Explore

A trigonometric equation is an equation that includes the trigonometric ratio of a variable. Note these examples:

• 3 tan x − 7 = 5 is a trigonometric equation because you need to find the tangent ratio for x.
• x tan 7 − 5 = 3 is not a trigonometric equation because you are finding the tangent ratio of 7, not the tangent of a variable. 

In Try This 1 you began to look at the solutions to a trigonometric equation. You may have found that a trigonometric equation will often have more than one solution between 0 and 2π and may have unlimited solutions in the real numbers.

The next activity leads you through solving a trigonometric equation that has a reciprocal trigonometric ratio and a domain in radians.

Try This 2

To calculate a reference angle with your calculator for a trigonometric equation, you can enter the ratio and use the sin−1 , cos−1, or tan−1 button.

 

Example

tan θ = 0.50, so tan−1 0.50 = 25.560 511 8…°

 

θ ≈ 25.6° (Calculator is in degree mode.)

tan θ = 0.50, so tan−1 0.50 = 0.463 647 609… rad.

 

θ ≈ 0.46 (Calculator is in radian mode.)

Complete Solving Trigonometric Equations 1. Pay attention to how a reference angle is used to solve the equation.