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Degrees are not the only units we use to measure angles.  We also use radians.  Just as there are $360^\circ$ in a circle, there are $2\pi$ radians in a circle. Compute $\cos \frac{2 \pi}{7},$ where the angle $\frac{2 \pi}{7}$ is in radians.

 Oct 28, 2023
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To computecos72 p.m​, we can use the cosine addition formula: \begin{align*} \cos (x + y) &= \cos x \cos y - \sin x \sin y\ &= \cos x \cos y + \sqrt{1 - \cos^2 x} \sqrt{1 - \cos^2 y}, \end{align*}where xandyare any angles.

We know thatcos7Pi​=748​​andcos73 p.m​=721​​,so we can plug these values into the cosine addition formula to get: \begin{align*} \cos \frac{2 \pi}{7} &= \cos \left( \frac{\pi}{7} + \frac{3 \pi}{7} \right)\ &= \cos \frac{\pi}{7} \cos \frac{3 \pi}{7} - \sin \frac{\pi}{7} \sin \frac{3 \pi}{7}\ &= \frac{\sqrt{48}}{7} \cdot \frac{\sqrt{21}}{7} - \sqrt{1 - \left( \frac{\sqrt{48}}{7} \right)^2} \sqrt{1 - \left( \frac{\sqrt{21}}{7} \right)^2}\ &= \frac{6 \sqrt{3} - \sqrt{21}}{7}. \end{align*}Therefore, cos72 p.m​=763​−21​​​.

 Oct 28, 2023

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