# Difference between revisions of "Trigonometric identities"

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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> | | <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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+ | Another useful identity that isn't a reciprocal relation is that <math> \tan A =\frac{\sin A}{\cos A} </math>. | ||

== Pythagorean Identities == | == Pythagorean Identities == | ||

− | + | Using the [[Pythagorean Theorem]] on our triangle above, we know that <math>\displaystyle a^2 + b^2 = c^2 </math>. If we divide by <math> c^2 </math> we get <math> \left(\frac ac\right)^2 + \left(\frac bc\right)^2 = 1 </math> which is just <math> \sin^2 A + \cos^2 A =1 </math>. Dividing by <math> a^2 </math> or <math> b^2 </math> instead produces two other similar identities. The Pythagorean Identities are listed below: | |

− | + | ||

− | + | {| style="height:150px; margin: 1em auto 1em auto" | |

+ | |- | ||

+ | |<math>\displaystyle \sin^2x + \cos^2x = 1</math> | ||

+ | |- | ||

+ | |<math>\displaystyle 1 + \cot^2x = \csc^2x</math> | ||

+ | |- | ||

+ | |<math>\displaystyle \tan^2x + 1 = \sec^2x</math> | ||

+ | |} | ||

== Angle Addition Identities == | == Angle Addition Identities == | ||

*<math>\displaystyle \sin \theta \cos \gamma + \sin \gamma \cos \theta = \sin \left(\theta+\gamma\right)</math> | *<math>\displaystyle \sin \theta \cos \gamma + \sin \gamma \cos \theta = \sin \left(\theta+\gamma\right)</math> | ||

− | *<math>\displaystyle \cos \theta \cos \gamma - \sin theta \sin gamma = \cos \left(\theta+\gamma\right)</math> | + | *<math>\displaystyle \cos \theta \cos \gamma - \sin theta \sin \gamma = \cos \left(\theta+\gamma\right)</math> |

− | *<math>\displaystyle \frac{\tan \theta + \tan gamma}{1-\tan\theta\tan\gamma}=\tan\left(\theta+\gamma\right)</math> | + | *<math>\displaystyle \frac{\tan \theta + \tan \gamma}{1-\tan\theta\tan\gamma}=\tan\left(\theta+\gamma\right)</math> |

== Even-Odd Identities == | == Even-Odd Identities == |

## Revision as of 07:48, 24 June 2006

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

## Contents

## Basic Definitions

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

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 ). They are defined as follows:

## Reciprocal Relations

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

Another useful identity that isn't a reciprocal relation is that .

## Pythagorean Identities

Using the Pythagorean Theorem on our triangle above, we know that . If we divide by we get which is just . Dividing by or instead produces two other similar identities. The Pythagorean Identities are listed below:

## Angle Addition Identities

## Even-Odd Identities

## Prosthaphaersis Indentities

(Otherwise known as sum-to-product identities)

## Other Identities

## See also

*This article is a stub. Help us out by expanding it.*