5. Solving Trigonometric Equations

by M. Bourne

Trigonometric equations can be solved using the algebraic methods and trigonometric identities and values discussed in earlier sections. You may wish to go back and have a look at Trigonometric Functions of Any Angle, where we see the background to the following solutions.

A painless way to solve these is using a graph. Where the graph cuts the x-axis, that's where you'll find your solutions (the x-values that "work"). Graphs also help you to understand why sometimes there is one answer, and sometimes many answers. I use Scientific Notebook or similar math software to graph the functions for me.

You can use this Online Graphing Calculator to solve the following equations (or check your solutions) .

Example 1

Solve the equation 2 cos θ − 1 = 0 for 0 ≤ θ < 2π.

Example 2

Revision Tip

Getting lost in this section? See the background at Trigonometric Functions of Any Angle

Graphically solve the equation

2 cos2x − sin x − 1 = 0

such that 0 ≤ θ < 2π.

Solving Equations Involving Multiples of θ

Example 3

Solve the equation sin 2θ = 0.8 for 0 ≤ θ < 2π.

Example 4

Solve the equation

`cos^2theta=1/16`

for 0 ≤ θ < 2π.

Example 5

Solve the equation

6 sin2θ − sin θ − 1 = 0

for 0 ≤ θ < 2π.

Example 6

Solve the equation

`cos{:x/2:}=1+cos\ x`

for 0 ≤ θ < 2π.

Example 7

Solve the equation

tan 2θ − cot 2θ = 0

for 0 ≤ θ < 2π.

Exercises

Note 1: "Analytically" means use the methods and formulas from previous sections. It means don't just use a graph to solve it.

Note 2: However, I always use a graph to check my analytical work. I can see immediately if some error has occurred. I encourage you to do the same!

1. Solve the trigonometric equation analytically

4 tan x − sec2 x = 0 (for 0 ≤ x < 2π)

2. Solve the trigonometric equation analytically for 0 ≤ x < 2π:

sin 2x cos x − cos 2x sin x = 0

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3. Solve the given trigonometric equation analytically and by graphical method (for 0 ≤ x < 2π):

sin 4x − cos 2x = 0