Difference between revisions of "Trigonometric identities"

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Trigonometric identities are used to manipulate trig equations in certain ways.  Here is a list of them:
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'''Trigonometric identities''' are used to manipulate trig equations in certain ways.  Here is a list of them:
  
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== Basic Definitions ==
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The six basic trigonometric functions can be defined using a right triangle:
  
== Reciprocal Identities ==
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<center>[[Image:righttriangle.png]]</center>
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The six trig functions are sine, cosine, tangent, cosecant, secant, and cotangent.  They are abbreviated by using the first three letters of their name (except for cosecant which uses <math>\csc</math>).  They are defined as follows:
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{| style="width:75%; height:200px; margin: 1em auto 1em auto" border="0"
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|-
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| <math>\sin A = \frac ac</math> || <math>\csc A = \frac ca</math>
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|-
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| <math> \cos A = \frac bc</math> || <math>\sec A = \frac cb</math>
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|-
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| <math> \tan A = \frac ab</math> || <math> \cot A = \frac ba</math>
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|}
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== Reciprocal Relations ==
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From the last section, it is easy to see that the following hold:
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{| style="width:75%; margin: 1em auto 1em auto"
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|-
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| <math> \sin A = \frac 1{\csc A}</math> || <math> \cos A = \frac 1{\sec A}</math> || <math> \tan A = \frac 1{\cot A}</math>
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|}
  
 
== Pythagorean Identities ==
 
== Pythagorean Identities ==
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*<math>|1-e^{i\theta}|=2\sin\frac{\theta}{2}</math>
 
*<math>|1-e^{i\theta}|=2\sin\frac{\theta}{2}</math>
 
This page is incomplete--if you know of a trigonometric identity, add it.
 
  
 
==See also==
 
==See also==

Revision as of 07:34, 24 June 2006

Trigonometric identities are used to manipulate trig equations in certain ways. Here is a list of them:

Basic Definitions

The six basic trigonometric functions can be defined using a right triangle:

Righttriangle.png

The six trig functions are sine, cosine, tangent, cosecant, secant, and cotangent. They are abbreviated by using the first three letters of their name (except for cosecant which uses $\csc$). They are defined as follows:

$\sin A = \frac ac$ $\csc A = \frac ca$
$\cos A = \frac bc$ $\sec A = \frac cb$
$\tan A = \frac ab$ $\cot A = \frac ba$

Reciprocal Relations

From the last section, it is easy to see that the following hold:

$\sin A = \frac 1{\csc A}$ $\cos A = \frac 1{\sec A}$ $\tan A = \frac 1{\cot A}$

Pythagorean Identities

  • $\displaystyle \sin^2x + \cos^2x = 1$
  • $\displaystyle 1 + \cot^2x = \csc^2x$
  • $\displaystyle \tan^2x + 1 = \sec^2x$

Angle Addition Identities

  • $\displaystyle \sin \theta \cos \gamma + \sin \gamma \cos \theta = \sin \left(\theta+\gamma\right)$
  • $\displaystyle \cos \theta \cos \gamma - \sin theta \sin gamma = \cos \left(\theta+\gamma\right)$
  • $\displaystyle \frac{\tan \theta + \tan gamma}{1-\tan\theta\tan\gamma}=\tan\left(\theta+\gamma\right)$

Even-Odd Identities

Prosthaphaersis Indentities

(Otherwise known as sum-to-product identities)

Other Identities

  • $|1-e^{i\theta}|=2\sin\frac{\theta}{2}$

See also

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