Answer
The required value of $\sin 15{}^\circ $ is $\frac{\sqrt{2-\sqrt{3}}}{2}$.
Work Step by Step
We have to find the value of $\sin 15{}^\circ $; the sine formulas are used which represent the cos identity. The term $\sin 15{}^\circ $ comes under the first quadrant where the value of all trigonometric functions is positive.
$\begin{align}
& \sin 15{}^\circ =\sin \frac{30{}^\circ }{2} \\
& =\sqrt{\frac{1-\cos 30{}^\circ }{2}} \\
& =\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}} \\
& =\sqrt{\frac{2-\sqrt{3}}{4}}
\end{align}$
Now, taking the square root of 4, that is 2, we get:
$\sin 15{}^\circ =\frac{\sqrt{2-\sqrt{3}}}{2}$
Thus, the value of $\sin 15{}^\circ $ is $\frac{\sqrt{2-\sqrt{3}}}{2}$.
