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Lesson 20: Trigonometric Equations

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?

Lesson NOtes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Basic Trigonometric Equations

Equations of the form tan(x)=c

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

VIDEO: Example 20.1, tan(x)=c

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)=cSolve: \cos(x)=cSolve: \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 \piThe general solution is: x=\pm \cos^{-1}(c)+2 n \piThe general solution is: x=\tan ^{-1}(c)+n \pi
where n=0,\pm 1,\pm 2, \ldotswhere n=0,\pm 1,\pm 2, \ldotswhere 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

VIDEO: Example 20.11 – equations with trig functions (linear)

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

VIDEO: Example 20.12 – equations with trig functions (quadratic)

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