# Difference between revisions of "Trigonometric identities"

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− | '''Trigonometric | + | '''Trigonometric Identities''' are used to manipulate [[trigonometry]] [[equation]]s in certain ways. Here is a list of them: |

== Basic Definitions == | == Basic Definitions == | ||

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*<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> | ||

*<math> \cos \frac{\theta}2 = \pm \sqrt{\frac{1+\cos \theta}2} </math> | *<math> \cos \frac{\theta}2 = \pm \sqrt{\frac{1+\cos \theta}2} </math> | ||

− | *<math> \tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} </math> | + | *<math> \tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=\frac{\sin \theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin \theta} </math> |

== Prosthaphaeresis Identities == | == Prosthaphaeresis Identities == | ||

(Otherwise known as sum-to-product identities) | (Otherwise known as sum-to-product identities) | ||

− | * <math>\sin \theta \ | + | * <math>{\sin \theta + \sin \gamma = 2 \sin \frac{\theta + \gamma}2 \cos \frac{\theta - \gamma}2}</math> |

− | * <math>\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2</math> | + | * <math>{\sin \theta - \sin \gamma = 2 \sin \frac{\theta - \gamma}2 \cos \frac{\theta + \gamma}2}</math> |

− | * <math>\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2</math> | + | * <math>{\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2}</math> |

+ | * <math>{\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2}</math> | ||

== Law of Sines == | == Law of Sines == | ||

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The extended [[Law of Sines]] states | The extended [[Law of Sines]] states | ||

− | *<math>\frac a{\sin A} = \frac b{\sin B} = \frac c{\sin C} = 2R | + | *<math>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R</math> |

== Law of Cosines == | == Law of Cosines == | ||

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*<math>\cos(\theta) = \frac{\sin(\theta)}{\tan(\theta)}</math> | *<math>\cos(\theta) = \frac{\sin(\theta)}{\tan(\theta)}</math> | ||

*<math>\sec(\theta) = \frac{\tan(\theta)}{\sin(\theta)}</math> | *<math>\sec(\theta) = \frac{\tan(\theta)}{\sin(\theta)}</math> | ||

+ | *<math>\arctan(x) + \arctan(y) = \arctan \left( \dfrac{x+y}{1-xy} \right)</math> | ||

*<math>\sin^2(\theta) + \cos^2(\theta) + \tan^2(\theta) = \sec^2(\theta)</math> | *<math>\sin^2(\theta) + \cos^2(\theta) + \tan^2(\theta) = \sec^2(\theta)</math> | ||

*<math>\sin^2(\theta) + \cos^2(\theta) + \cot^2(\theta) = \csc^2(\theta)</math> | *<math>\sin^2(\theta) + \cos^2(\theta) + \cot^2(\theta) = \csc^2(\theta)</math> | ||

− | The two identities | + | The two identities above are derived from the Pythagorean Identities. |

*<math>\cos(2\theta) = (\cos(\theta) + \sin(\theta))(\cos(\theta) - \sin(\theta))</math> | *<math>\cos(2\theta) = (\cos(\theta) + \sin(\theta))(\cos(\theta) - \sin(\theta))</math> | ||

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* [[Trigonometry]] | * [[Trigonometry]] | ||

* [[Trigonometric substitution]] | * [[Trigonometric substitution]] | ||

− | + | ||

+ | ==External Links== | ||

+ | [http://www.sosmath.com/trig/Trig5/trig5/trig5.html Trigonometric Identities] | ||

[[Category:Trigonometry]] | [[Category:Trigonometry]] |

## Latest revision as of 11:03, 30 July 2020

**Trigonometric Identities** are used to manipulate trigonometry 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:

## Even-Odd Identities

### Further Conclusions

Based on the above identities, we can also claim that

This is only true when is in the domain of .

## Reciprocal Relations

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

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

Note that ; the former refers to the inverse trigonometric functions.

## 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:

(Note that the last two are easily derived by dividing the first by and , respectively.)

## 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.

We can prove easily by using and .

## Double Angle Identities

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

# Further Conclusions

We can see from the above that

## Half Angle Identities

Using the double angle identities, we can derive half angle identities. The double angle formula for cosine tells us . Solving for we get where we look at the quadrant of to decide if it's positive or negative. Likewise, we can use the fact that to find a half angle identity for sine. Then, to find a half angle identity for tangent, we just use the fact that and plug in the half angle identities for sine and cosine.

To summarize:

## Prosthaphaeresis Identities

(Otherwise known as sum-to-product identities)

## Law of Sines

*Main article: Law of Sines*

The extended Law of Sines states

## Law of Cosines

*Main article: Law of Cosines*

The Law of Cosines states

## Law of Tangents

*Main article: Law of Tangents*

The Law of Tangents states that if and are angles in a triangle opposite sides and respectively, then

A further extension of the Law of Tangents states that if , , and are angles in a triangle, then

## Other Identities

- (This is also written as )

The two identities above are derived from the Pythagorean Identities.