Module 4: Now You Try II

Compute $LaTeX: \sin\left(\frac{\pi}{8}\right)$ $\sin\left(\frac{\pi}{8}\right)$ without a calculator.

Solution:

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Using the half angle trig identity for sine, we can write $LaTeX: \sin\left(\frac{\pi}{8}\right)=\sin\left(\frac{\pi}{2(4)}\right)=\sin\left(\frac{1}{2}(\frac{\pi}{4})\right)= {\color{red}+}\sqrt{\frac{1-\cos(\frac{\pi}{4}) }{2} } =\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2} }.\\$ $\sin\left(\frac{\pi}{8}\right)=\sin\left(\frac{\pi}{2(4)}\right)=\sin\left(\frac{1}{2}(\frac{\pi}{4})\right)= {\color{red}+}\sqrt{\frac{1-\cos(\frac{\pi}{4}) }{2} } =\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2} }.\\$ We chose the positive solution, since $LaTeX: \frac{\pi}{8}$ $\frac{\pi}{8}$ is in the first quadrant and has a positive sine value.