Lesson 20: Trigonometric Equations

Table of Contents

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Topic. This lesson covers Chapter 20: Trigonometric Equations.

WeBWorK. There is one WeBWorK assignment on today’s material:

Trigonometry - Equations

Lesson Notes. (download pdf)

Question of the Day: If we know $\sin(x)=\frac{1}{2}$ , what do we know about the angle $x$ ?

Basic Trigonometric Equations

Equations of the form tan(x)=c

Example 20.1. Solve for $x$ : $\tan(x)=\sqrt{3}$

Observation 20.2. To solve $\tan(x)=c$ , we first determine one solution $x=\tan^{-1}(c).$ Then the general solution is given by:

$x=\tan^{-1}(c)+n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, ...$

EQUATIONS OF THE FORM cos(x)=c

Example 20.4. Solve for $x$ : $\cos(x)=\frac{1}{2}$

VIDEO: Example 20.4, cos(x)=c

Observation 20.5. To solve $\cos(x)=c$ , we first determine one solution $x=\cos^{-1}(c)$ . Then the general solution is given by:

$x=\pm\cos^{-1}(c)+2n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, ...$

EQUATIONS OF THE FORM sin(x)=c

Example 20.7. Solve for $x$ : $\sin(x)=\frac{\sqrt{2}}{2}$

VIDEO: Example 20.7, sin(x)=c

Observation 20.8. To solve $\sin(x)=c$ , we first determine one solution $x=\sin^{-1}(c)$ . Then the general solution is given by:

$x=(-1)^{n}\cdot\sin^{-1}(c)+n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, ...$

Summary: We summarize the different formulas used to solve the basic trigonometric equations in the following table.

Solve: $\sin(x)=c$	Solve: $\cos(x)=c$	Solve: $\tan(x)=c$
First, find one solution, that is: $\sin^{-1}(c)$ . Use: $\sin^{-1}(-c)=-\sin^{-1}(c)$	First, find one solution, that is: $\cos^{-1}(c)$ . Use: $\cos^{-1}(-c)=\pi-\cos^{-1}(c)$	First, find one solution, that is: $\tan^{-1}(c)$ . Use: $\tan^{-1}(-c)=-\tan^{-1}(c)$
The general solution is: $x=(-1)^{n} \sin^{-1}(c)+n \pi$	The general solution is: $x=\pm \cos^{-1}(c)+2 n \pi$	The general solution is: $x=\tan ^{-1}(c)+n \pi$
where $n=0,\pm 1,\pm 2, \ldots$	where $n=0,\pm 1,\pm 2, \ldots$	where $n=0,\pm 1,\pm 2, \ldots$

Example 20.10. Find the general solution of the equation, and state at least $5$ distinct solutions.
a) $\sin(x)=-\frac{1}{2}$
b) $\cos(x)=-\frac{\sqrt{3}}{2}$