Question

What are other methods for solving equations that can be adapted to solving trigonometric equations?

Answer 1

Solving concept. To solve a trig equation, transform it into one, or many, basic trig equations. Solving a trig equation, finally, results in solving various basic trig equations.
There are 4 main basic trig equations:
sin x = a; cos x = a; tan x = a; cot x = a.
Exp. Solve sin 2x - 2sin x = 0
Solution. Transform the equation into 2 basic trig equations:
2sin x.cos x - 2sin x = 0
2sin x(cos x - 1) = 0.
Next, solve the 2 basic equations: sin x = 0, and cos x = 1.
Transformation process.
There are 2 main approaches to solve a trig function F(x).
1. Transform F(x) into a product of many basic trig functions.
Exp. Solve F(x) = cos x + cos 2x + cos 3x = 0.
Solution. Use trig identity to transform (cos x + cos 3x):
F(x) = 2cos 2x.cos x + cos 2x = cos 2x(2cos x + 1 ) = 0.
Next, solve the 2 basic trig equations.
2. Transform a trig equation F(x) that has many trig functions as variable, into a equation that has only one variable. The common variables to be chosen are: cos x, sin x, tan x, and tan (x/2)
Exp Solve `sin ^2 x + sin^4 x = cos^2 x`
Solution. Call cos x = t, we get
`(1 - t^2)(1 + 1 - t^2) = t^2`.
Next, solve this equation for t.
Note . There are complicated trig equations that require special transformations.