(20 intermediate revisions by 5 users not shown)
Line 1: Line 1:
 +
[[Category:Formulas]]
 +
 +
keywords: double-angle, triple-angle, angle sum
 +
 +
<center><font size= 4>
 +
'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
 +
</font size>
 +
 +
'''Trigonometric Identities'''
 +
 +
click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
 +
 +
</center>
 +
 +
----
 +
 +
 
{|
 
{|
 
|-
 
|-
Line 6: Line 23:
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |Definition of tangent  
 
| align="right" style="padding-right: 1em;" |Definition of tangent  
−
|<math>  \tan \theta = \frac{\sin \theta}{\cos\theta} </math> [[User:Kumar51formulas|credit]]
+
|<math>  \tan \theta = \frac{\sin \theta}{\cos\theta} </math>  
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | Definition of cotangent
 
| align="right" style="padding-right: 1em;" | Definition of cotangent
−
| <math>  \cot \theta = \frac{\cos \theta}{\sin\theta} \ </math>  [[User:Kumar51formulas|credit]]
+
| <math>  \cot \theta = \frac{\cos \theta}{\sin\theta} \ </math>   
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | Definition of secant
 
| align="right" style="padding-right: 1em;" | Definition of secant
Line 18: Line 35:
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | Definition of versed sine (versine)
 
| align="right" style="padding-right: 1em;" | Definition of versed sine (versine)
−
| <math>\text{ver } \theta = 1- \cos \theta \ </math>
+
| <math>\text{versin } \theta = 1- \cos \theta \ </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | Definition of versed cosine (versine)
 
| align="right" style="padding-right: 1em;" | Definition of versed cosine (versine)
−
| <math>\text{vercosine } \theta = 1+ \cos \theta \ </math>
+
| <math>\text{vercosin } \theta = 1+ \cos \theta \ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | please continue
+
| align="right" style="padding-right: 1em;" | Definition of coversed sine (coversine)
−
| place formula here
+
| <math>\text{coversin } \theta = \text{cvs } \theta = 1- \sin \theta \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of coversed cosine (covercosine)
 +
| <math>\text{covercosin } \theta = 1+ \sin \theta \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of haversed sine (haversine)
 +
| <math>\text{haversin } \theta = \frac{1- \cos \theta}{2}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of haversed cosine (havercosine)
 +
| <math>\text{havercosin } \theta = \frac{1+ \cos \theta}{2}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of hacoversed sine (hacoversin)
 +
| <math>\text{hacoversin } \theta = \frac{1 - \sin \theta}{2}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of hacoversed cosine (hacovercosin)
 +
| <math>\text{hacovercosin } \theta = \frac{1 + \sin \theta}{2}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of exterior secant (exsec)
 +
| <math>\text{exsec } \theta = \sec \theta - 1 \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of exterior cosecant (excosec)
 +
| <math>\text{excosec } \theta = \csc \theta - 1 \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Definition of chord (crd)
 +
| <math>\text{crd } \theta = 2 \sin(\frac{\theta}{2})</math>
 
|-
 
|-
 
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Pythagorean identity and other related identities
 
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Pythagorean identity and other related identities
Line 42: Line 83:
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
 
| <math>\csc^2 \theta = 1+\cot^2 \theta \ </math>
 
| <math>\csc^2 \theta = 1+\cot^2 \theta \ </math>
−
|-
 
−
| align="right" style="padding-right: 1em;" | write name here
 
−
| place formula here
 
 
|-
 
|-
 
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Half-Angle Formulas  
 
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Half-Angle Formulas  
Line 51: Line 89:
 
