Question

Find the value of $\sin \left({105}^o\right)$.

Answer 1

We have that

$sin\left({105}^{\circ }\right)=sin({60}^{\circ }+{45}^{\circ })$

Because

$sin(a+b)=sina\cdot cosb+cosa\cdot sinb$

Hence,

$sin{105}^{\circ }=sin({60}^{\circ }+{45}^{\circ })=sin{60}^{\circ }\cdot cos{45}^{\circ }+cos{60}^{\circ }\cdot sin{45}^{\circ }=\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}+\frac{1}{2}.\frac{\sqrt{2}}{2}$

$=\frac{1}{4}\left(\sqrt{6}+\sqrt{2}\right)$