# Difference between revisions of "Trigonometric identities"

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− | == Angle Addition Identities == | + | == Angle Addition/Subtraction Identities == |

− | + | Once we have formulas for angle addition, angle subtraction is rather easy to derive. For example, we just look at <math> \sin(\alpha+(-\beta))</math> and we can derive the sine angle subtraction formula using the sine angle addition formula. | |

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+ | {| style="width:100%; height:130px; margin: 1em auto 1em auto" | ||

+ | |- | ||

+ | | <math> \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> || <math> \sin(\alpha - \beta) = \sin \alpha \cos \beta - \sin \beta \cos \alpha</math> | ||

+ | |- | ||

+ | | <math> \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> || <math> \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta</math> | ||

+ | |- | ||

+ | | <math> \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta} </math> || <math> \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta} </math> | ||

+ | |} | ||

+ | |||

+ | == Double/Half Angle Identities == | ||

+ | Double angle identities are easily derived from the angle addition formulas by just letting <math> \alpha = \beta </math>. Doing so yields: | ||

+ | |||

+ | {| style="height:200px; margin: 1em auto 1em auto" | ||

+ | |- | ||

+ | | <math> \sin 2\alpha </math> || = || <math>2\sin \alpha \cos \alpha</math> | ||

+ | |- | ||

+ | | <math> \cos 2\alpha </math> || = || <math> \cos^2 \alpha - \sin^2 \alpha</math> | ||

+ | |- | ||

+ | | || = || <math> 2\cos^2 \alpha - 1</math> | ||

+ | |- | ||

+ | | || = || <math> 1-2\sin^2 \alpha</math> | ||

+ | |- | ||

+ | | <math> \tan 2\alpha </math> || = || <math>\frac{2\tan \alpha}{1-\tan^2\alpha} </math> | ||

+ | |} | ||

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

## Revision as of 09:22, 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/Subtraction Identities

Once we have formulas for angle addition, angle subtraction is rather easy to derive. For example, we just look at and we can derive the sine angle subtraction formula using the sine angle addition formula.

## Double/Half Angle Identities

Double angle identities are easily derived from the angle addition formulas by just letting . Doing so yields:

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## 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.*