Question

How to solve trig equations?

Answer 1

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 4x = sin (pi/2 - 3x)`
a. `4x = pi/2 - 3x`
b. `4x = pi - (pi/2 - 3x)`
Example 3. Solve: `f(x) = cos x + sin (x/2) = 1`
Use trig identity: `cos x = 1 - 2sin^2 (x/2)`
`f(x) = 1 - 2sin^2 (x/2) + sin (x/2) = 1`
`f(x) = sin (x/2)(-2sin (x/2) + 1) = 0`