Question

Exact value of addition and subtraction formulas / sin cos tan

Answer 1

Take into account that:

$\frac{5\pi}{4}+\frac{\pi}{3}=\frac{15\pi+4\pi}{12}=\frac{19}{12}\pi$

Then, for the given sine value, you can write:

$\sin (\frac{19}{12}\pi)=\sin (\frac{5}{4}\pi+\frac{1}{3}\pi)$

Now, consider that the sine of a sum is:

$\sin (a+b)=\sin a\cdot\cos b+\cos a\cdot\sin b$

Then, by applying the previous identity to the given expression, you obtain:

$\sin (\frac{5}{4}\pi+\frac{1}{3}\pi)=\sin (\frac{5}{4}\pi)\cos (\frac{1}{3}\pi)+\cos (\frac{5}{4}\pi)\sin (\frac{1}{3}\pi)$

Consider now that:

$\begin{gathered} \sin (\frac{5}{4}\pi)=-\frac{\sqrt[]{2}}{2} \\ \cos (\frac{1}{3}\pi)=\frac{1}{2} \\ \sin (\frac{1}{3}\pi)=\frac{\sqrt[]{3}}{2} \\ \cos (\frac{5}{4}\pi)=-\frac{\sqrt[]{2}}{2} \end{gathered}$

Then, for the given expression of the question, you get:

$\begin{gathered} \sin (\frac{5}{4}\pi+\frac{1}{3}\pi)=(-\frac{\sqrt[]{2}}{2})(\frac{1}{2})+(-\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2}) \\ =-\frac{\sqrt[]{2}}{4}-\frac{\sqrt[]{2}\sqrt[]{3}}{4}=-\frac{(1+\sqrt[]{3})\sqrt[]{2}}{4} \end{gathered}$

The pervious result is the answer to the given expression.