Alyssa Hanson

2023-03-18

How to find the exact values of cos(5pi/12) using the half angle formula?

Krasiniecb8rw

Using the formula for a half-angle:
$\text{XXXX}$$\mathrm{cos}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1+\mathrm{cos}\left(\theta \right)}{2}}$
If $\frac{\theta }{2}=\frac{5\pi }{12}$
$\text{XXXX}$then $\theta =\frac{5\pi }{6}$
Note that $\frac{5\pi }{6}$ is a standard angle in quadrant 2 with a reference angle of $\frac{\pi }{6}$
so $\mathrm{cos}\left(\frac{5\pi }{6}\right)=-\mathrm{cos}\left(\frac{\pi }{6}\right)=-\frac{\sqrt{3}}{2}$
Therefore
$\text{XXXX}\mathrm{cos}\left(\frac{5\pi }{12}\right)=±\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}$
$\text{XXXXXXXXXXX}=±\sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}}$
$\text{XXXXXXXXXXX}=±\sqrt{\frac{2-\sqrt{3}}{4}}$
$\text{XXXXXXXXXXX}=±\frac{\sqrt{2-\sqrt{3}}}{2}$
Since $\frac{5\pi }{12}<\frac{\pi }{2}$
$\text{XXXX}$$\frac{5\pi }{12}$ is in quadrant 1
$\text{XXXX}$$\to \mathrm{cos}\left(\frac{5\pi }{12}\right)$ is positive
$\text{XXXX}$(the negative solution is extraneous)

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