Problem 116 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \(y=\sin x\) has an inverse function if \(x\) is restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\) they should make restrictions easier to remember by also using \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) as the restriction for \(y=\cos x\).

Step-by-Step Solution

Answer

The statement does not make sense. Using the same restriction as the sine function for the inverse cosine function would be incorrect as it would yield wrong values because inverse cosine function is properly defined for the interval \([0, \pi]\), not \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 1: Understand the statement

The statement says that since the inverse sine function has a restriction of \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the inverse cosine should be restricted to the same range for consistency.

Step 2: Analyze the cosine function

The cosine function produces its maximum and minimum values of 1 and -1 at \(0\) and \(\pi\) respectively within the interval \([0, \pi]\). So the appropriate restriction for the inverse cosine function is \([0, \pi]\) to make it a function.

Step 3: Compare restrictions

Comparing the restrictions, we see that they are not the same. The restriction for the inverse sine function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\) and for the inverse cosine function is \([0, \pi]\). Therefore, using the same restrictions would not work because it would yield incorrect outputs for the inverse cosine.