\( \sin ^{2}\left(\frac{\pi}{8}\right)+\sin ^{2}\left(\frac{3 \pi}{8}\right)+\sin ^{2}\left(\frac{5 \pi}{8}\right)+\sin ^{2}\left(\frac{7 \pi}{8}\right) \) Correction

Did you know that the angles \( \frac{\pi}{8} \), \( \frac{3\pi}{8} \), \( \frac{5\pi}{8} \), and \( \frac{7\pi}{8} \) correspond to specific points on the unit circle? Their sine values represent the vertical distances from the x-axis to the circle at those angles, making it easy to visualize why these particular sine squares are often analyzed together. Add some geometry to your math! When calculating \( \sin^{2}\left(\frac{\pi}{8}\right) + \sin^{2}\left(\frac{3 \pi}{8}\right) + \sin^{2}\left(\frac{5 \pi}{8}\right) + \sin^{2}\left(\frac{7 \pi}{8}\right) \), you might notice some repeated values. Specifically, \( \sin\left(\frac{5\pi}{8}\right) = \sin\left(\frac{3\pi}{8}\right) \) and \( \sin\left(\frac{7\pi}{8}\right) = \sin\left(\frac{\pi}{8}\right) \). This can simplify calculations and reduce potential mistakes—always good to double-check your inputs!