4.3 Solving The Most Complex Trigonometric Equations

In this section,

We will be taking a look at the most complex equations. When solving the most complex trig equations, it is important to make sure the equation cannot be simplified. one way you could simplify a trig equation is by factoring and using the zero product property. Take a look:

Solving these equations:

Simplify the equation
Determine if the equation can be factored
Use the zero product property

Examples:

Ex.1

3 Tan²x + 5Tanx = 0

Tanx (3Tanx + 5) = 0 ←Factor out GCF

Tanx = 0 Tanx = -5/3 ←Set equal to 0

X=0,π Tanx-1=-5/3

Quadrants: II, IV ←Find quadrants
X¹=1.03
II = 2.11
IV = 5.25

Solutions: {2.11, 5.25}

Ex.2
6sin²x + 7sinx - 5 = 0

(3sinx + 5) (2sinx - 1) = 0 ←Factor

3sinx + 5 = 0 2sinx - 1 = 0 ←Set equal to 0

sinx = -5/3 sinx = ½

Quadrants: I, II ← Find quadrants
X^I = π/6
I = π/6
II = 5π/6

Solutions: {π/6, 5π/6}

Practice Problems:

ANSWER KEY
1) Tan²x + 2tanx = -1

2) 2sin²x - 3 sinx + 1 = 0