Compound Angles and Double Angles

In short: Compound angle formulas expand the trig of a sum or difference, like $sin (A + B) = sin A cos B + cos A sin B$ . Double angle formulas follow from them: $sin 2 A = 2 sin A cos A$ and $cos 2 A = 2 cos^{2} A - 1$ , and $a sin x + b cos x$ can be written as $R sin (x + α)$ .

Starting point: what you already know

Exact values for $\frac{π}{6}$ , $\frac{π}{4}$ and $\frac{π}{3}$ , 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 $0 \leq x \leq 2 π$ .
$sin (- x) = - sin x$ 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:

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

One wave: the auxiliary angle

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

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

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

Further trig identities

Compound angles

sin(A ± B)
cos(A ± B) swaps the sign
tan(A ± B)

Double angles

sin 2A = 2 sin A cos A
cos 2A has three forms
sin² A and cos² A in terms of cos 2A

Auxiliary angle

a sin x + b cos x = R sin(x + α)
R = √(a² + b²)

Uses

Exact values
Solving equations
Integrating sin² and cos²

Quick summary

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

Quiz

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$sin (A + B)$ equals:
1. A $sin A + sin B$
2. B $sin A cos B + cos A sin B$
3. C $sin A sin B + cos A cos B$
4. D $cos A cos B - sin A sin B$
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B: $sin A cos B + cos A sin B$
Sine mixes: sin cos plus cos sin.
$cos (A + B)$ equals:
1. A $cos A cos B + sin A sin B$
2. B $cos A cos B - sin A sin B$
3. C $cos A + cos B$
4. D $sin A cos B - cos A sin B$
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B: $cos A cos B - sin A sin B$
Cosine keeps cos cos and sin sin, and swaps the sign.
$sin 2 A$ equals:
1. A $2 sin A$
2. B $2 sin A cos A$
3. C $sin^{2} A - cos^{2} A$
4. D $sin A cos A$
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B: $2 sin A cos A$
Put $B = A$ in $sin (A + B)$ .