; text-align: left;" |<math>\arccot (x)</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
|}
Berikut untuk persamaan dengan kedua ruas berupa fungsi trigonometri, tinjau sudut <math>\theta</math> dan <math>\varphi</math>.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!Persamaan
! rowspan="8" |{{math|[[Logical equality|<math>\iff</math>]]}}
! colspan="6" |Penyelesaian
! rowspan="8" |untuk suatu <math>k \in \Z</math>
!Juga berlaku untuk persamaan
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\sin \theta = \sin \varphi</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>(-1)^k</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\varphi</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\csc \theta = \csc \varphi</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\cos \theta = \cos \varphi</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>\pm\,</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\varphi</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |<math>2</math>
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\sec \theta = \sec \varphi</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\tan \theta = \tan \varphi</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |
| style="border-style: solid none solid none; text-align: left;" |<math>\varphi</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\cot \theta = \cot \varphi</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>-\sin \theta = \sin \varphi</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>(-1)^{k+1}</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\varphi</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>-\csc \theta = \csc \varphi</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>-\cos \theta = \cos \varphi</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>\pm\,</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\varphi</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |<math>2</math>
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k + \pi</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>-\sec \theta = \sec \varphi</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>-\tan \theta = \tan \varphi</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>-</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\varphi</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>-\cot \theta = \cot \varphi</math>
|-
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\left| \sin \theta \right| = \left| \sin \varphi \right|</math>
| style="border-style: solid none solid none; text-align: left; padding-left: 2em;" |<math>\theta =\,</math>
| style="border-style: solid none solid none; text-align: right;" |<math>\pm</math>
| style="border-style: solid none solid none; text-align: left;" |<math>\varphi</math>
| style="border-style: solid none solid none;" |<math>+</math>
| style="border-style: solid none solid none;" |
| style="border-style: solid none solid none; padding-right: 2em;" |<math>\pi k</math>
| style="text-align: center; padding: 0.5% 2em 0.5% 2em;" |<math>\begin{align}
\left| \cos \theta \right| &= \left| \cos \varphi \right| \\
\left| \tan \theta \right| &= \left| \tan \varphi \right| \\
\left| \csc \theta \right| &= \left| \csc \varphi \right| \\
\left| \sec \theta \right| &= \left| \csc \varphi \right| \\
\left| \cot \theta \right| &= \left| \csc \varphi \right| 
\end{align}</math>
|}

== Beberapa fungsi trigonometri lainnya ==
[[Berkas:Circle-trig6.svg|jmpl|280x280px|Beberapa fungsi trigonometri lainnya.]]
Beberapa fungsi trigonometri antara lain: fungsi yang jarang digunakan seperti '''versin''', '''coversin''', '''vercosin''', '''covercosin''', '''haversin''', '''havercosin''', '''hacoversin''', '''hacovercosin''', '''exsec''', dan '''excsc'''. Tabel di bawah menunjukkan fungsi trigonometri yang jarang digunakan beserta dengan grafiknya, antara lain sebagai berikut.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
|<math>\textrm{versin} (\theta) := 2\sin^2\!\left(\frac{\theta}{2}\right) = 1 - \cos (\theta) \,</math><ref name=":1">{{Cite web|title=Abramowitz and Stegun. Page 78|url=https://personal.math.ubc.ca/~cbm/aands/page_78.htm|website=personal.math.ubc.ca|access-date=2021-12-05}}</ref>
|[[Berkas:Versin plot 2.svg|300x300px]]
|-
|<math>\textrm{coversin}(\theta) := \textrm{versin}\!\left(\frac{\pi}{2} - \theta\right) = 1 - \sin(\theta) \,</math><ref name=":1" />
|[[Berkas:Coversin plot 2.svg|300x300px]]
|-
|<math>\textrm{vercosin} (\theta) := 2\cos^2\!\left(\frac{\theta}{2}\right) = 1 + \cos (\theta) \,</math><ref name=":1" />
|[[Berkas:Vercosin plot 2.svg|300x300px]]
|-
|<math>\textrm{covercosin}(\theta) := \textrm{vercosin}\!\left(\frac{\pi}{2} - \theta\right) = 1 + \sin(\theta) \,</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Covercosine|url=https://mathworld.wolfram.com/Covercosine.html|website=mathworld.wolfram.com|language=en|access-date=2021-12-05}}</ref>
|[[Berkas:Covercosin plot 2.svg|300x300px]]
|-
|<math>\textrm{haversin}(\theta) := \frac {\textrm{versin}(\theta)} {2} = \sin^2\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos (\theta)}{2} \,</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Haversine|url=https://mathworld.wolfram.com/Haversine.html|website=mathworld.wolfram.com|language=en|access-date=2021-12-05}}</ref>
|[[Berkas:Haversin plot 2.svg|300x300px]]
|-
|<math>\textrm{hacoversin}(\theta) := \frac {\textrm{coversin}(\theta)} {2} = \frac{1 - \sin (\theta)}{2} \,</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Hacoversine|url=https://mathworld.wolfram.com/Hacoversine.html|website=mathworld.wolfram.com|language=en|access-date=2021-12-05}}</ref>
|[[Berkas:Hacoversin plot 2.svg|300x300px]]
|-
|<math>\textrm{havercosin}(\theta) := \frac {\textrm{vercosin}(\theta)} {2} = \cos^2\!\left(\frac{\theta}{2}\right) = \frac{1 + \cos (\theta)}{2} \,</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Havercosine|url=https://mathworld.wolfram.com/Havercosine.html|website=mathworld.wolfram.com|language=en|access-date=2021-12-07}}</ref>
|[[Berkas:Havercosin plot 2.svg|300x300px]]
|-
|<math>\textrm{hacovercosin}(\theta) := \frac {\textrm{covercosin}(\theta)} {2} = \frac{1 + \sin (\theta)}{2} \,</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Hacovercosine|url=https://mathworld.wolfram.com/Hacovercosine.html|website=mathworld.wolfram.com|language=en|access-date=2021-12-05}}</ref>
|[[Berkas:Hacovercosin plot 2.svg|300x300px]]
|}
Selain fungsi yang jarang digunakan, terdapat fungsi trigonometri lainnya. Berikut di antaranya: [[Tali busur (geometri)|tali busur]] disingkat '''crd''', dan '''gd''' mengindikasikan [[fungsi Gudermann]]. Masing-masing dirumuskan sebagai berikut.

