Solving Trigonometric Equations Using Identities

Section 5.3 Solving Trigonometric Equations Using Identities

Goals:

Be able to solve a trigonometric equation by first transforming the equation using trigonometric identities.
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One way in which trigonometric identities are used is in solving equations. Sometimes, an equation can be transformed to one that is easier to solve using a trigonometric identity.

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Example 5.3.1.

Solve \(\cos^2 x - 2 \sin x = -2\) on the interval \([0, 2\pi)\text{.}\)
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Solve \(\cos^2 x - 2 \sin x = -2\) on the interval \((-\infty, \infty)\text{.}\)
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Problem 5.3.2.

Solve \(2\cos x - \sec x = 0\) on the interval \([0, 2\pi)\text{.}\)
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Solve \(2\cos x - \sec x = 0\) on \([0, 2\pi)\) on the interval \((-\infty, \infty)\text{.}\)
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Problem 5.3.3.

Use trigonometric identities to solve each equation on the interval \([0, 2\pi)\text{.}\)

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\(\displaystyle 2=4+\csc(-\theta)\)
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\(\displaystyle \sin(\theta-\pi)=\frac{\sqrt{2}}{2}\)
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\(\displaystyle \cos 2\theta=\cos \theta\)
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\(\displaystyle \sin^2 \theta - \sin^2 2\theta=0\)
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\(\displaystyle 0=\cos \theta - \sin \frac{\theta}{2}\)
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Problem 5.3.4.

Solve each equation below on the interval \([0, 2\pi)\text{.}\) Some equations will require you to use one or more trigonometric identities.

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\(\displaystyle -2=-1-4\cos^2\theta\)
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\(\displaystyle 3+\tan^2\theta=3\sec\theta\)
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\(\displaystyle -\tan\theta=-\tan\theta \sin\theta\)
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\(\displaystyle 2\cos^2\theta=1-\cos 2\theta\)
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\(\displaystyle -5=-4+\sec\theta\)
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\(\displaystyle -\tan^2\theta+2\tan\theta+3=4\)
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