# How to solve trig equations?

Jul 4, 2017

To solve a trig equation, the common approach is to transform it into one or many Basic Trig Equations. Solving trig equations, finally, results in solving basic trig equations.
There are 4 types of basic trig equations:
sin x = a , cos x = a ; tan x = a ; cot x = a
See trig books to know how to solve basic trig equations.

#### Explanation:

To order to transform a complex trig equation into many basic trig equations we use:
Trig identities, factoring, definitions and properties of trig functions ...
Example 1. Solve: f(x) = sin 2x - sin x = 0.
Use trig identity: sin 2x = sin x.cos x to transform.
f(x) = 2sin x.cos x - sin x = sin x(2cos x - 1) = 0
Example 2. Solve: sin 4x = cos 3x.
Use property of complementary arcs to transform:
$\sin 4 x = \sin \left(\frac{\pi}{2} - 3 x\right)$
a. $4 x = \frac{\pi}{2} - 3 x$
b. $4 x = \pi - \left(\frac{\pi}{2} - 3 x\right)$
Example 3. Solve: $f \left(x\right) = \cos x + \sin \left(\frac{x}{2}\right) = 1$
Use trig identity: $\cos x = 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)$
$f \left(x\right) = 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right) + \sin \left(\frac{x}{2}\right) = 1$
$f \left(x\right) = \sin \left(\frac{x}{2}\right) \left(- 2 \sin \left(\frac{x}{2}\right) + 1\right) = 0$