Special Angles

Exact values from squares and triangles

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

• 45-45-90: sides $1, 1, \sqrt2$, so $\sin 45° = \cos 45° = \tfrac{\sqrt2}{2}$.
• 30-60-90: sides $1, \sqrt3, 2$, so $\sin 30° = \tfrac12$ and $\sin 60° = \tfrac{\sqrt3}{2}$.
• $\sin^2\theta + \cos^2\theta = 1$ is Pythagoras in disguise.

Steps

1. Half a square
2. Half an equilateral triangle
3. Read it off
4. The Pythagorean identity
5. Use the identity
6. Equilateral height