Solving trigonometric equations | Trigonometry

The goal: reduce to basic equations

Almost every trig equation is solved by turning it into one or more basic equations like $\sin x = a$ . The tools are the same as in algebra: isolate, factor, substitute, plus the trig identities.

Factoring

Quadratic in form

If an equation involves one trig function squared and to the first power, treat it as a quadratic.

Using identities

When an equation mixes sine and cosine, an identity can rewrite it in terms of one function.

Multiple angles

For an equation like $\sin 2x = \frac{\sqrt{3}}{2}$ , solve for $$2x$$ first, then divide. Because dividing shrinks the period, you'll get more solutions in $[0, 2\pi)$ .

Equations with sine and cosine to the first power

An equation like $\sin x = \cos x$ can be divided by $\cos x$ : values with $\cos x = 0$ can't be solutions here, since then $\sin x = \pm 1 \neq 0$ .

Practice

Quick check

You substitute $u = \sin x$ and get $$u = 3$$ or $u = \frac{1}{2}$ . Which values lead to solutions?

Quick check

How many solutions does $\sin 2x = 1$ have on $[0, 2\pi)$ ?