Compound Angles and Double Angles

Starting point: what you already know

• Exact values for $\frac{\pi }{6}$, $\frac{\pi }{4}$ and $\frac{\pi }{3}$, and the unit circle.
• ${\sin }^{2}x+{\cos }^{2}x=1$ and $\tan x=\frac{\sin x}{\cos x}$.
• Solving equations like $\sin x=\frac{1}{2}$ for $0\le x\le 2\pi$.
• $\sin (-x)=-\sin x$ and $\cos (-x)=\cos x$.

Compound angle formulas

Notice that the cosine formula swaps the sign: $\cos (A+B)$ has a minus in the middle.

Double angle formulas

Put $B=A$ in the compound formulas.

Rearranging the cosine versions gives two formulas you will use in calculus:

$${\sin }^{2}A=\frac{1}{2}(1-\cos 2A){\cos }^{2}A=\frac{1}{2}(1+\cos 2A)$$

One wave: the auxiliary angle

Any $a\sin x+b\cos x$ can be written as a single wave $R\sin (x+\alpha )$, where

$$R=\sqrt{a^2+b^2},\cos \alpha =\frac{a}{R},\sin \alpha =\frac{b}{R}$$

This works because $R\sin (x+\alpha )=R\cos \alpha \sin x+R\sin \alpha \cos x$.

Quick summary

• $\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$ and $\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$.
• $\sin 2A=2\sin A\cos A$ and $\cos 2A=2{\cos }^{2}A-1=1-2{\sin }^{2}A$.
• ${\sin }^{2}A=\frac{1}{2}(1-\cos 2A)$ and ${\cos }^{2}A=\frac{1}{2}(1+\cos 2A)$.
• $a\sin x+b\cos x=R\sin (x+\alpha )$ with $R=\sqrt{a^2+b^2}$.
• Factorise, do not divide, when solving trig equations.

Quiz