Editing Trigonometric functions (section)

== Right-angled triangle definitions ==
[[File:TrigonometryTriangle.svg|thumb|In this right triangle, denoting the measure of angle BAC as A: {{math|1=sin ''A'' = {{sfrac|''a''|''c''}}}}; {{math|1=cos ''A'' = {{sfrac|''b''|''c''}}}}; {{math|1=tan ''A'' = {{sfrac|''a''|''b''}}}}.]]
[[File:TrigFunctionDiagram.svg|thumb|Plot of the six trigonometric functions, the unit circle, and a line for the angle {{math|1=''θ'' = 0.7 radians}}. The points labeled {{color|#D00|1}}, {{color|#02D|Sec(''θ'')}}, {{color|#0D1|Csc(''θ'')}} represent the length of the line segment from the origin to that point. {{color|#D00|Sin(''θ'')}}, {{color|#02D|Tan(''θ'')}}, and {{color|#0D1|1}} are the heights to the line starting from the {{mvar|x}}-axis, while {{color|#D00|Cos(''θ'')}}, {{color|#02D|1}}, and {{color|#0D1|Cot(''θ'')}} are lengths along the {{mvar|x}}-axis starting from the origin.]]

If the acute angle {{mvar|θ}} is given, then any right triangles that have an angle of {{mvar|θ}} are [[similarity (geometry)|similar]] to each other. This means that the ratio of any two side lengths depends only on {{mvar|θ}}. Thus these six ratios define six functions of {{mvar|θ}}, which are the trigonometric functions. In the following definitions, the [[hypotenuse]] is the length of the side opposite the right angle, ''opposite'' represents the side opposite the given angle {{mvar|θ}}, and ''adjacent'' represents the side between the angle {{mvar|θ}} and the right angle.<ref>{{harvtxt|Protter|Morrey|1970|pp=APP-2, APP-3}}</ref><ref>{{Cite web|title=Sine, Cosine, Tangent|url=https://www.mathsisfun.com/sine-cosine-tangent.html|access-date=29 August 2020|website=www.mathsisfun.com}}</ref>
{|
| style="padding-left: 2em; padding-right: 2em; |
;sine: <math>\sin \theta = \frac \mathrm{opposite}\mathrm{hypotenuse}</math>
| style="padding-left: 2em; padding-right: 2em; |
;cosecant: <math>\csc \theta = \frac \mathrm{hypotenuse}\mathrm{opposite}</math>
|-
| style="padding-left: 2em; padding-right: 2em; |
;cosine: <math>\cos \theta = \frac \mathrm{adjacent}\mathrm{hypotenuse}</math>
| style="padding-left: 2em; padding-right: 2em; |
;secant: <math>\sec \theta = \frac \mathrm{hypotenuse}\mathrm{adjacent}</math>
|-
| style="padding-left: 2em; padding-right: 2em; |
;tangent: <math>\tan \theta = \frac \mathrm{opposite}\mathrm{adjacent}</math>
| style="padding-left: 2em; padding-right: 2em; |
;cotangent: <math>\cot \theta = \frac \mathrm{adjacent}\mathrm{opposite}</math>
|}
[[mnemonics in trigonometry|Various mnemonics]] can be used to remember these definitions.

In a right-angled triangle, the sum of the two acute angles is a right angle, that is, {{math|90°}} or {{math|{{sfrac|π|2}} [[radian]]s}}. Therefore <math>\sin(\theta)</math> and <math>\cos(90^\circ - \theta)</math> represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.

[[File:Periodic sine.svg|thumb|'''Top:''' Trigonometric function {{math|sin ''θ''}} for selected angles {{math|''θ''}}, {{math|{{pi}} − ''θ''}}, {{math|{{pi}} + ''θ''}}, and {{math|2{{pi}} − ''θ''}} in the four quadrants.<br>'''Bottom:''' Graph of sine versus angle. Angles from the top panel are identified.]]

{| class="wikitable sortable"
|+ Summary of relationships between trigonometric functions<ref>{{harvtxt|Protter|Morrey|1970|p=APP-7}}</ref>
|-
! rowspan=2 | Function
! rowspan=2 | Description
! colspan=2 | [[List of trigonometric identities|Relationship]]
|-
! using [[radian]]s
! using [[Degree (angle)|degree]]s
|-
! sine
|align=center|{{math|{{sfrac|opposite|hypotenuse}}}}
| <math>\sin \theta = \cos\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\csc \theta}</math>
| <math>\sin x = \cos\left(90^\circ - x \right) = \frac{1}{\csc x}</math>
|-
! cosine
|align=center|{{math|{{sfrac|adjacent|hypotenuse}}}}
| <math>\cos \theta = \sin\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sec \theta}\,</math>
| <math>\cos x = \sin\left(90^\circ - x \right) = \frac{1}{\sec x}\,</math>
|-
! tangent
|align=center|{{math|{{sfrac|opposite|adjacent}}}}
| <math>\tan \theta = \frac{\sin \theta}{\cos \theta} = \cot\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cot \theta} </math>
| <math>\tan x = \frac{\sin x}{\cos x} = \cot\left(90^\circ - x \right) = \frac{1}{\cot x} </math>
|-
! cotangent
|align=center|{{math|{{sfrac|adjacent|opposite}}}}
| <math>\cot \theta = \frac{\cos \theta}{\sin \theta} = \tan\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\tan \theta} </math>
| <math>\cot x = \frac{\cos x}{\sin x} = \tan\left(90^\circ - x \right) = \frac{1}{\tan x} </math>
|-
! secant
|align=center|{{math|{{sfrac|hypotenuse|adjacent}}}}
| <math>\sec \theta = \csc\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\cos \theta} </math>
| <math>\sec x = \csc\left(90^\circ - x \right) = \frac{1}{\cos x} </math>
|-
! cosecant
|align=center|{{math|{{sfrac|hypotenuse|opposite}}}}
| <math>\csc \theta = \sec\left(\frac{\pi}{2} - \theta \right) = \frac{1}{\sin \theta} </math>
| <math>\csc x = \sec\left(90^\circ - x \right) = \frac{1}{\sin x} </math>
|}