: <math>\operatorname{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} =2 \sin \left(\frac{\theta }{2}\right) </math>.
: <math> \operatorname{gd} x=\int_0^x\operatorname{sech} t \, \mathrm dt
</math>.<ref name="weinstein">{{mathworld|urlname=Gudermannian|title=Gudermannian}}</ref>

== Identitas Pythagoras ==
{{Main|Identitas Pythagoras}}
[[Identitas Pythagoras]] adalah identitas trigonometri yang diturunkan dari [[teorema Pythagoras]].<ref>{{Cite web|title=Trigonometric Identities {{!}} Boundless Algebra|url=https://courses.lumenlearning.com/boundless-algebra/chapter/trigonometric-identities/|website=courses.lumenlearning.com|access-date=2021-11-26}}</ref> Dengan kata lain, identitas Pythagoras merupakan konsep teorema Pythagoras melalui fungsi trigonometri. Berikut adalah identitas Pythagoras beserta buktinya, antara lain:
{{Equation box 1|border|indent=:|title=|equation=<math>\sin^2 A + \cos^2 A = 1 </math>|cellpadding=6|border colour=#0073CF|background colour=#F5FFFA}}{{collapse top|title=Klik "tampil" 'tuk melihat bukti}}
Dengan menggunakan definisi dari fungsi sinus dan kosinus, maka
:<math>\sin^2 A + \cos^2 A = \left(\frac{b}{c}\right)^2 + \left(\frac{a}{c}\right)^2 = \frac{a^2 + b^2}{c^2}</math>
Karena berupa segitiga siku-siku, maka menurut teorema Pythagoras, <math>a^2 + b^2 = c^2</math>. Jadi, 
:<math>\sin^2 A + \cos^2 A = \frac{c^2}{c^2} = 1</math>. <math>\blacksquare</math>
{{collapse bottom}}{{Equation box 1|border|indent=:|title=|equation=<math>1 + \tan^2 A = \sec^2 A </math>|cellpadding=6|border colour=#0073CF|background colour=#F5FFFA}}{{collapse top|title=Klik "tampil" 'tuk melihat bukti}}
:<math>1 + \tan^2 A = \frac{\cos^2 A}{\cos^2 A} + \frac{\sin^2 A}{\cos^2 A} = \frac{1}{\cos^2 A} = \sec^2 A</math>. <math>\blacksquare</math>
{{collapse bottom}}{{Equation box 1|border|indent=:|title=|equation=<math>1 + \cot^2 A = \csc^2 A </math>|cellpadding=6|border colour=#0073CF|background colour=#F5FFFA}}{{collapse top|title=Klik "tampil" 'tuk melihat bukti}}
:<math>1 + \cot^2 A = \frac{\sin^2 A}{\sin^2 A} + \frac{\cos^2 A}{\sin^2 A} = \frac{1}{\sin^2 A} = \csc^2 A</math>. <math>\blacksquare</math>
{{collapse bottom}}

