Find the exact value, if any, of the following composite function. Do not use a calculator. \( \cos ^{-1}\left[\cos \left(\frac{7 \pi}{10}\right)\right] \) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. \( \cos ^{-1}\left[\cos \left(\frac{7 \pi}{10}\right)\right]= \) (Simplify your answer. Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the expression.) B. It is not defined.
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Bonus Knowledge
To evaluate \( \cos ^{-1}\left[\cos \left(\frac{7 \pi}{10}\right)\right] \), we first need to recognize that the function \( \cos^{-1}(x) \), also known as the arccosine function, returns values in the range \( [0, \pi] \). The angle \( \frac{7 \pi}{10} \) is approximately \( 1.57 \times 7 = 3.49 \) radians, which falls within this range. However, the value of \( \frac{7 \pi}{10} \) can be expressed in a more familiar angle. Since \( \cos(x) \) is periodic and symmetric, \( \cos \left(\frac{7 \pi}{10}\right) \) does yield a valid output, specifically the corresponding angle in the range of the arccos function remains \( \frac{7 \pi}{10} \). Therefore, the exact value is: A. \( \cos^{-1}\left[\cos\left(\frac{7 \pi}{10}\right)\right] = \frac{7\pi}{10} \)
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