# Sections 2.3 Solving Trigonometric Equations

Section Objectives

1. Use standard algebraic techniques to solve trigonometric equations.
2. Use inverse trig functions to solve trigonometric equations.
3. Solve trig equations that are quadratic in form.
4. Use simple substitutions to solve trig equations involving a multiple angle.

### Solving Trig Equations

Trigonometric equations come in lots of varieties. Here are some helpful tips to keep in mind...

1. Expect to use basic trig identities as well as standard algebraic techniques (.e.g, isolating the variable, combining like terms, factoring, expanding, extracting roots, multiplying by conjugates, getting common denominators, simplifying, quadratic formula, etc.).
2. Expect to think graphically.
3. Expect to use the inverse trig functions.
4. Expect to use simple substitutions to simplify expressions (e.g, $u=2\theta$, $u=x/2$).
5. To find all solutions, find the solutions inside one period, then add (or subtract) multiples of the period.

#### Examples

• Find all solutions of $\displaystyle \sin x = \frac{1}{2}$.

• Find all solutions: $\quad \tan^2 x = 1 - 2 \tan^2 x$

• Find all solutions: $\quad \sin^2 x = 2 \sin x$

• Find all solutions: $\quad 3\sec^2 x - 2 \tan^2 x - 4 =0$

• Find the solutions in $[0,2\pi)$: $\quad \cos x + 1 = \sin x$

• Find all solutions: $\quad 2 \cos 3t -1=0$

• Find all solutions: $\quad 4\tan^2x + 5\tan x-6=0$