5.6.1: Solving Trigonometric Equations (Exercises)

Activity 4.6.1

Solving Trigonometric Equations

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I. For the following exercises, find all exact solutions on the interval \([0,2 \pi)\).

1. \(\tan x+1=0\)

2. \(2 \sin (2 x)+\sqrt{2}=0\)

II. For the following exercises, find all exact solutions on the interval \([0,2 \pi)\).

1. \(2 \sin ^{2} x-\sin x=0\)

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

3. \(2 \sin ^{2} x+5 \sin x+3=0\)

4. \(\cos x-5 \sin (2 x)=0\)

5. \(\frac{1}{\sec ^{2} x}+2+\sin ^{2} x+4 \cos ^{2} x=0\)

III. For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval \([0,2 \pi)\). Round to four decimal places.

1. \(\sqrt{3} \cot ^{2} x+\cot x=1\)

2. \(\csc ^{2} x-3 \csc x-4=0\)

IV. For the following exercises, graph each side of the equation to find the approximate solutions on the interval \([0,2 \pi)\).

1. \(20 \cos ^{2} x+21 \cos x+1=0\)

2. \(\sec ^{2} x-2\) sec \(x=15\)

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