|<math>  \sin \frac{\theta}{2} = \pm \sqrt{ \frac{1-\cos  \theta}{2} } \ </math>  
 
|<math>  \sin \frac{\theta}{2} = \pm \sqrt{ \frac{1-\cos  \theta}{2} } \ </math>  
 
|-
 
|-
−
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Angle sum and differences identities
+
| align="right" style="padding-right: 1em;" |Half-angle for cosine
 +
|<math>  \cos \frac{\theta}{2} = \pm \sqrt{ \frac{1+\cos  \theta}{2} } \ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" |Angle sum for sine
+
| align="right" style="padding-right: 1em;" |Half-angle for tangent
 +
|<math>  \tan \frac{\theta}{2} = \csc \theta - \cot \theta \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |Half-angle for tangent
 +
|<math>  \tan \frac{\theta}{2} =\pm\sqrt{\frac{1-\cos \theta}{ 1+\cos \theta }} \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |Half-angle for tangent
 +
|<math>  \tan \frac{\theta}{2} =\frac{\sin  \theta}{ 1+\cos \theta } \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |Half-angle for tangent
 +
|<math>  \tan \frac{\theta}{2} =\frac{1-\cos  \theta}{ \sin \theta } \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Half-angle for cotangent
 +
|  <math>\cot \frac{\theta}{2} = \csc \theta + \cot \theta </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Half-angle for cotangent
 +
|  <math>\cot \frac{\theta}{2} = \frac{1 + \cos \theta}{\sin \theta} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Half-angle for cotangent
 +
|  <math>\cot \frac{\theta}{2} = \pm \sqrt{1 + \cos \theta \over 1 - \cos \theta} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Half-angle for cotangent
 +
|  <math>\cot \frac{\theta}{2} = \frac{\sin \theta}{1 - \cos \theta}  </math>
 +
|-
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Double-Angle Formulas
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for sine
 +
| <math>\sin 2 \theta = 2 \sin \theta \cos \theta \  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for sine
 +
|  <math>\sin 2 \theta = \frac{ 2 \tan \theta}{1+  \tan^2 \theta } \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for cosine
 +
|  <math>\cos 2 \theta =\cos^2 \theta - \sin^2 \theta \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for cosine
 +
|  <math>\cos 2 \theta =2 \cos^2 \theta - 1 \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for cosine
 +
|  <math>\cos 2 \theta =1- 2 \sin^2 \theta  \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for cosine
 +
|  <math>\cos 2 \theta =\frac{1-  \tan^2 \theta}{ 1+\tan^2 \theta }  \ </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for tangent
 +
| <math>\tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta}\, </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | double-angle for cotangent
 +
| <math>\cot 2\theta = \frac{\cot^2 \theta - 1}{2 \cot \theta}\,</math>
 +
|-
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Triple-Angle Formulas
 +
|-
 +
| align="right" style="padding-right: 1em;" | triple-angle for sine
 +
|<math>\begin{align}\sin 3\theta & = 3 \cos^2\theta \sin\theta - \sin^3\theta \\
 +
& = 3\sin\theta - 4\sin^3\theta \end{align}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | triple-angle for cosine
 +
|<math>\begin{align}\cos 3\theta & = \cos^3\theta - 3 \sin^2 \theta\cos \theta \\
 +
& = 4 \cos^3\theta - 3 \cos\theta\end{align}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | triple-angle for tangent
 +
|<math>\tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | tripe-angle for cotangent
 +
|<math>\cot 3\theta = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta}</math>
 +
|-
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Angle sum and difference identities
 +
|-
 +
| align="right" style="padding-right: 1em;" |Sine
 
|<math>  \sin \left( \theta\pm \alpha \right)=\sin \theta \cos \alpha \pm \cos \theta \sin \alpha</math>   
 
|<math>  \sin \left( \theta\pm \alpha \right)=\sin \theta \cos \alpha \pm \cos \theta \sin \alpha</math>   
 +
|-
 +
| align="right" style="padding-right: 1em;" | Cosine
 +
|<math> \cos \left(\theta\pm \alpha \right)= \cos \theta \cos \alpha \mp \sin \theta \sin \alpha</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Tangent
 +
|<math> \tan \left(\theta\pm \alpha \right)= \frac {\tan \theta \pm \tan \alpha}{1 \mp \tan \theta \tan \alpha}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Arcsine
 +
| <math>\arcsin\alpha \pm \arcsin\beta = \arcsin(\alpha\sqrt{1-\beta^2} \pm \beta\sqrt{1-\alpha^2})</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Arccosine
 +
| <math>\arccos\alpha \pm \arccos\beta = \arccos(\alpha\beta \mp \sqrt{(1-\alpha^2)(1-\beta^2)})</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | Arctangent
 +
| <math>\arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right)</math>
 +
|-
 
