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

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

$$2\pi=360º$$

$$\frac{11\pi}{12}=345º$$

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

how do we get an angle like this?

$$30º-45º=-15º$$

then

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

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

now we replace this values

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

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

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

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