Use the Half-angle Formulas to find the exact value of the expression \( \sin \left(-\frac{7 \pi}{8}\right) \) \( \sin \left(-\frac{7 \pi}{8}\right)=\square \) (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

To find \( \sin \left(-\frac{7 \pi}{8}\right) \), let's first rewrite the angle using the positive angle. We know that \( \sin(-x) = -\sin(x) \): \[ \sin \left(-\frac{7 \pi}{8}\right) = -\sin \left(\frac{7 \pi}{8}\right) \] Now, to evaluate \( \sin \left(\frac{7 \pi}{8}\right) \), we can use the half-angle formula. We have: \[ \frac{7 \pi}{8} = \frac{1}{2} \times \frac{7 \pi}{4} \] We can express \( \frac{7 \pi}{8} \) as \( \pi - \frac{\pi}{8} \), which implies: \[ \sin \left(\frac{7 \pi}{8}\right) = \sin \left(\pi - \frac{\pi}{8}\right) = \sin \left(\frac{\pi}{8}\right) \] Next, we can apply the half-angle formula for sine: \[ \sin \left(\frac{\pi}{8}\right) = \sqrt{\frac{1 - \cos \left(\frac{\pi}{4}\right)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \] Thus, substituting back into our original expression: \[ \sin \left(-\frac{7 \pi}{8}\right) = -\frac{\sqrt{2 - \sqrt{2}}}{2} \] The final answer is: \[ \sin \left(-\frac{7 \pi}{8}\right) = -\frac{\sqrt{2 - \sqrt{2}}}{2} \]