Question

Without using a trigonometric table find $\sin {105}^{\circ }$.

Answer 1

$\sin {105}^{\circ }$

$=\sin ({90}^{\circ }+{15}^{\circ })$

$=\cos {15}^{\circ }$

$=\cos ({45}^{\circ }-{30}^{\circ })$

$=\cos {45}^{\circ }\cos {30}^{\circ }+\sin {45}^{\circ }\sin {30}^{\circ }$

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

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