5.3 Solving Trigonometric Equations

Some important considerations:

* Standard algebraic techniques are to be used
* The goal is to isolate the trigonometric function, not necessarily just the variable
* Your last step should involve knowledge of the unit circle

Start with something simple:

2 cos x - 1 = 0
2 cos x      = 1
        cos x = 1/2 <----- As I said before, this is how the last step of most trigonometric
                                      equations should look
Solutions:

-If only along the interval [0, 2π) : x = π/3, x = 5π/3
- If the interval is not specified: x = π/3 + 2nπ, x = 5π/3 + 2nπ, where n is an integer.
Why?
Cos x = 1/2 has indefinitely many solutions. The period of y = cos x is 2π, so every time the period is repeated (integer n), there is another pair of solutions for cos x = 1/2
          (For solutions to equations involving tangent and cotangent functions, πn would be added
           instead of 2πn)

Basic algebraic concepts are used in trigonometric equations:

Examples:


Combining like terms:
tan x + 2 = -tan x
2 tan x    = -2
     tan x  = -1

x = 3π/4, x = 7π/4

Factoring:
tan x(sin x) = tan x
tan x(sin x) - tan x = 0
tan x (sin x - 1) = 0
tan x = 0, sin x = 1


x = 0, π/2, π


Factoring Quadratics

tan²x - 2tanx +1 = 0
(tan x -1) (tan x - 1) = 0
tan x = 1
x= π/4, 5π/4