1. Tardigrade
  2. Question
  3. Mathematics
  4. What is the value of ( sin 22 (1°/2) + cos 22 (1° /2) )4 ?

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

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

Trigonometric Functions

Solution:

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} $