Dengan menggunakan ketiga identitas di atas, kita dapat menentukan identitas trigonometri lainnya. Tabel berikut menunjukkannya.<ref>Abramowitz and Stegun, hlm.&nbsp;73, 4.3.45</ref>
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!
! scope="col" |<math>\sin \theta</math>
! scope="col" |<math>\cos \theta</math>
! scope="col" |<math>\tan \theta</math>
! scope="col" |<math>\csc \theta</math>
! scope="col" |<math>\sec \theta</math>
! scope="col" |<math>\cot \theta</math>
|-
!<math>\sin \theta</math>
|<math>\sin \theta</math>
|<math>\pm\sqrt{1 - \cos^2 \theta}</math>
|<math>\pm\frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}</math>
|<math>\frac{1}{\csc \theta}</math>
|<math>\pm\frac{\sqrt{\sec^2 \theta - 1}}{\sec \theta}</math>
|<math>\pm\frac{1}{\sqrt{1 + \cot^2 \theta}}</math>
|-
!<math>\cos \theta</math>
|<math>\pm\sqrt{1 - \sin^2\theta}</math>
|<math>\cos \theta</math>
|<math>\pm\frac{1}{\sqrt{1 + \tan^2 \theta}}</math>
|<math>\pm\frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta}</math>
|<math>\frac{1}{\sec \theta}</math>
|<math>\pm\frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}</math>
|-
!<math>\tan \theta</math>
|<math>\pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}</math>
|<math>\pm\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}</math>
|<math>\tan \theta</math>
|<math>\pm\frac{1}{\sqrt{\csc^2 \theta - 1}}</math>
|<math>\pm\sqrt{\sec^2 \theta - 1}</math>
|<math>\frac{1}{\cot \theta}</math>
|-
!<math>\csc \theta</math>
|<math>\frac{1}{\sin \theta}</math>
|<math>\pm\frac{1}{\sqrt{1 - \cos^2 \theta}}</math>
|<math>\pm\frac{\sqrt{1 + \tan^2 \theta}}{\tan \theta}</math>
|<math>\csc \theta</math>
|<math>\pm\frac{\sec \theta}{\sqrt{\sec^2 \theta - 1}}</math>
|<math>\pm\sqrt{1 + \cot^2 \theta}</math>
|-
!<math>\sec \theta</math>
|<math>\pm\frac{1}{\sqrt{1 - \sin^2 \theta}}</math>
|<math>\frac{1}{\cos \theta}</math>
|<math>\pm\sqrt{1 + \tan^2 \theta}</math>
|<math>\pm\frac{\csc \theta}{\sqrt{\csc^2 \theta - 1}}</math>
|<math>\sec \theta</math>
|<math>\pm\frac{\sqrt{1 + \cot^2 \theta}}{\cot \theta}</math>
|-
!<math>\cot \theta</math>
|<math>\pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}</math>
|<math>\pm\frac{\cos \theta}{\sqrt{1 - \cos^2 \theta}}</math>
|<math>\frac{1}{\tan \theta}</math>
|<math>\pm\sqrt{\csc^2 \theta - 1}</math>
|<math>\pm\frac{1}{\sqrt{\sec^2 \theta - 1}}</math>
|<math>\cot \theta</math>
|}

== Refleksi dan putaran sudut ==
[[Berkas:Unit Circle - symmetry.svg|al=Unit circle with a swept angle theta plotted at coordinates (a,b). As the angle is reflected in increments of one-quarter pi (45 degrees), the coordinates are transformed. For a transformation of one-quarter pi (45 degrees, or 90 - theta), the coordinates are transformed to (b,a). Another increment of the angle of reflection by one-quarter pi (90 degrees total, or 180 - theta) transforms the coordinates to (-a,b). A third increment of the angle of reflection by another one-quarter pi (135 degrees total, or 270 - theta) transforms the coordinates to (-b,-a). A final increment of one-quarter pi (180 degrees total, or 360 - theta) transforms the coordinates to (a,-b).|ka|jmpl|Transformasi koordinat <math>(a,b)</math> ketika putaran sudut refleksi <math>\alpha</math> bertambah <math>\frac{\pi}{4}</math> radian.]]
Kita dapat menentukan pencerminan dan putaran sudut bila kita meneliti [[satuan lingkaran]]. Berikut adalah tabel-tabel mengenai pencerminan dan putaran sudut.

=== Refleksi sudut ===
Berikut adalah tabel-tabel mengenai pencerminan sudut. Misal <math>\alpha</math> adalah suatu sudut sembarang yang mencerminkan atau merefleksikan sudut <math>\theta</math>. Tabel berikut hanya menjelaskan refleksi <math>\theta</math> terhadap <math>\alpha</math> yang bernilaikan satuan radian, <math>0</math>, <math display="inline">\frac{\pi}{4}</math>, <math display="inline">\frac{\pi}{2}</math>, <math display="inline">\frac{3\pi}{4}</math>, dan <math>\pi</math>. Sudut dengan nilai radian <math>\pi</math> dapat kita bandingkan dengan sudut <math>0</math>

