Overview

Angle addition formulas let you compute exact values for angles like $15^\circ$ or $75^\circ$, and double/half angles drive many contest identities.

Core Skills

Choose a Formula Direction

Use sum/difference to build special angles, and double/half-angle to rewrite expressions into squares.

Track Signs Carefully

For half-angle formulas, decide the sign using the quadrant of $\theta/2$.

Simplify Before Expanding

If the angle is special (like $15^\circ$ or $75^\circ$), write it as a sum of $30^\circ$ and $45^\circ$ to keep radicals clean.

Sum and Difference

$\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$

$\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$

$\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}$

Double Angle

$\sin 2\theta = 2\sin\theta\cos\theta$

$\cos 2\theta = \cos^2\theta-\sin^2\theta = 2\cos^2\theta-1 = 1-2\sin^2\theta$

$\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}$

Half Angle

$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}$

$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}$

$\tan\frac{\theta}{2} = \frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta}$

Worked Example

Compute $\sin 75^\circ$.

$75^\circ = 45^\circ + 30^\circ$, so

$\sin 75^\circ = \frac{\sqrt{6}+\sqrt{2}}{4}.$

More Examples

Example 1: Cosine of $15^\circ$

$\cos 15^\circ = \cos(45^\circ-30^\circ) = \frac{\sqrt{6}+\sqrt{2}}{4}$.

Example 2: Double Angle

If $\sin\theta=3/5$ and $\cos\theta=4/5$, find $\sin 2\theta$.

$\sin 2\theta = 2\sin\theta\cos\theta = 24/25$.

Example 3: Half Angle

If $\cos\theta = -1/2$ and $\theta$ is in quadrant II, find $\sin(\theta/2)$.

$\theta/2$ is in quadrant I, so $\sin(\theta/2)=\sqrt{(1-\cos\theta)/2} =\sqrt{(1+1/2)/2}=\sqrt{3/4}=\sqrt{3}/2$.

Strategy Checklist

• Choose sum/difference or double/half-angle based on the target.
• Use quadrant signs for half-angle formulas.
• Keep radicals simplified.

Common Pitfalls

• Using $\sin(\alpha+\beta)=\sin\alpha+\sin\beta$ (false).
• Forgetting quadrant signs on half-angle formulas.
• Mixing degrees and radians in the same calculation.

Practice Problems