|}
 
|}
  

Latest revision as of 13:28, 26 February 2015


keywords: double-angle, triple-angle, angle sum

Collective Table of Formulas

Trigonometric Identities

click here for more formulas


Trigonometric Identities
Basic Definitions
Definition of tangent $ \tan \theta = \frac{\sin \theta}{\cos\theta} $
Definition of cotangent $ \cot \theta = \frac{\cos \theta}{\sin\theta} \ $
Definition of secant $ \sec \theta = \frac{1}{\cos \theta} \ $
Definition of cosecant $ \csc \theta = \frac{1}{\sin \theta} \ $
Definition of versed sine (versine) $ \text{versin } \theta = 1- \cos \theta \ $
Definition of versed cosine (versine) $ \text{vercosin } \theta = 1+ \cos \theta \ $
Definition of coversed sine (coversine) $ \text{coversin } \theta = \text{cvs } \theta = 1- \sin \theta \ $
Definition of coversed cosine (covercosine) $ \text{covercosin } \theta = 1+ \sin \theta \ $
Definition of haversed sine (haversine) $ \text{haversin } \theta = \frac{1- \cos \theta}{2} $
Definition of haversed cosine (havercosine) $ \text{havercosin } \theta = \frac{1+ \cos \theta}{2} $
Definition of hacoversed sine (hacoversin) $ \text{hacoversin } \theta = \frac{1 - \sin \theta}{2} $
Definition of hacoversed cosine (hacovercosin) $ \text{hacovercosin } \theta = \frac{1 + \sin \theta}{2} $
Definition of exterior secant (exsec) $ \text{exsec } \theta = \sec \theta - 1 \ $
Definition of exterior cosecant (excosec) $ \text{excosec } \theta = \csc \theta - 1 \ $
Definition of chord (crd) $ \text{crd } \theta = 2 \sin(\frac{\theta}{2}) $
Pythagorean identity and other related identities
Pythagorean identity $ \cos^2 \theta+\sin^2 \theta =1 \ $
$ \sin^2 \theta = 1-\cos^2 \theta \ $
$ \cos^2 \theta = 1-\sin^2 \theta \ $
$ \sec^2 \theta = 1+\tan^2 \theta \ $
$ \csc^2 \theta = 1+\cot^2 \theta \ $
Half-Angle Formulas
Half-angle for sine $ \sin \frac{\theta}{2} = \pm \sqrt{ \frac{1-\cos \theta}{2} } \ $
Half-angle for cosine $ \cos \frac{\theta}{2} = \pm \sqrt{ \frac{1+\cos \theta}{2} } \ $
Half-angle for tangent $ \tan \frac{\theta}{2} = \csc \theta - \cot \theta \ $
Half-angle for tangent $ \tan \frac{\theta}{2} =\pm\sqrt{\frac{1-\cos \theta}{ 1+\cos \theta }} \ $
Half-angle for tangent $ \tan \frac{\theta}{2} =\frac{\sin \theta}{ 1+\cos \theta } \ $
Half-angle for tangent $ \tan \frac{\theta}{2} =\frac{1-\cos \theta}{ \sin \theta } \ $
Half-angle for cotangent $ \cot \frac{\theta}{2} = \csc \theta + \cot \theta $
Half-angle for cotangent $ \cot \frac{\theta}{2} = \frac{1 + \cos \theta}{\sin \theta} $
Half-angle for cotangent $ \cot \frac{\theta}{2} = \pm \sqrt{1 + \cos \theta \over 1 - \cos \theta} $
Half-angle for cotangent $ \cot \frac{\theta}{2} = \frac{\sin \theta}{1 - \cos \theta} $
Double-Angle Formulas
double-angle for sine $ \sin 2 \theta = 2 \sin \theta \cos \theta \ $
double-angle for sine $ \sin 2 \theta = \frac{ 2 \tan \theta}{1+ \tan^2 \theta } \ $
double-angle for cosine $ \cos 2 \theta =\cos^2 \theta - \sin^2 \theta \ $
double-angle for cosine $ \cos 2 \theta =2 \cos^2 \theta - 1 \ $
double-angle for cosine $ \cos 2 \theta =1- 2 \sin^2 \theta \ $
double-angle for cosine $ \cos 2 \theta =\frac{1- \tan^2 \theta}{ 1+\tan^2 \theta } \ $
double-angle for tangent $ \tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta}\, $
double-angle for cotangent $ \cot 2\theta = \frac{\cot^2 \theta - 1}{2 \cot \theta}\, $
Triple-Angle Formulas
triple-angle for sine $ \begin{align}\sin 3\theta & = 3 \cos^2\theta \sin\theta - \sin^3\theta \\ & = 3\sin\theta - 4\sin^3\theta \end{align} $
triple-angle for cosine $ \begin{align}\cos 3\theta & = \cos^3\theta - 3 \sin^2 \theta\cos \theta \\ & = 4 \cos^3\theta - 3 \cos\theta\end{align} $
triple-angle for tangent $ \tan 3\theta = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta} $
tripe-angle for cotangent $ \cot 3\theta = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta} $
Angle sum and difference identities
Sine $ \sin \left( \theta\pm \alpha \right)=\sin \theta \cos \alpha \pm \cos \theta \sin \alpha $
Cosine $ \cos \left(\theta\pm \alpha \right)= \cos \theta \cos \alpha \mp \sin \theta \sin \alpha $
Tangent $ \tan \left(\theta\pm \alpha \right)= \frac {\tan \theta \pm \tan \alpha}{1 \mp \tan \theta \tan \alpha} $
Arcsine $ \arcsin\alpha \pm \arcsin\beta = \arcsin(\alpha\sqrt{1-\beta^2} \pm \beta\sqrt{1-\alpha^2}) $
Arccosine $ \arccos\alpha \pm \arccos\beta = \arccos(\alpha\beta \mp \sqrt{(1-\alpha^2)(1-\beta^2)}) $
Arctangent $ \arctan\alpha \pm \arctan\beta = \arctan\left(\frac{\alpha \pm \beta}{1 \mp \alpha\beta}\right) $

Back to Collective Table

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Trigonometric_Identities&oldid=69228"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

Followed her dream after having raised her family.

Ruth Enoch, PhD Mathematics