Question

Find the value of sin $\left(22{\frac{1}{2}}^o\right)$

Answer 1

$\sin {\left(22\frac{1}{2}\right)}^{\circ }=\sin {\left(\frac{45}{2}\right)}^{\circ }$

We know that $2{\sin }^{2}A=1-\cos 2A$

So,

$2{\sin }^2{\left(\frac{45}{2}\right)}^{\circ }=1-\cos {45}^{\circ }$

$2{\sin }^2{\left(\frac{45}{2}\right)}^{\circ }=1-\frac{1}{\sqrt{2}}$

$2{\sin }^2{\left(\frac{45}{2}\right)}^{\circ }=1-\frac{\sqrt{2}}{2}$

${\sin }^2{\left(\frac{45}{2}\right)}^{\circ }=\frac{1}{2}-\frac{\sqrt{2}}{4}$

$\sin {\left(\frac{45}{2}\right)}^{\circ }=\sqrt{\frac{1}{2}-\frac{\sqrt{2}}{4}}$

$=\frac{\sqrt{2-\sqrt{2}}}{2}$