Lesson 20: Trigonometric Equations

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

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 $x$: $\cos(x)=\frac{1}{2}$

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 $x$: $\sin(x)=\frac{\sqrt{2}}{2}$

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) $\cos(x)=-\frac{\sqrt{3}}{2}$

Equations involving trigonometric functions

Example 20.11. Solve for $x$
a) $2 \sin (x)-1=0$
b) $\sec (x)=-\sqrt{2}$
c) $7 \cot (x)+3=0$

Example 20.12. Solve for $x$.
a) $\tan ^{2}(x)+2 \tan (x)+1=0 \quad$ b) $2 \cos ^{2}(x)-1=0$