Questions: Find the exact value of the real number y if it exists. Do not use a calculator. y = csc^(-1)(-1) Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. y = csc^(-1)(-1) = (Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.) B. csc^(-1)(-1) does not exist.

$Find the exact value of the real number y if it exists. Do not use a calculator. y = csc^(-1)(-1) Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. y = csc^(-1)(-1) = (Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.) B. csc^(-1)(-1) does not exist.$

Transcript text: Find the exact value of the real number $y$ if it exists. Do not use a calculator. \[ y=\csc ^{-1}(-1) \] Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. $y=\csc ^{-1}(-1)=$ $\square$ (Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.) B. $\csc ^{-1}(-1)$ does not exist.

Solution

Solution Steps

Step 1: Understand the Problem

We are tasked with finding the exact value of $ y = \csc^{-1}(-1) $. The inverse cosecant function, $ \csc^{-1}(x) $, returns the angle whose cosecant is $ x $. The range of $ \csc^{-1}(x) $ is typically $ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] $.

Step 2: Recall the Definition of Cosecant

The cosecant function is defined as: \[ \csc(\theta) = \frac{1}{\sin(\theta)} \] We need to find an angle $ \theta $ such that: \[ \csc(\theta) = -1 \] This implies: \[ \sin(\theta) = -1 \]

Step 3: Determine the Angle

The sine function equals $-1$ at: \[ \theta = \frac{3\pi}{2} + 2\pi k \quad \text{for integer } k \] However, the principal value of $ \csc^{-1}(x) $ lies in the range $ \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] $. The angle $ \frac{3\pi}{2} $ is not within this range, but its equivalent angle $ -\frac{\pi}{2} $ is.