Dalam tabel yang bersubjudulkan <math>\alpha = 0</math> merupakan identitas [[fungsi ganjil dan genap]] terhadap fungsi trigonometri.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!<math>\theta</math> refleksi terhadap <math>\alpha = 0</math><ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.13–15</ref>
!<math>\theta</math> refleksi terhadap <math>\alpha = \frac{\pi}{4}</math>
!<math>\theta</math> refleksi terhadap <math>\alpha = \frac{\pi}{2}</math>
!<math>\theta</math> refleksi terhadap <math>\alpha = \frac{3\pi}{4}</math>
!<math>\theta</math> refleksi terhadap <math>\alpha = \pi</math><span style="font-weight:normal">(Bandingkan dengan <math>\alpha = 0</math>)</span>
|-
|<math>\sin(-\theta) = -\sin \theta</math>
|<math>\sin\left(\tfrac{\pi}{2} - \theta\right) =\cos \theta</math>
|<math>\sin(\pi - \theta) = +\sin \theta</math>
|<math>\sin\left(\tfrac{3\pi}{2} - \theta\right) =-\cos \theta</math>
|<math>\sin(2\pi - \theta) = -\sin(\theta) = \sin(-\theta)</math>
|-
|<math>\cos(-\theta) =+ \cos \theta</math>
|<math>\cos\left(\tfrac{\pi}{2} - \theta\right) = \sin \theta</math>
|<math>\cos(\pi - \theta) = -\cos \theta</math>
|<math>\cos\left(\tfrac{3\pi}{2} - \theta\right) = -\sin \theta</math>
|<math>\cos(2\pi - \theta) = +\cos(\theta) = \cos(-\theta)</math>
|-
|<math>\tan(-\theta) = -\tan \theta</math>
|<math>\tan\left(\tfrac{\pi}{2} - \theta\right) = \cot \theta</math>
|<math>\tan(\pi - \theta) = -\tan \theta</math>
|<math>\tan\left(\tfrac{3\pi}{2} - \theta\right) = +\cot \theta</math>
|<math>\tan(2\pi - \theta) = -\tan(\theta) = \tan(-\theta)</math>
|-
|<math>\csc(-\theta) = -\csc \theta</math>
|<math>\csc\left(\tfrac{\pi}{2} - \theta\right) = \sec \theta</math>
|<math>\csc(\pi - \theta) =+ \csc \theta</math>
|<math>\csc\left(\tfrac{3\pi}{2} - \theta\right) = -\sec \theta</math>
|<math>\csc(2\pi - \theta) = -\csc(\theta) = \csc(-\theta)</math>
|-
|<math>\sec(-\theta) = +\sec \theta</math>
|<math>\sec\left(\tfrac{\pi}{2} - \theta\right) = \csc \theta</math>
|<math>\sec(\pi - \theta) = -\sec \theta</math>
|<math>\sec\left(\tfrac{3\pi}{2} - \theta\right) = -\csc \theta</math>
|<math>\sec(2\pi - \theta) = +\sec(\theta) = \sec(-\theta)</math>
|-
|<math>\cot(-\theta) = -\cot \theta</math>
|<math>\cot\left(\tfrac{\pi}{2} - \theta\right) = \tan \theta</math>
|<math>\cot(\pi - \theta) = -\cot \theta</math>
|<math>\cot\left(\tfrac{3\pi}{2} - \theta\right) = +\tan \theta</math>
|<math>\cot(2\pi - \theta) = -\cot(\theta) = \cot(-\theta)</math>
|}
[[Berkas:Unit Circle - shifts.svg|al=Unit circle with a swept angle theta plotted at coordinates (a,b). As the swept angle is incremented by one-half pi (90 degrees), the coordinates are transformed to (-b,a). Another increment of one-half pi (180 degrees total) transforms the coordinates to (-a,-b). A final increment of one-half pi (270 degrees total) transforms the coordinates to (b,a).|ka|jmpl|Transformasi koordinat <math>(a,b)</math> ketika putaran sudut <math>\theta</math> bertambah <math>\frac{\pi}{2}</math> radian.]]

=== Putaran sudut ===
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!Putaran seperempat radian
!Putaran setengan radian
!Putaran satu radian<ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.7–9</ref>
!Periode
|-
|<math>\sin(\theta \pm \tfrac{\pi}{2}) = \pm\cos \theta</math>
|<math>\sin(\theta + \pi) = -\sin \theta</math>
|<math>\sin(\theta + k\cdot 2\pi) = +\sin \theta</math>
| style="text-align: center;" |<math>2\pi</math>
|-
|<math>\cos(\theta \pm \tfrac{\pi}{2}) = \mp\sin \theta</math>
|<math>\cos(\theta + \pi) = -\cos \theta</math>
|<math>\cos(\theta + k\cdot 2\pi) = +\cos \theta</math>
| style="text-align: center;" |<math>2\pi</math>
|-
|<math>\csc(\theta \pm \tfrac{\pi}{2}) = \pm\sec \theta</math>
|<math>\csc(\theta + \pi) = -\csc \theta</math>
|<math>\csc(\theta + k\cdot 2\pi) = +\csc \theta</math>
| style="text-align: center;" |<math>2\pi</math>
|-
|<math>\sec(\theta \pm \tfrac{\pi}{2}) = \mp\csc \theta</math>
|<math>\sec(\theta + \pi) = -\sec \theta</math>
|<math>\sec(\theta + k\cdot 2\pi) = +\sec \theta</math>
| style="text-align: center;" |<math>2\pi</math>
|-
|<math>\tan(\theta \pm \tfrac{\pi}{4}) =  \tfrac{\tan \theta \pm 1}{1\mp \tan \theta}</math>
|<math>\tan(\theta + \tfrac{\pi}{2}) = -\cot \theta</math>
|<math>\tan(\theta + k\cdot \pi) = +\tan \theta</math>
| style="text-align: center;" |<math>\pi</math>
|-
|<math>\cot(\theta \pm \tfrac{\pi}{4}) =  \tfrac{\cot \theta \mp 1}{1\pm \cot \theta}</math>
|<math>\cot(\theta + \tfrac{\pi}{2}) = -\tan\theta</math>
|<math>\cot(\theta + k\cdot \pi) = +\cot \theta</math>
| style="text-align: center;" |<math>\pi</math>
|}

== Definisi eksponensiasi ==
Untuk suatu fungsi trigonometri dasar beserta inversnya, dapat didefinisikan melalui eksponensiasi. Tabel berikut menunjukkannya.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!Fungsi
!Fungsi invers<ref>Abramowitz and Stegun, hlm.&nbsp;80, 4.4.26–31</ref>
|-
|<math>\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} </math>
|<math>\arcsin \theta = -i\, \ln \left(i\theta + \sqrt{1 - \theta^2}\right) </math>
|-
|<math>\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} </math>
|<math>\arccos \theta = -i\,\ln\left(\theta+\,\sqrt{\theta^2-1}\right) </math>
|-
|<math>\tan \theta = -i\, \frac{e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}} </math>
|<math>\arctan \theta = \frac{i}{2} \ln \left(\frac{i + \theta}{i - \theta}\right) </math>
|-
|<math>\csc \theta = \frac{2i}{e^{i\theta} - e^{-i\theta}} </math>
|<math>\arccsc \theta = -i\, \ln \left(\frac{i}{\theta} + \sqrt{1 - \frac{1}{\theta^2}}\right) </math>
|-
|<math>\sec \theta = \frac{2}{e^{i\theta} + e^{-i\theta}} </math>
|<math>\arcsec \theta = -i\, \ln \left(\frac{1}{\theta} + i \sqrt{1 - \frac{1}{\theta^2}}\right) </math>
|-
|<math>\cot \theta = i\, \frac{e^{i\theta} + e^{-i\theta}}{e^{i\theta} - e^{-i\theta}} </math>
|<math>\arccot \theta = \frac{i}{2} \ln \left(\frac{\theta - i}{\theta + i}\right) </math>
|-
!
!
|-
|[[Cis (mathematics)|<math>\operatorname{cis} \theta = e^{i\theta} </math>]]
|<math>\operatorname{arccis} \theta = -i \ln \theta </math>
|}
Disini, <math>e</math> adalah [[E (konstanta matematika)|konstanta]] dengan nilai <math>2.718281845\dots</math>, <math>i</math> adalah [[bilangan imajiner]], dan <math>\operatorname{cis}</math> merupakan fungsi trigonometri kosinus ditambahkan oleh fungsi trigonometri sinus yang dikali oleh imajiner, yaitu

