Double & Half Angles

Squares of sines are waves too

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $\cos 2x = \cos^2 x - \sin^2 x = 1 - 2\sin^2 x = 2\cos^2 x - 1$.
• Power reduction: $\sin^2 x = \frac{1 - \cos 2x}{2}$, so $\sin^2$ averages $\frac12$.
• Half-angle formulas give exact values for smaller and smaller angles.

Steps

1. Three versions of cos 2x
2. Squaring a sine wave
3. The average of sin²
4. Half angle
5. Same range
6. Triple angle