Question

Evaluate $\sin {105}^0$

Answer 1

To find the value of $\sin {105}^0$

${105}^0={60}^0+{45}^0$

⇒$\sin {105}^0$$=\sin ({60}^0+{45}^0)$

Now, applying the identity $\sin (\mathrm{A}+\mathrm{B})=\sin \mathrm{A}\times \cos \mathrm{B}+\cos \mathrm{A}\times \sin \mathrm{B}$, we get

$\sin {105}^0$ $=\sin ({60}^0+{45}^0)$

$=\sin {60}^0\cos {45}^0+\cos {60}^0\sin {45}^0$

$=\frac{\surd 3}{2}\times \frac{1}{\surd 2}+\frac{1}{2}\times \frac{1}{\surd 2}$

$=\frac{\surd 3}{2\surd 2}+\frac{1}{2\surd 2}$

$=\frac{\surd 3+1}{2\surd 2}$

Thus, required value of $\sin {105}^0$$=\frac{\surd 3+1}{2\surd 2}$.