Evaluate the following without a calculator. csc(-(3pi )/(2)) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. D A. csc(-(3pi )/(2))=square (Simplify your answer.) B. The expression is undefined.

To evaluate \( \csc\left(-\frac{3\pi}{2}\right) \) without a calculator, we need to understand the properties of the cosecant function and the unit circle.<br /><br />The cosecant function is defined as:<br />\[ \csc(\theta) = \frac{1}{\sin(\theta)} \]<br /><br />First, let's find the angle \( -\frac{3\pi}{2} \) on the unit circle. This angle is equivalent to rotating \( \frac{3\pi}{2} \) radians clockwise from the positive x-axis. <br /><br />The angle \( \frac{3\pi}{2} \) radians corresponds to 270 degrees, which points to the negative y-axis. Therefore, \( -\frac{3\pi}{2} \) radians also points to the negative y-axis.<br /><br />Next, we need to determine the sine of \( -\frac{3\pi}{2} \):<br />\[ \sin\left(-\frac{3\pi}{2}\right) = \sin\left(\frac{3\pi}{2}\right) = -1 \]<br /><br />Now, we can find the cosecant:<br />\[ \csc\left(-\frac{3\pi}{2}\right) = \frac{1}{\sin\left(-\frac{3\pi}{2}\right)} = \frac{1}{-1} = -1 \]<br /><br />So, the correct choice is:<br />A. \( \csc\left(-\frac{3\pi}{2}\right) = -1 \)