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### Course: Trigonometry>Unit 4

Lesson 1: Inverse trigonometric functions

# Trigonometric equations and identities: FAQ

## What's the difference between a trigonometric equation and a trigonometric identity?

A trigonometric equation is just that — an equation that uses trigonometric functions. We try to solve these equations to find the value or values that make them true.
A trigonometric identity, on the other hand, is an equation that is always true, no matter what values we plug in.

## What are inverse trigonometric functions?

Inverse trigonometric functions are the inverse functions of the trigonometric functions. For example, the inverse of the sine function is the arcsine function, written as ${\mathrm{sin}}^{-1}$ or $\mathrm{arcsin}$.
Inverse trigonometric functions can be helpful for solving equations. For example, if we know that $\mathrm{sin}\left(x\right)=0.5$, we can use the inverse sine function, ${\mathrm{sin}}^{-1}$, to find that $x=\frac{\pi }{6}$ or $x=\frac{5\pi }{6}$.

## How can we use sinusoidal models in the real world?

Sinusoidal models can be useful for modeling periodic phenomena. For example, we might use a sinusoidal model to describe the height of a point on a wheel over time, or the amount of daylight in a given location over the course of a year.

## What are angle addition identities?

Angle addition identities are formulas that allow us to find the sine or cosine of the sum or difference of two angles. They are useful for simplifying trigonometric expressions, solving trigonometric equations, and proving trigonometric identities.
$\begin{array}{rl}\mathrm{sin}\left(A+B\right)& =\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{sin}\left(A-B\right)& =\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)-\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{cos}\left(A+B\right)& =\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)-\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{cos}\left(A-B\right)& =\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{tan}\left(A+B\right)& =\frac{\mathrm{tan}\left(A\right)+\mathrm{tan}\left(B\right)}{1-\mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}\\ \\ \mathrm{tan}\left(A-B\right)& =\frac{\mathrm{tan}\left(A\right)-\mathrm{tan}\left(B\right)}{1+\mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}\end{array}$

## What's the best way to get better at working with trigonometric identities?

Practice, practice, practice! The more you use them, the more comfortable you'll get with manipulating and solving equations involving trigonometric functions.

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• why are these identities looking scary
• Third time is a charm!
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