Answer
The required value of $\sin 105{}^\circ $ is $\frac{\sqrt{2+\sqrt{3}}}{2}$.
Work Step by Step
We have to find the value of $\sin 105{}^\circ $; the formula is used which represents the sin identity. The term $\sin 105{}^\circ $ comes under the second quadrant where the value of sine and cosec functions is positive and the rest of the functions are negative. And the value of the second quadrant angle is between $90{}^\circ \,\text{to}\,18\text{0}{}^\circ $.
$\begin{align}
& \sin 105{}^\circ =\sin \frac{210{}^\circ }{2} \\
& =\sqrt{\frac{1-\cos 210{}^\circ }{2}} \\
& =\sqrt{\frac{1-\left( -\frac{\sqrt{3}}{2} \right)}{2}} \\
& =\sqrt{\frac{2+\sqrt{3}}{4}}
\end{align}$
Now, taking the square root of 4, that is 2, we get:
$\sin 105{}^\circ =\frac{\sqrt{2+\sqrt{3}}}{2}$
Thus, the value of $\sin 105{}^\circ $ is $\frac{\sqrt{2+\sqrt{3}}}{2}$.
