Give your answer in degrees: [tex]$\cos ^{-1}\left(\cos \left(\cos ^{-1}(90)\right)\right)$[/tex]
Answer (1)
cos − 1 ( 90 ) is undefined because 90 is not within the domain of the inverse cosine function, which is [ − 1 , 1 ] . Therefore, the expression cos − 1 ( cos ( cos − 1 ( 90 ))) is undefined. u n d e f in e d
Explanation
Understanding the Problem We are asked to evaluate the expression cos − 1 ( cos ( cos − 1 ( 90 ))) and give the answer in degrees.
The domain of cos − 1 ( x ) is − 1 ≤ x ≤ 1 . This means that the input to the inverse cosine function must be between -1 and 1, inclusive. Since 90 is not in this interval, cos − 1 ( 90 ) is undefined.
Therefore, the entire expression cos − 1 ( cos ( cos − 1 ( 90 ))) is undefined.
Invalid Input Since the domain of the inverse cosine function, cos − 1 ( x ) , is − 1 ≤ x ≤ 1 , and we are given cos − 1 ( 90 ) as part of the expression, we immediately recognize that this is not a valid input because 90 is not within the domain of the inverse cosine function.
Conclusion Because cos − 1 ( 90 ) is undefined, the entire expression cos − 1 ( cos ( cos − 1 ( 90 ))) is undefined. There is no further calculation needed.
Examples
Consider a scenario where you're designing a sound system. The inverse cosine function helps determine the angle at which a speaker should be placed to achieve optimal sound distribution. However, if you input a value outside the function's domain (between -1 and 1), like trying to find an angle based on an impossible sound intensity ratio, the calculation becomes invalid, indicating a flaw in your setup parameters. This highlights the importance of understanding function domains in practical applications.
