Questions: Use reference angles to find the exact value of the following expression. csc (-π/4) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. csc (-π/4)= (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize the denominator.) B. The answer is undefined.

Solution Steps

To find \( \csc \left(-\frac{\pi}{4}\right) \), we first identify the reference angle. The reference angle for \( -\frac{\pi}{4} \) is \( \frac{\pi}{4} \).

The sine of the reference angle \( \frac{\pi}{4} \) is given by: \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

The cosecant function is the reciprocal of the sine function: \[ \csc\left(-\frac{\pi}{4}\right) = \frac{1}{\sin\left(-\frac{\pi}{4}\right)} \] Since \( \sin\left(-x\right) = -\sin\left(x\right) \), we have: \[ \sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \] Thus, \[ \csc\left(-\frac{\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \]

Final Answer