Tag: trigonometric equations

Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.

Lesson Date: Thursday, April 23rd.

Topic: This lesson covers Chapter 20: Trigonometric Equations.

WeBWorK: There is one WeBWorK assignment on today’s material, due next Thursday 4/30:

Trigonometry – Equations

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) $2 \cos ^{2}(x)-1=0$