Trigonometric identities and equations

What you’ll learn

• Use basic identities to reduce equations to one trigonometric function.
• Find every solution in a stated interval.

Understand the idea

The identity $\sin^2x+\cos^2x=1$ lets you replace a squared sine by a squared cosine, or the reverse. The identity $\tan x=\sin x/\cos x$ is valid where cosine is non-zero. Simplifying first can turn an equation into an ordinary quadratic in a trigonometric value.

After solving for a trigonometric value, find all corresponding angles in the requested interval. Inverse trigonometric functions return one principal angle, so use symmetry and periodicity to find the others. For a multiplied or shifted angle, transform the interval first and convert back at the end.

Key formulae

$$\sin^2x+\cos^2x=1$$

$$\tan x=\frac{\sin x}{\cos x}$$

$$\sin x=\sin\alpha\Rightarrow x=\alpha+2k\pi\text{ or }x=\pi-\alpha+2k\pi$$

$$\tan x=\tan\alpha\Rightarrow x=\alpha+k\pi\quad(k\in\mathbb Z)$$

Worked example

Solve $2\sin^2x-\sin x=0$ for $0\leq x<360^\circ$.

1. Factorise: $\sin x(2\sin x-1)=0$.
2. From $\sin x=0$, obtain $x=0^\circ,180^\circ$ in the interval.
3. From $\sin x=1/2$, obtain $x=30^\circ,150^\circ$.

Answer: $x=0^\circ,30^\circ,150^\circ,180^\circ$.