Double-Angle and Half-Angle Identities

Study Guide

Key Definition

The double-angle and half-angle identities are trigonometric identities that allow simplification of expressions such as $\sin(2u) = 2 \sin(u) \cos(u)$, $\cos(2u) = \cos^2(u) - \sin^2(u)$, $\sin^2(u) = \frac{1 - \cos(2u)}{2}$, $\cos(u/2) = \pm\sqrt{\frac{1 + \cos(u)}{2}}$, and $\sin(u/2) = \pm\sqrt{\frac{1 - \cos(u)}{2}}$.

Important Notes

Double-angle identities are derived from basic angle-addition formulas
The identity $\cos(2u) = \cos^2(u) - \sin^2(u)$ is one of three forms for $\cos(2u)$
Use $\cos(2u) = 1 - 2 \sin^2(u)$ when the expression involves sine
$\tan(2u) = \frac{2 \tan(u)}{1 - \tan^2(u)}$ helps simplify tangent expressions
Identities can be used to convert expressions into simpler forms

Mathematical Notation

$\sin$ means sine function

$\cos$ means cosine function

$\tan$ means tangent function

$\frac{a}{b}$ represents a fraction

$\sqrt{x}$ represents square root

$u/2$ denotes a half-angle

$±$ indicates plus or minus

Remember to use proper notation when solving problems

Why It Works

Double-angle and half-angle identities work because they are derived from basic trigonometric identities, allowing for transformation of complex expressions into simpler ones using $trigonometric formulas$.

Remember

Key formulas: $\sin(2u) = 2 \sin(u) \cos(u)$, $\cos(2u) = \cos^2(u) - \sin^2(u)$, $\sin^2(u) = \frac{1 - \cos(2u)}{2}$, $\cos(u/2) = \pm\sqrt{\frac{1 + \cos(u)}{2}}$, $\sin(u/2) = \pm\sqrt{\frac{1 - \cos(u)}{2}}$

Quick Reference

Sine Double-Angle:$\sin(2u) = 2 \sin(u) \cos(u)$

Cosine Double-Angle (Form 1):$\cos(2u) = \cos^2(u) - \sin^2(u)$

Cosine Double-Angle (Form 2):$\cos(2u) = 2 \cos^2(u) - 1$

Cosine Double-Angle (Form 3):$\cos(2u) = 1 - 2 \sin^2(u)$

Tangent Double-Angle:$\tan(2u) = \frac{2 \tan(u)}{1 - \tan^2(u)}$

Sine Half-Angle:$\sin(u/2) = \pm\sqrt{\frac{1 - \cos(u)}{2}}$

Cosine Half-Angle:$\cos(u/2) = \pm\sqrt{\frac{1 + \cos(u)}{2}}$

Understanding Double-Angle and Half-Angle Identities

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Beginner

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Beginner Explanation

The double-angle identity for sine, $\sin(2u) = 2 \sin(u) \cos(u)$, lets you express sin(2u) using sin(u) and cos(u). The half-angle identity, $\sin(u/2) = \pm \sqrt{(1 - \cos(u))/2}$, shows how to write sin of half an angle.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is $\sin(2u)$ if $\sin(u) = 1/2$ and $\cos(u) = \sqrt{3}/2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder launches off a ramp at an angle u = 30° above the horizontal. What is the value of $\cos(2u)$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove that $\tan(2u) = \frac{2 \tan(u)}{1 - \tan^2(u)}$ using trigonometric identities.

Challenge Quiz

Single Choice Quiz

Advanced

If $\tan(u) = 1$, what is $\tan(2u)$?

Recap

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