5.2.1: Solving Trigonometric Equations with Identities (Exercises)

Activity 4.2.1

Solving Trigonometric Equations with Identities

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

1. \(\csc ^{2} t=3\)

2. \(\cos ^{2} x=\frac{1}{4}\)

3. \(2 \sin \theta=-1\)

4. \(\tan x \sin x+\sin (-x)=0\)

5. \(9 \sin \omega-2=4 \sin ^{2} \omega\)

6. \(1-2 \tan (\omega)=\tan ^{2}(\omega)\)

II. For the following exercises, use basic identities to simplify the expression.

7. \(\sec x \cos x+\cos x-\frac{1}{\sec x}\)

8. \(\sin ^{3} x+\cos ^{2} x \sin x\)

III. For the following exercises, determine if the given identities are equivalent.

9. \(\sin ^{2} x+\sec ^{2} x-1=\frac{\left(1-\cos ^{2} x\right)\left(1+\cos ^{2} x\right)}{\cos ^{2} x}\)

10. \(\tan ^{3} x \csc ^{2} x \cot ^{2} x \cos x \sin x=1\)

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