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Math  /  Trigonometry

QuestionFind the exact value. Write your answer using a simplified fraction and rationalize the denominator, if necessary. cos⁡−1(cos⁡7π6)=\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)= □\square

Studdy Solution

STEP 1

1. We are given the expression cos⁡−1(cos⁡7π6)\cos^{-1}\left(\cos \frac{7\pi}{6}\right).
2. The range of the cos⁡−1\cos^{-1} function is [0,π][0, \pi].
3. We need to find the exact value using a simplified fraction and rationalize the denominator if necessary.

STEP 2

1. Determine the angle 7π6\frac{7\pi}{6} in terms of its position on the unit circle.
2. Use the properties of the inverse cosine function to find the principal value.
3. Simplify the expression to find the exact value.

STEP 3

Determine the angle 7π6\frac{7\pi}{6} on the unit circle:
The angle 7π6\frac{7\pi}{6} is in the third quadrant because it is greater than π\pi but less than 3π2\frac{3\pi}{2}.

STEP 4

Find the reference angle for 7π6\frac{7\pi}{6}:
The reference angle is found by subtracting π\pi from 7π6\frac{7\pi}{6}:
7π6−π=7π6−6π6=π6 \frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}

STEP 5

Use the properties of the inverse cosine function:
Since cos⁡−1\cos^{-1} returns values in the range [0,π][0, \pi], and cos⁡7π6=−cos⁡π6\cos \frac{7\pi}{6} = -\cos \frac{\pi}{6}, we need to find an angle in the range [0,π][0, \pi] that has the same cosine value as −cos⁡π6-\cos \frac{\pi}{6}.
The angle in the second quadrant with the same cosine value is:
π−π6=6π6−π6=5π6 \pi - \frac{\pi}{6} = \frac{6\pi}{6} - \frac{\pi}{6} = \frac{5\pi}{6}

STEP 6

Simplify the expression:
The exact value of cos⁡−1(cos⁡7π6)\cos^{-1}\left(\cos \frac{7\pi}{6}\right) is:
5π6 \frac{5\pi}{6}
The exact value is:
5π6 \boxed{\frac{5\pi}{6}}

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