Evaluate the following without a calculator. csc(-(5pi )/(2)) Select the correct choice below and, if necessary fill in the answer box to complete your choice. A. csc(-(5pi )/(2))=square (Simplify your answer.) B. The expression is undefined

To evaluate \( \csc\left(-\frac{5\pi}{2}\right) \) without a calculator, we need to understand the properties of the cosecant function and the angle involved.<br /><br />First, let's recall that the cosecant function is the reciprocal of the sine function:<br />\[ \csc(x) = \frac{1}{\sin(x)} \]<br /><br />Next, we need to find the sine of the angle \( -\frac{5\pi}{2} \).<br /><br />The angle \( -\frac{5\pi}{2} \) can be converted to a positive angle by adding \( 2\pi \) (since adding \( 2\pi \) to an angle results in a coterminal angle):<br />\[ -\frac{5\pi}{2} + 2\pi = -\frac{5\pi}{2} + \frac{4}{2} = -\frac{\pi}{2} \]<br /><br />So, \( \sin\left(-\frac{5\pi}{2}\right) = \sin\left(-\frac{\pi}{2}\right) \).<br /><br />We know that:<br />\[ \sin\left(-\frac{\pi}{2}\right) = -1 \]<br /><br />Now, we can find the cosecant:<br />\[ \csc\left(-\frac{5\pi}{2}\right) = \csc\left(-\frac{\pi}{2}\right) = \frac{1}{\sin\left(-\frac{\pi}{2}\right)} = \frac{1}{-1} = -1 \]<br /><br />Therefore, the correct choice is:<br />A. \( \csc\left(-\frac{5\pi}{2}\right) = -1 \)