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 $(-\mathrm{\infty },-\pi /2]\cup [\pi /2,\mathrm{\infty })$ in radians, or $(-\mathrm{\infty },-{90}^{\circ }]\cup [{90}^{\circ },\mathrm{\infty })$ 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: $\boxed{{\displaystyle \text{The answer doesn't exist.}}}$