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Sum and Difference Identities

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Sum and Difference Identities

Study Guide

Key Definition

The sum and difference identities allow the computation of trigonometric functions of sums or differences of angles: $\sin(u+v) = \sin(u)\cos(v) + \cos(u)\sin(v)$, $\cos(u+v) = \cos(u)\cos(v) - \sin(u)\sin(v)$, $\tan(u+v) = \frac{\tan(u) + \tan(v)}{1 - \tan(u)\tan(v)}$, and for differences: $\sin(u-v) = \sin(u)\cos(v) - \cos(u)\sin(v)$, $\cos(u-v) = \cos(u)\cos(v) + \sin(u)\sin(v)$, $\tan(u-v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)}$.

Important Notes

These identities are useful for simplifying expressions involving angles.
Remember: $\sin(u+v)$ combines sine and cosine.
$\cos(u+v)$ combines cosine and sine subtraction.
$\tan(u+v)$ involves division.
Use these identities to transform complex expressions into simpler forms.
For differences: $\sin(u-v) = \sin(u)\cos(v) - \cos(u)\sin(v)$.
For differences: $\cos(u-v) = \cos(u)\cos(v) + \sin(u)\sin(v)$.
For differences: $\tan(u-v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)}$.

Mathematical Notation

$\sin(x)$ is the sine function

$\cos(x)$ is the cosine function

$\tan(x)$ is the tangent function

$+$ represents addition

$-$ represents subtraction

Remember to use proper notation when solving problems

Why It Works

The identities allow transformation of sums and differences of angles into sums and products of trigonometric functions, making calculations easier with $\sin$, $\cos$, and $\tan$.

Remember

These identities are crucial for simplifying trigonometric expressions: $\sin(u\pm v)$, $\cos(u\pm v)$, $\tan(u\pm v)$.

Quick Reference

Sine Sum:$\sin(u+v) = \sin(u)\cos(v) + \cos(u)\sin(v)$

Cosine Sum:$\cos(u+v) = \cos(u)\cos(v) - \sin(u)\sin(v)$

Tangent Sum:$\tan(u+v) = \frac{\tan(u) + \tan(v)}{1 - \tan(u)\tan(v)}$

Sine Difference:$\sin(u-v) = \sin(u)\cos(v) - \cos(u)\sin(v)$

Cosine Difference:$\cos(u-v) = \cos(u)\cos(v) + \sin(u)\sin(v)$

Tangent Difference:$\tan(u-v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)}$

Understanding Sum and Difference Identities

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $\sin(u+v) = \sin(u)\cos(v) + \cos(u)\sin(v)$ and $\sin(u-v) = \sin(u)\cos(v) - \cos(u)\sin(v)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Simplify $\sin(2x)\cos(5x) + \cos(2x)\sin(5x)$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are measuring angles in a skateboard ramp. You need to find the sine of the sum of two angles: $30^\circ$ and $45^\circ$. What is $\sin(75^\circ)$?

Thinking Challenge

Thinking Exercise