A. $\cos ^{-1}\left[\cos \left(\frac{4 \pi}{5}\right)\right]=$
(Simplify your answer. Type an exact answer, using $\pi$ as need
Answer (1)
The problem asks to evaluate cos − 1 ( cos ( 5 4 π )) .
The range of cos − 1 ( x ) is [ 0 , π ] .
Since 5 4 π is within the range [ 0 , π ] , the expression simplifies to 5 4 π .
The final answer is 5 4 π .
Explanation
Understanding the Problem We are asked to evaluate the expression cos − 1 ( cos ( 5 4 π )) . The inverse cosine function, denoted as cos − 1 ( x ) , has a range of [ 0 , π ] . This means that the output of the cos − 1 function must be between 0 and π .
Checking the Range We need to check if 5 4 π lies within the range of the inverse cosine function, which is [ 0 , π ] . Since 5 4 is a fraction between 0 and 1, 5 4 π is a positive value. Also, 5 4 π is less than π because 5 4 < 1 . Therefore, 0 ≤ 5 4 π ≤ π .
Evaluating the Expression Since 5 4 π is within the range of the inverse cosine function, we can directly evaluate the expression: cos − 1 ( cos ( 5 4 π )) = 5 4 π .
Final Answer Therefore, the simplified answer is 5 4 π .
Examples
Imagine you are designing a satellite dish. The angle at which the signal is received is 5 4 π radians. To optimize the signal processing, you need to find the equivalent angle within the range of 0 to π radians, which is the range of the inverse cosine function. This problem demonstrates how the inverse cosine function helps you find equivalent angles within a specific range, which is crucial in various engineering and physics applications.
