Find the exact value of \( \sin \left(\frac{13 \pi}{8}\right) \). A. \( \sqrt{\frac{2+\sqrt{2}}{4}} \) B. \( \sqrt{\frac{2-\sqrt{2}}{4}} \) C. \( -\sqrt{\frac{2+\sqrt{2}}{4}} \) D. \( -\sqrt{\frac{2-\sqrt{2}}{4}} \)

To find the exact value of \( \sin \left(\frac{13 \pi}{8}\right) \), we can first rewrite \( \frac{13 \pi}{8} \) in terms of an angle we can easily work with. We notice that: \[ \frac{13 \pi}{8} = \pi + \frac{5 \pi}{8} \] Since \( \sin(\pi + x) = -\sin(x) \), we have: \[ \sin\left(\frac{13 \pi}{8}\right) = -\sin\left(\frac{5 \pi}{8}\right) \] Now, \( \frac{5 \pi}{8} \) is in the second quadrant where the sine function is positive, and we can express it as: \[ \frac{5 \pi}{8} = \frac{\pi}{2} - \frac{3 \pi}{8} \] Using the sine identity, \( \sin\left(\frac{\pi}{2} - x\right) = \cos(x) \), we have: \[ \sin\left(\frac{5 \pi}{8}\right) = \cos\left(\frac{3 \pi}{8}\right) \] Next, we find \( \cos\left(\frac{3 \pi}{8}\right) \) using the half-angle formula. We know: \[ \cos\left(\frac{3 \pi}{8}\right) = \cos\left(\frac{\pi}{4} + \frac{\pi}{8}\right) \] Using the cosine addition formula: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] Substituting \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{8} \): \[ \cos\left(\frac{3 \pi}{8}\right) = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{\pi}{8}\right) - \sin\left(\frac{\pi}{4}\right) \sin\left(\frac{\pi}{8}\right) \] Knowing that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) and \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), we can find \( \sin(\frac{\pi}{8}) \) and \( \cos(\frac{\pi}{8}) \) using their half-angle formulas: \[ \sin\left(\frac{\pi}{8}\right) = \sqrt{\frac{1 - \cos\left(\frac{\pi}{4}\right)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \] \[ \cos\left(\frac{\pi}{8}\right) = \sqrt{\frac{1 + \cos\left(\frac{\pi}{4}\right)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2} \] Now, substituting back into our cosine addition formula: \[ \cos\left(\frac{3 \pi}{8}\right) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2 + \sqrt{2}}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2 - \sqrt{2}}}{2} \] \[ = \frac{\sqrt{2}}{4} \left( \sqrt{2 + \sqrt{2}} - \sqrt{2 - \sqrt{2}} \right) \] However, since we already know \( \sin\left(\frac{13 \pi}{8}\right) = -\sin\left(\frac{5 \pi}{8}\right) = -\cos\left(\frac{3 \pi}{8}\right) \), and looking for the exact answers from the options provided, simplifying leads us to conclude: \[ \sin\left(\frac{13 \pi}{8}\right) = -\sqrt{\frac{2 - \sqrt{2}}{4}} \quad \text{(Choice D)} \] Thus, the exact value of \(