Chapter Contents

• Analytic Trigonometry
• 1. Trigonometric Identities
• 2. Sin, cos tan of Sum of Two Angles
• 3. Double Angle Formulas
• 4. Half Angle Formulas
• 5. Trigonometric Equations
• 6. Express in the form R sin (θ + α)
• 7. Inverse Trigonometric Functions
• Analytic Trigonometry Problem Solver

• Comments, Questions?

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From the math blog...

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 LiveMath to graph the functions for me.

Example 1

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

Answer

Example 2

Graphically solve the equation

2 cos²x − sin x − 1 = 0

such that 0 ≤ θ < 2π.

Answer

LiveMath answer:

LIVEMath

Solving Equations Involving Multiples of θ

Example 3

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

Answer

Example 4

Solve the equation for 0 ≤ θ < 2π.

Answer

Example 5

Solve the equation 6 sin²θ − sin θ − 1 = 0 for 0 ≤ θ < 2π.

Answer

Example 6

Solve the equation for 0 ≤ θ < 2π.

Answer

Example 7

Solve the equation tan 2θ − cot 2θ = 0 for 0 ≤ θ < 2π.

Answer

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!

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1. Solve the trigonometric equation analytically

4 tan x − sec² x = 0 (for 0 ≤ x < 2π)

Answer

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

sin 2x cos x − cos 2x sin x = 0

Answer

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

sin 4x − cos 2x = 0

Answer