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)

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

Answer 2

Let $x = \left(\sin22 \frac{1^{\circ}}{2} + \cos^{2} 22 \frac{1^{\circ}}{2}\right)^{4}$ = \left\\{\left(\sin22 \frac{1^{\circ}}{2} + \cos22 \frac{1}{2}^{\circ}\right)^{2}\right\\}^{2} $= \left(\sin^{2} 22 \frac{1^{\circ}}{2} + \cos^{2} 22 \frac{1^{\circ}}{2} 2 \sin 22 \frac{1^{\circ}}{2} \cos22 \frac{1^{\circ}}{2}\right)^{2}$ $= \left(1+\sin45^{\circ}\right)^{2}$ $= \left(1 + \frac{1}{\sqrt{2}}\right)^{2} = \left( \frac{\sqrt{2} + 1}{\sqrt{2}}\right)^{2}$ $= \frac{2+1+2\sqrt{2}}{2} = \frac{3+2\sqrt{2}}{2}$