: <math>\operatorname{cis} \theta = \cos \theta + i \sin \theta  </math>.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Cis|url=https://mathworld.wolfram.com/Cis.html|website=mathworld.wolfram.com|language=en|access-date=2021-11-29}}</ref><ref>{{Cite web|title=Mathwords: Cis|url=http://www.mathwords.com/c/cis.htm|website=www.mathwords.com|access-date=2021-11-29}}</ref>

Pada tabel terakhir, baris awal dan kolom akhir, tepat di bawah kiri sel, rumus tersebut disebut juga sebagai [[rumus Euler]].

== Jumlah dan selisih sudut ==
Jumlah sudut dimana ketika suatu fungsi trigonometri dengan variabel merupakan jumlah sudut-sudut. Sebagai permisalan, diberikan <math>\alpha</math> dan <math>\beta</math> adalah sudut sembarang, kita rumuskan untuk suatu fungsi trigonometri. Berikut di antaranya,<ref>{{Cite web|date=2015-10-31|title=7.2: Sum and Difference Identities|url=https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(OpenStax)/07%3A_Trigonometric_Identities_and_Equations/7.02%3A_Sum_and_Difference_Identities|website=Mathematics LibreTexts|language=en|access-date=2021-12-02}}</ref>
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!Sinus
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\sin(\alpha \pm \beta)</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\sin \alpha \cos \beta \pm \cos \alpha \sin \beta</math><ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.16</ref><ref name="mathworld_addition">{{MathWorld|title=Trigonometric Addition Formulas|urlname=TrigonometricAdditionFormulas}}</ref>
|-
!Kosinus
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\cos(\alpha \pm \beta)</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\cos \alpha \cos \beta \mp \sin \alpha \sin \beta</math><ref name="mathworld_addition" /><ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.17</ref>
|-
!Tangen
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\tan(\alpha \pm \beta)</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}</math><ref name="mathworld_addition" /><ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.18</ref>
|-
!Kosekan
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\csc(\alpha \pm \beta)</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\frac{\sec \alpha \sec \beta \csc \alpha \csc \beta}{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta}</math><ref name=":0">{{Cite web|title=Angle Sum and Difference Identities|url=http://www.milefoot.com/math/trig/22anglesumidentities.htm|website=www.milefoot.com|access-date=2019-10-12}}</ref>
|-
!Sekan
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\sec(\alpha \pm \beta)</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\frac{\sec \alpha \sec \beta \csc \alpha \csc \beta}{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta}</math><ref name=":0" />
|-
!Kotangen
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\cot(\alpha \pm \beta)</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\frac{\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha}</math><ref name="mathworld_addition" /><ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.19</ref>
|-
!Invers sinus
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\arcsin x \pm \arcsin y</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\arcsin\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2}\right)</math><ref>Abramowitz and Stegun, hlm.&nbsp;80, 4.4.32</ref>
|-
!Invers kosinus
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\arccos x \pm \arccos y</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\arccos\left(xy \mp \sqrt{\left(1-x^2\right)\left(1-y^2\right)}\right)</math><ref>Abramowitz and Stegun, hlm.&nbsp;80, 4.4.33</ref>
|-
!Invers tangen
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\arctan x \pm \arctan y</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\arctan\left(\frac{x \pm y}{1 \mp xy}\right)</math><ref>Abramowitz and Stegun, hlm.&nbsp;80, 4.4.34</ref>
|-
!Invers kotangen
| colspan="3" style="border-style: solid none solid solid; text-align: right;" |<math>\arccot x \pm \arccot y</math>
| style="border-style: solid none solid none; text-align: center;" |<math>=</math>
| style="border-style: solid solid solid none; text-align: left;" |<math>\arccot\left(\frac{xy \mp 1}{y \pm x}\right)</math>
|}
Jumlah dan selisih sudut sekan juga dirumuskan sebagai

:<math>\sec(\alpha \pm \beta) = \frac{\sec \alpha \sec \beta}{1 \mp \tan \alpha \tan \beta}</math>.

