Question

NEED HELP ASAP!!!!
Find the exact value by using a half-angle identity.


sin five pi divided by twelve

Answer 1

Answer:

$\frac{1}{2}\sqrt{2+\sqrt{3}}$

Step-by-step explanation:

we know that

An half-angle identity is equal to

$sin(\frac{\theta}{2})=(+/-)\sqrt{\frac{1-cos(\theta)}{2}}$

we have

$sin(\frac{5\pi}{12})$

The angle $\frac{5\pi}{12}=75^o$  ----> belong to the First Quadrant, so the value of the sine is positive

Let

$\frac{\theta}{2}=\frac{5\pi}{12}$

so

${\theta=\frac{5\pi}{6}$

$sin(\frac{5\pi}{12})=\sqrt{\frac{1-cos(\theta)}{2}}$

$cos(\theta)=cos(\frac{5\pi}{6})=cos(150^o)=-\frac{\sqrt{3}}{2}$

substitute

$sin(\frac{5\pi}{12})=\sqrt{\frac{1-(-\frac{\sqrt{3}}{2})}{2}}$

$sin(\frac{5\pi}{12})=\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$

$sin(\frac{5\pi}{12})=\sqrt{\frac{2+\sqrt{3}}{4}$

$sin(\frac{5\pi}{12})=\frac{1}{2}\sqrt{2+\sqrt{3}}$