SOLUTION: What is the exact value of sin(-75) using the half-angle formula?

Question 712635: What is the exact value of sin(-75) using the half-angle formula?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
$\sin\left(\frac{\theta}{2}\right) = -\sqrt{\frac{1-\cos(\theta)}{2}}$ ... Half Angle Formula (for this given angle)

$\sin\left(\frac{-150}{2}\right) = -\sqrt{\frac{1-\cos(-150)}{2}}$

$\sin\left(\frac{-150}{2}\right) = -\sqrt{\frac{1+\cos(150)}{2}}$

$\sin\left(\frac{-150}{2}\right) = -\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$

$\sin\left(\frac{-150}{2}\right) = -\sqrt{\frac{1}{2}+\frac{\sqrt{3}}{4}}$

$\sin\left(\frac{-150}{2}\right) = -\sqrt{\frac{2}{4}+\frac{\sqrt{3}}{4}}$

$\sin\left(\frac{-150}{2}\right) = -\sqrt{\frac{2+\sqrt{3}}{4}}$

$\sin\left(\frac{-150}{2}\right) = -\frac{\sqrt{2+\sqrt{3}}}{\sqrt{4}}$

$\sin\left(\frac{-150}{2}\right) = -\frac{\sqrt{2+\sqrt{3}}}{2}$

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

Answer: $\sin\left(-75\right) = -\frac{\sqrt{2+\sqrt{3}}}{2}$
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