== Sudut rangkap ==
{{Sidebox|above==== Sudut dua rangkap ===
:<math>\sin (2\theta) = 2 \sin \theta \cos \theta = \frac{2\tan \theta}{1 + \tan^2 \theta}</math>
:<math>\cos (2\theta) = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta = \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta}</math>
:<math>\tan (2\theta) = \frac{2 \tan \theta} {1 - \tan^2 \theta}</math>
:<math>\cot (2\theta) = \frac{\cot^2 \theta - 1}{2 \cot \theta}</math>
:<math>\sec (2\theta) = \frac{\sec^2 \theta}{2 - \sec^2 \theta}</math>
:<math>\csc (2\theta) = \frac{\sec \theta \csc \theta}{2}</math>
=== Sudut tiga rangkap ===
:<math>\sin (3\theta) =3\sin\theta - 4\sin^3\theta = 4\sin\theta\sin\left(\frac{\pi}{3} -\theta\right)\sin\left(\frac{\pi}{3} + \theta\right)</math>
:<math>\cos (3\theta) = 4 \cos^3\theta - 3 \cos\theta =4\cos\theta\cos\left(\frac{\pi}{3} -\theta\right)\cos\left(\frac{\pi}{3} + \theta\right)</math>
:<math>\tan (3\theta) = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta} = \tan \theta\tan\left(\frac{\pi}{3} - \theta\right)\tan\left(\frac{\pi}{3} + \theta\right)</math>
:<math>\cot (3\theta) = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta}</math>
:<math>\sec (3\theta) = \frac{\sec^3\theta}{4-3\sec^2\theta}</math>
:<math>\csc (3\theta) = \frac{\csc^3\theta}{3\csc^2\theta-4}</math>}}Sudut rangkap merupakan sudut yang dimana suatu variabel yang sama ditambahkan oleh variabel tersendiri. Sudut rangkap dapat dibuktikan melalui sifat jumlah sudut. Sebagai contoh, ketika kita ingin mencari <math>\sin 2x</math>, maka kita gunakan rumus jumlah sudut untuk memperoleh rumus sudut sinus dua rangkap ini.

: <math>\begin{aligned}
\sin 2\theta &= \sin({\color{red}{\theta}} + {\color{green}{\theta}}) \\
 &= \sin {\color{red}{\theta}} \cos {\color{green}{\theta}} + \sin {\color{green}{\theta}} \cos {\color{red}{\theta}} \\
 &= 2 \sin \theta \cos \theta
\end{aligned}</math>

Rumus jumlah sudut tersebut juga kita pakai untuk mencari sudut rangkap tiga. Andaikan kita diminta untuk mencari <math>\sin 3x</math>, maka dengan menggunakan rumus jumlah sudut.

: <math>\begin{aligned}
\sin 3\theta &= \sin(2\theta + \theta) \\
&= \sin 2\theta \cos \theta + \sin \theta \cos 2\theta \\
&= 2 \sin \theta \cos^2 \theta + \sin \theta (1 - 2\sin^2 \theta) \\
&= 2 \sin \theta (1 - \sin^2 \theta) + \sin \theta - 2 \sin^3 \theta \\
&= 2 \sin \theta - 2 \sin^3 \theta + \sin \theta - 2 \sin^3 \theta \\
&= 3 \sin \theta - 4 \sin^3 \theta
\end{aligned}</math>

Dengan cara yang serupa, kita dapat mencari rumus untuk fungsi trigonometri sudut rangkap lainnya, seperti kosinus, tangen, kotangen, sekan, serta dengan kosekan.

Kita telah memperoleh rumus sudut rangkap dua dan sudut rangkap tiga (pada kotak di samping), maka kita beralih ke sudut <math>n</math>-rangkap, dimana <math>n = 1,2,3\dots</math>. Dengan kata lain, rumus sudut <math>n</math>-rangkap dapat kita pakai untuk nilai <math>n</math> sembarang. Sebagai contoh, ketika <math>n = 2</math>, maka kita memperoleh sudut dua rangkap dan begitu pula seterusnya.

Tanpa basa-basi, berikut adalah rumus sudut <math>n</math>-rangkap.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Multiple-Angle Formulas|url=https://mathworld.wolfram.com/Multiple-AngleFormulas.html|website=mathworld.wolfram.com|language=en|access-date=2021-11-29}}</ref>

: <math>\sin(nx) = \sum_{k=0}^n \binom{n}{k}\cos^k x \sin^{n-k} x \sin \left(\frac{\pi}{2}(n-k)\right)</math><math>\cos(nx) = \sum_{k=0}^n \binom{n}{k}\cos^k x \sin^{n-k} x \cos \left(\frac{\pi}{2}(n-k)\right)</math>
=== Metode Chebyshev ===
[[Metode Chebyshev]] adalah [[Algoritma|algoritme]] rekursif yang mencari rumus sudut <math>n</math>-rangkap dengan diketahui nilai ke-<math>(n-1)</math> dan ke-<math>(n-2)</math>. Metode Chebyshev dapat dirumuskan untuk sudut rangkap fungsi sinus dan kosinus.<ref>{{Cite web|title=Cosine, Sine and Tangent of Multiple Angles (Recursive Formula)|url=https://trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm#Recursive_Formula|website=trans4mind.com|access-date=2021-12-02}}</ref>

: <math>\begin{aligned}
\sin(nx) &= 2 \cos x \sin((n-1)x) - \sin((n-2)x) \\
\cos(nx) &= 2 \cos x \cos((n-1)x) - \cos((n-2)x) \\
\tan(nx) &= \frac{\tan((n-1)x) + \tan x}{1 - \tan((n-1)x)\tan x}
\end{aligned}</math>

