How do you find the exact values of sin 75 degrees using the half angle formula?

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Nghi N. Share
Jul 15, 2015

Find: sin 75 deg
Answer: $\sin 75 = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$

Explanation:

Call sin 75 = sin t --> cos 150 = cos 2t
On the trig unit circle,
cos (150) = cos (180 - 30) = - cos 30 =$- \frac{\sqrt{3}}{2}$
$\cos 150 = \frac{- \sqrt{3}}{2} = 1 - 2 {\sin}^{2} t$
$2 {\sin}^{2} t = \frac{2 + \sqrt{3}}{2}$

$\sin \left(75\right) = \sin t = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$

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