Question

Suppose theta= 11pi/12. How do you use the sum identity to find the exact value of sin theta?

Answer 1

The better way is, first we have to find the equivalent in degrees

$2\pi=360\º$

$\frac{11\pi}{12}=345\º$

now we can change this value to $-15\º$

how do we get an angle like this?!

$30\º-45\º=-15\º$

then

$sin(30\º-45\º)=sin(30\º)*cos(45\º)-sin(45\º)*cos(30\º)$

$\begin{Bmatrix}sin(30\º)&=&\frac{1}{2}\\\\sin(45\º)&=&cos(45\º)&=&\frac{\sqrt{2}}{2}}\end{matrix}\\\\cos(30\º)&=&\frac{\sqrt{3}}{2}\end{matrix}$

now we replace this values

$sin(-15\º)=\frac{1}{2}*\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}*\frac{\sqrt{3}}{2}$

$sin(-15\º)=\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}$

$\boxed{\boxed{sin(-15\º)=sin(345\º)=\frac{\sqrt{2}-\sqrt{6}}{4}}}$