Question

(Sin 11 pi/36)(cos pi/18) - (cos 11pi/36)(sin pi/18). I have to find the exact value

Answer 1

From the trigonometric identity of the sum of angles

`\sin (\alpha-\beta)=\sin \alpha\cos \beta-\cos \alpha\sin \beta`

we ca note that

`\begin{gathered} \alpha=(11\pi)/(36) \\ \beta=(\pi)/(18) \end{gathered}`

Then, by substituting these values into the formula, we have

`\sin ((11\pi)/(36)-(\pi)/(18))=\sin (11\pi)/(36)\cos (\pi)/(18)-\cos (11\pi)/(36)\sin (\pi)/(18)`

which is the same given expression. Then, we have that

`\sin ((11\pi)/(36)(\pi)/(18))=\sin ((11\pi)/(36)-(2\pi)/(36))=\sin ((11\pi)/(36)-(2\pi)/(36))=\sin (9\pi)/(36)=\sin (\pi)/(4)`

Then, the answer is:

`\sin (\pi)/(4)=\frac{\sqrt[]{2}}{2}`