=== Sudut setengah rangkap ===
Berikut adalah sudut setengah rangkap, antara lain<ref name="ReferenceA">Abramowitz and Stegun, hlm.&nbsp;72, 4.3.20–22</ref><ref name="mathworld_half_angle">{{MathWorld|title=Half-Angle Formulas|urlname=Half-AngleFormulas}}</ref>

: <math>\begin{align}
   \sin \frac{\theta}{2} &= \pm \sqrt{\frac{1 - \cos \theta}{2}} \\[3pt]
   \cos \frac{\theta}{2} &= \pm \sqrt{\frac{1 + \cos\theta}{2}} \\[3pt]
   \tan \frac{\theta}{2} &= \csc \theta - \cot \theta = \pm\, \sqrt\frac{1 - \cos \theta}{1 + \cos \theta} = \frac{\sin \theta}{1 + \cos \theta} \\[3pt]
                         &= \frac{1 - \cos \theta}{\sin \theta} = \frac{-1 \pm \sqrt{1+\tan^2\theta}}{\tan\theta} = \frac{\tan\theta}{1 + \sec{\theta}} \\[3pt]
   \cot \frac{\theta}{2} &= \csc \theta + \cot \theta = \pm\, \sqrt\frac{1 + \cos \theta}{1 - \cos \theta} = \frac{\sin \theta}{1 - \cos \theta} = \frac{1 + \cos \theta}{\sin \theta}
\end{align}</math>

Masih terdapat rumus-rumus lainnya berkaitan dengan sudut setengah rangkap. Berikut di antaranya:

: <math>\begin{align}
  \tan\frac{\eta\pm\theta}{2} &= \frac{\sin\eta \pm \sin\theta}{\cos\eta + \cos\theta} \\[3pt]
  \tan\left(\frac{\theta}{2} + \frac{\pi}{4}\right) &= \sec\theta + \tan\theta \\[3pt]
       \sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}} &= \frac{\left|1 - \tan\frac{\theta}{2}\right|}{\left|1 + \tan\frac{\theta}{2}\right|} 
\end{align}</math>

== Penjumlahan dan perkalian fungsi trigonometri ==
{| class="wikitable" style="float:right; margin-left:1em; text-align:center; font-size:90%"
!Perkalian ke penjumlahan dan penjumlahan ke perkalian<ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.31–33</ref><ref>Abramowitz and Stegun, hlm.&nbsp;72, 4.3.34–39</ref>
|-
|<math>2\cos \theta \cos \varphi = {{\cos(\theta - \varphi) + \cos(\theta + \varphi)}}</math>
|-
|<math>2\sin \theta \sin \varphi = {{\cos(\theta - \varphi) - \cos(\theta + \varphi)} }</math>
|-
|<math>2\sin \theta \cos \varphi = {{\sin(\theta + \varphi) + \sin(\theta - \varphi)} }</math>
|-
|<math>2\cos \theta \sin \varphi = {{\sin(\theta + \varphi) - \sin(\theta - \varphi)} }</math>
|-
|<math>\tan \theta \tan \varphi =\frac{\cos(\theta-\varphi)-\cos(\theta+\varphi)}{\cos(\theta-\varphi)+\cos(\theta+\varphi)}</math>
|-
|<math>\begin{align} \prod_{k=1}^n \cos \theta_k & = \frac{1}{2^n}\sum_{e\in S} \cos(e_1\theta_1+\cdots+e_n\theta_n) \\[6pt]
& \text{dimana }S=\{1,-1\}^n
\end{align}
</math>
|-
|<math>
\prod_{k=1}^n \sin(\theta_k)=\frac{(-1)^{\lfloor\frac
        {n}{2}\rfloor}}{2^n}\begin{cases}
\sum_{e\in S}\cos(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j \;\text{jika}\; n\; \text{ genap},\\
\sum_{e\in S}\sin(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j \;\text{jika}\; n\; \text{ ganjil}
\end{cases}
</math>
|-
|<math>\sin \theta \pm \sin \varphi = 2 \sin\left( \frac{\theta \pm \varphi}{2} \right) \cos\left( \frac{\theta \mp \varphi}{2} \right)</math>
|-
|<math>\cos \theta + \cos \varphi = 2 \cos\left( \frac{\theta + \varphi} {2} \right) \cos\left( \frac{\theta - \varphi}{2} \right)</math>
|-
|<math>\cos \theta - \cos \varphi = -2\sin\left( \frac{\theta + \varphi}{2}\right) \sin\left(\frac{\theta - \varphi}{2}\right)</math>
|}
Suatu penjumlahan fungsi trigonometri dapat dikonversikan menjadi perkalian fungsi trigonometri. Sebaliknya, perkalian fungsi trigonometri juga dapat dikonversikan menjadi penjumlahan fungsi trigonometri. Tabel berikut menunjukkan perkalian ke penjumlahan suatu fungsi trigonometri dan begitu juga dengan penjumlahan ke perkalian suatu fungsi trigonometri.

== Kalkulus ==

=== Limit ===
Contoh limit fungsi trigonometri yang paling sering digunakan adalah

: <math>\lim_{x \to 0} \frac{\sin x}{x} = 1</math>

Kita dapat membuktikan contoh pertama dengan menggunakan [[satuan lingkaran]] dan [[teorema apit]]. Terdapat contoh limit yang juga paling sering dipakai,

: <math>\lim_{x \to 0} \frac{1 - \cos x}{x} = 0</math>

Limit tersebut dapat dibuktikan melalui fungsi trigonometri tangen rangkap setengah. Untuk limit fungsi trigonometri lainnya, berikut adalah limit fungsi trigonometri beserta dengan pembuktiannya.

