Find the exact value, if any, of the following composite function. Do not use a calculator. [tex]$\cos ^{-1}\left[\cos \left(\frac{7 \pi}{8}\right)\right]$[/tex]
A. [tex]$\cos ^{-1}\left[\cos \left(\frac{7 \pi}{8}\right)\right]=$[/tex] (Simplify your answer. Type an exact answer, using [tex]$\pi$[/tex] as needed. Use integers or fractions.)
B. It is not defined.
Answer (1)
Check if 8 7 π is within the range of cos − 1 ( x ) , which is [ 0 , π ] .
Since 0 ≤ 8 7 π ≤ π , the composite function simplifies to cos − 1 ( cos ( 8 7 π )) = 8 7 π .
The exact value of the composite function is 8 7 π .
Therefore, the final answer is 8 7 π .
Explanation
Understanding the Problem We are asked to find the exact value of the composite function cos − 1 [ cos ( 8 7 π ) ] . The inverse cosine function, denoted as cos − 1 ( x ) , has a range of [ 0 , π ] . This means that the output of the inverse cosine function must be between 0 and π (inclusive).
Checking the Range We need to check if 8 7 π lies within the range of the inverse cosine function, which is [ 0 , π ] . Since 8 7 is between 0 and 1, 8 7 π is indeed between 0 and π . Therefore, we can proceed with evaluating the composite function.
Evaluating the Composite Function Since 8 7 π is within the range [ 0 , π ] , we can directly evaluate the composite function using the property that cos − 1 ( cos ( x )) = x when x is in the range [ 0 , π ] . In this case, x = 8 7 π , so we have:
Final Calculation cos − 1 [ cos ( 8 7 π ) ] = 8 7 π
Conclusion Therefore, the exact value of the composite function is 8 7 π .
Examples
Imagine you're designing a satellite dish. The angle at which the signal is received, represented by 8 7 π in this problem, needs to be correctly interpreted by the receiver. The receiver uses the inverse cosine function to determine the angle. Understanding how the composite function cos − 1 ( cos ( x )) works ensures the receiver accurately calculates the angle, allowing the satellite dish to properly focus and amplify the signal. This is crucial for clear communication and data transmission.
