Question

Use a half-angled identity to find the exact value of sin 75 degrees

Answer 1

Answer:

$\sin 75\degree=\frac{\sqrt{2+\sqrt{3}}}{2}$

Step-by-step explanation:

The haf-angle formula is given by:

$\sin \frac{1}{2}\theta =\sqrt{\frac{1-\cos \theta}{2} }$

$\sin 75\degree=\sin \frac{1}{2}(150\degree)$.

This implies that:

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

$\sin \frac{1}{2}(150\degree) =\sqrt{\frac{1--\frac{\sqrt{3}}{2}}{2}}$

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

We simplify the square root to obtain:

$\sin \frac{1}{2}(150\degree)=\frac{\sqrt{2+\sqrt{3}}}{2}$

Answer 2

Answer:

[A]    $\frac{\sqrt{2 + \sqrt{3} } }{2}$