Question

What is the value of ${\left(\sin 22\frac{1^{\circ }}{2}+\cos 22\frac{1^{\circ }}{2}\right)}^4$ ?

• A. $\frac{3+2\sqrt{2}}{2}$
• B. $\frac{1+2\sqrt{2}}{2}$
• C. $\frac{3\sqrt{2}+2}{2}$
• D. 1

Answer 1

Answer: (A)

Let $x={\left(\sin 22\frac{1^{\circ }}{2}+{\cos }^{2}22\frac{1^{\circ }}{2}\right)}^4$
$={\left\{{\left(\sin 22\frac{1^{\circ }}{2}+\cos 22{\frac{1}{2}}^{\circ }\right)}^2\right\}}^2$
$={\left({\sin }^{2}22\frac{1^{\circ }}{2}+{\cos }^{2}22\frac{1^{\circ }}{2}2\sin 22\frac{1^{\circ }}{2}\cos 22\frac{1^{\circ }}{2}\right)}^2$
$={\left(1+\sin 45^{\circ }\right)}^2$
$={\left(1+\frac{1}{\sqrt{2}}\right)}^2={\left(\frac{\sqrt{2}+1}{\sqrt{2}}\right)}^2$
$=\frac{2+1+2\sqrt{2}}{2}=\frac{3+2\sqrt{2}}{2}$