* <math>\lim_{x \to 0} \frac{\sin ax}{bx} = \lim_{x \to 0} \frac{ax}{\sin bx} = \lim_{x \to 0} \frac{\sin ax}{\sin bx} = \frac{a}{b}</math>{{Refn|Sifat berikut juga memiliki beragam limit fungsi trigonometri yang sama dengan <math>\frac{a}{b}</math>. Limit fungsi di antaranya ialah <math>\lim_{x \to 0} \frac{\tan ax}{bx}</math>, <math>\lim_{x \to 0} \frac{ax}{\tan bx} </math>, dan <math>\lim_{x \to 0} \frac{\tan ax}{\tan bx}</math>. Beberapa limit fungsi trigonometri ini serupa juga dengan <math>\lim_{x \to 0} \frac{\sin ax}{\tan bx} = \lim_{x \to 0} \frac{\tan ax}{\sin bx}</math>.|group=nb}}

=== [[Turunan]] dan [[antiturunan]] ===
Untuk suatu fungsi trigonometri, terdapat [[turunan]] dan [[antiturunan]]. Walakin, halaman ini hanya menjelaskan turunan dan antiturunan terhadap fungsi trigonometri yang bersifat dasar beserta fungsi inversnya. Untuk mengenai antiturunan fungsi trigonometri lainnya, lihat [[Daftar integral dari fungsi trigonometri]] dan [[Daftar integral dari fungsi invers trigonometri]]. Tabel berikut ini menunjukkan turunan dan antiturunan, antara lain:
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
!Turunan
!Integral
|-
|<math>\frac{\mathrm d}{\mathrm dx} \sin x = \cos x </math>
|<math>\int \sin x \, \mathrm dx = -\cos x + C</math>
|-
|<math>\frac{\mathrm d}{\mathrm dx} \cos x = -\sin x</math>
|<math>\int \cos \, \mathrm dx = \sin x + C</math>
|-
|<math>\frac{\mathrm d}{\mathrm dx} \tan x = \sec^2 x</math>
|<math>\int \tan x \, \mathrm dx = -\ln \left|\cos x \right| + C</math>
|-
|<math>\frac{\mathrm d}{\mathrm dx} \csc x = -\cot x \csc x</math>
|<math>\int \csc x \, \mathrm dx = -\ln \left|\csc x + \cot x \right| + C</math>
|-
|<math>\frac{\mathrm d}{\mathrm dx} \sec x = \tan x \sec x</math>
|<math>\int \sec x \, \mathrm dx = \ln \left|\sec x + \tan x \right| + C</math>
|-
|<math>\frac{\mathrm d}{\mathrm dx} \cot x = -\csc^2 x</math>
|<math>\int \cot x \, \mathrm dx = \ln \left|\sin x \right| + C</math>
|}

== Representasi deret ==
Suatu fungsi trigonometri dapat dikonversikan sebagai deret, dimana bentuk tersebut merupakan representasinya. Deret tersebut dapat merupakan representasi dari [[deret Maclaurin]] datau [[deret Laurent]]. Keterangan mengenai rumus-rumus di bawah, <math>B_n</math> adalah [[bilangan Bernoulli]] dan <math>E_n</math> adalah [[bilangan Euler]].
{{div col|colwidth=30em}}
* <math>\sin x = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^{2n-1}}{(2n-1)!}</math>
* <math>\cos x = \sum_{n=1}^\infty \frac{(-1)^{n} x^{2n}}{(2n)!}</math>
* <math>\tan x = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Tangent|url=https://mathworld.wolfram.com/Tangent.html#eqn31|website=mathworld.wolfram.com|language=en|access-date=2021-12-07}}</ref>
* <math>\csc x = \sum_{n=0}^\infty \frac{(-1)^{n+1} 2^{2n} (2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Cosecant|url=https://mathworld.wolfram.com/Cosecant.html#eqn7|website=mathworld.wolfram.com|language=en|access-date=2021-12-07}}</ref>
* <math>\sec x = \sum_{n=0}^\infty \frac{(-1)^n E_{2n} x^{2n-1}}{(2n)!} </math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Secant|url=https://mathworld.wolfram.com/Secant.html#eqn7|website=mathworld.wolfram.com|language=en|access-date=2021-12-07}}</ref>
* <math>\cot x = \sum_{n=0}^\infty \frac{(-1)^{n-1} 2^{2n} x^{2n-1}}{(2n)!}</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Cotangent|url=https://mathworld.wolfram.com/Cotangent.html#eqn17|website=mathworld.wolfram.com|language=en|access-date=2021-12-07}}</ref>
{{div col end}}

== Lihat pula ==

* [[Bukti identitas trigonometri]]
* [[Daftar integral dari fungsi trigonometri]]
** [[Daftar integral dari fungsi invers trigonometri]]
* [[Fungsi trigonometri]]
* [[Trigonometri]]

== Catatan, rujukan, dan bibliografi ==

=== Catatan ===
<references group="nb" />

=== Rujukan ===
{{Reflist|30em}}

=== Bibliografi ===

* {{Cite book|year=1972|url=https://archive.org/details/handbookofmathe000abra|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables|location=New York|publisher=[[Dover Publications]]|isbn=978-0-486-61272-0|editor1-last=Abramowitz|editor1-first=Milton|editor1-link=Milton Abramowitz|editor2-last=Stegun|editor2-first=Irene A.|editor2-link=Irene Stegun}}