sin(A ± B) and friends: exact values without a calculator

8 minA Level3.3.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can use the expansions of $\sin(A \pm B)$ , $\cos(A \pm B)$ and $\tan(A \pm B)$ to find exact values like $\sin 75°$ , expand compound angles like $\sin(x - 60°)$ , and solve equations that contain them.

Key to the symbols

Symbol	Meaning
$A$ , $B$	any two angles
$A \pm B$	shorthand for two formulas at once: one with $A + B$ , one with $A - B$
$\mp$	the opposite sign to $\pm$ : when you pick $+$ for $\pm$ , use $-$ for $\mp$
compound angle	an angle made of two parts, like $x - 30°$

What they ask

Three jobs, one tool:

Exact values of angles not in your exact-value table, like $75° = 45° + 30°$ .
Expanding a compound angle, like $\sin(x - 60°)$ , into terms of $\sin x$ and $\cos x$ .
Collapsing an expression like $\cos\theta\cos 3\theta - \sin\theta\sin 3\theta$ back into one function, by reading a formula backwards.

First, a warning: $\sin(A + B)$ is not $\sin A + \sin B$ . Try $A = B = 45°$ : $\sin 90° = 1$ , but $\sin 45° + \sin 45° = \sqrt2 \approx 1.41$ .

Definition

The addition formulae

$\sin(A + B) = \sin A\cos B + \cos A\sin B$

$\sin(A - B) = \sin A\cos B - \cos A\sin B$

$\cos(A + B) = \cos A\cos B - \sin A\sin B$

$\cos(A - B) = \cos A\cos B + \sin A\sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

They are in the formula list. Patterns that help:

sin mixes the functions (sin·cos and cos·sin) and keeps the sign.
cos keeps the functions together (cos·cos and sin·sin) and flips the sign.
tan keeps the sign on top and flips it underneath.

Worked example: Example 1: the exact value of sin 75°

Find the exact value of $\sin 75°$ .

Have a go first?

Quick check

CalculateQuestion 1 · Recall

Answer the 3 questions above to continue.