Question

How do you find the exact values of cos 22.5 degree using the half angle formula?

Answer 1

Hint:
In this question, we would use the formula of $\cos (2x)=2{{\cos }^{2}}x-1$. This is a double angle formula for cos. For finding the exact value we will divide the 45 degrees by 2 so that we can get 22.5 degrees. So let us see how we can solve this problem.

Complete step by step solution:
For solving the above problem we will use the formula of cos double angle which is $\cos (2x)=2{{\cos }^{2}}x-1$ . On arranging this formula we will get ${{\cos }^{2}}x=\dfrac{1+\cos (2x)}{2}$ . On dividing the angles of both sides by we get ${{\cos }^{2}}(\dfrac{x}{2})=\dfrac{1+\cos x}{2}$
Applying square root on both sides,
 $\cos (\dfrac{x}{2})=\pm \sqrt{\dfrac{1+\cos x}{2}}$ , since we have to find the value of 22.5 degrees, therefore, it will lie on the first quadrant. Hence, we will only consider the plus sign because in the first quadrant all the trigonometric identities are positive.
$\Rightarrow \cos (\dfrac{{{45}^{\circ }}}{2})=\sqrt{\dfrac{1+\cos {{45}^{\circ }}}{2}}$
The value of cos45 is $\dfrac{1}{\sqrt{2}}$ , after multiplying $\sqrt{2}$ with the numerator and denominator we get $\dfrac{\sqrt{2}}{2}$
 $=\sqrt{\dfrac{1+(\dfrac{\sqrt{2}}{2})}{2}}$
 $=\sqrt{\dfrac{(\dfrac{2+\sqrt{2}}{2})}{2}}$
After solving we get,
 $=\sqrt{\dfrac{2+\sqrt{2}}{4}}$
 $=\dfrac{\sqrt{2+\sqrt{2}}}{2}$

Therefore, the value of $\cos {{22.5}^{\circ }}=\dfrac{\sqrt{2+\sqrt{2}}}{2}$.

Note:
In the above solution, we have used the formula $\cos (2x)=2{{\cos }^{2}}x-1$ . So let’s see how we have simplified this to ${{\cos }^{2}}x=\dfrac{1+\cos (2x)}{2}$ . From this formula $\cos (2x)=2{{\cos }^{2}}x-1$ ,
Take the -1 on the other side of equal to get
 $\cos (2x)+1=2{{\cos }^{2}}x$
 On dividing both sides with 2 we get,
 $\dfrac{\cos (2x)+1}{2}={{\cos }^{2}}x$
Therefore, we get ${{\cos }^{2}}x=\dfrac{1+\cos (2x)}{2}$.