Give the degree measure of $\theta$ if it exists. Do not use a calculator.
\[
\theta=\csc ^{-1}(-\sqrt{2})
\]
Select the correct choice below and fill in any answer boxes in your choice.
A. $\theta=$ (Type your answer in degrees.)
B. The answer doesn't exist.

Step 1 ：The cosecant function, \(\csc(\theta)\), is defined as \(1/\sin(\theta)\). Therefore, \(\csc^{-1}(-\sqrt{2})\) is asking for the angle whose sine is \(-1/\sqrt{2}\).

Step 2 ：The sine function takes on values between -1 and 1. The value \(-1/\sqrt{2}\) is within this range, so there should be an angle that satisfies this condition.

Step 3 ：However, we need to consider the range of the inverse cosecant function. The range of \(\csc^{-1}(x)\) is \((-\infty, -\pi/2] \cup [\pi/2, \infty)\) in radians, or \((-\infty, -90^\circ] \cup [90^\circ, \infty)\) in degrees. This means that the angle we're looking for must be either less than or equal to \(-90^\circ\) or greater than or equal to \(90^\circ\).

Step 4 ：The sine function is negative in the third and fourth quadrants. Therefore, the angle we're looking for must be in the third or fourth quadrant.

Step 5 ：In the unit circle, the sine of an angle is represented by the y-coordinate. The y-coordinate is \(-1/\sqrt{2}\) at two points: one in the third quadrant and one in the fourth quadrant.

Step 6 ：The angle in the third quadrant is \(180^\circ + 45^\circ = 225^\circ\), and the angle in the fourth quadrant is \(360^\circ - 45^\circ = 315^\circ\). Both of these angles are outside the range of the inverse cosecant function, so the answer doesn't exist.

Step 7 ：Final Answer: \(\boxed{\text{The answer doesn't exist.}}\)