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Math  /  Trigonometry

QuestionFind the exact solution of each xx for each inverse equation. a. 2cos⁡−1x=π2 \cos ^{-1} x=\pi b. 3tan⁡−1x=π3 \tan ^{-1} x=\pi

Studdy Solution

STEP 1

1. The equations involve inverse trigonometric functions.
2. We will use properties of inverse trigonometric functions to solve for x x .
3. The solutions will be in terms of exact values.

STEP 2

1. Solve part (a) for x x using the inverse cosine function.
2. Solve part (b) for x x using the inverse tangent function.

STEP 3

For part (a), start by isolating cos⁡−1x \cos^{-1} x :
2cos⁡−1x=π 2 \cos^{-1} x = \pi
Divide both sides by 2:
cos⁡−1x=π2 \cos^{-1} x = \frac{\pi}{2}

STEP 4

Use the property of the inverse cosine function to solve for x x . Recall that if cos⁡−1x=θ \cos^{-1} x = \theta , then cos⁡θ=x \cos \theta = x .
cos⁡(π2)=x \cos \left(\frac{\pi}{2}\right) = x
Since cos⁡(π2)=0 \cos \left(\frac{\pi}{2}\right) = 0 , we have:
x=0 x = 0

STEP 5

For part (b), start by isolating tan⁡−1x \tan^{-1} x :
3tan⁡−1x=π 3 \tan^{-1} x = \pi
Divide both sides by 3:
tan⁡−1x=π3 \tan^{-1} x = \frac{\pi}{3}

STEP 6

Use the property of the inverse tangent function to solve for x x . Recall that if tan⁡−1x=θ \tan^{-1} x = \theta , then tan⁡θ=x \tan \theta = x .
tan⁡(π3)=x \tan \left(\frac{\pi}{3}\right) = x
Since tan⁡(π3)=3 \tan \left(\frac{\pi}{3}\right) = \sqrt{3} , we have:
x=3 x = \sqrt{3}
The exact solutions for x x are: a. x=0 x = 0 b. x=3 x = \sqrt{3}

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