Trigonometry Sheet 5

Multiple and Half Angle Formulas

1. Double Angle Formulas ($$2\theta$$)

Function Standard Forms In terms of $$\tan \theta / \cot \theta$$
$$\sin 2\theta$$ $$2 \sin \theta \cos \theta$$ $$\frac{2\tan \theta}{1 + \tan^2 \theta}$$
$$\cos 2\theta$$ $$\cos^2 \theta – \sin^2 \theta$$
$$2\cos^2 \theta – 1$$
$$1 – 2\sin^2 \theta$$		 $$\frac{1 – \tan^2 \theta}{1 + \tan^2 \theta}$$
$$\tan 2\theta$$ — $$\frac{2\tan \theta}{1 – \tan^2 \theta}$$
$$\cot 2\theta$$ — $$\frac{\cot^2 \theta – 1}{2\cot \theta}$$

2. Power Reduction (+ / – Form)

Addition Form

$$1 + \cos 2\theta = 2\cos^2 \theta$$

Subtraction Form

$$1 – \cos 2\theta = 2\sin^2 \theta$$

3. Triple Angle Formulas ($$3\theta$$)

Ratio Formula
$$\sin 3\theta$$ $$3\sin \theta – 4\sin^3 \theta$$
$$\cos 3\theta$$ $$4\cos^3 \theta – 3\cos \theta$$
$$\tan 3\theta$$ $$\frac{3\tan \theta – \tan^3 \theta}{1 – 3\tan^2 \theta}$$
$$\cot 3\theta$$ $$\frac{\cot^3 \theta – 3\cot \theta}{3\cot^2 \theta – 1}$$

4. Half Angle Formulas ($$\theta/2$$)

$$\sin \frac{\theta}{2}$$ $$\pm \sqrt{\frac{1 – \cos \theta}{2}}$$
$$\cos \frac{\theta}{2}$$ $$\pm \sqrt{\frac{1 + \cos \theta}{2}}$$
$$\tan \frac{\theta}{2}$$ $$\frac{1 – \cos \theta}{\sin \theta}$$or$$\frac{\sin \theta}{1 + \cos \theta}$$
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