Question

How do you use a half – angle formula to find the exact value of $\cos 22.5$?

Answer 1

Hint: In this problem we have given some trigonometric functions. And we are asked to find the exact value of the given trigonometric function and also we have mentioned that we need to find the exact value of the given trigonometric function with the use of half – angle formula. Half – angle formulas allow the expression of trigonometric functions of angles to simply the functions and make it easier to perform more complex calculations.

Formula used: $\cos \dfrac{\theta }{2} = \pm \sqrt {\dfrac{{1 - \cos \theta }}{2}} $

Complete step-by-step solution:
Given trigonometric function is $\cos 22.5$
Using half – angle formula for $\cos $
$ \Rightarrow \cos \dfrac{\theta }{2} = \pm \sqrt {\dfrac{{1 - \cos \theta }}{2}} $
Let $\dfrac{\theta }{2} = 22.5 - - - - - (1)$
$ \Rightarrow \theta = 2 \times 22.5$
$ \Rightarrow \theta = {45^\circ }$
Now apply $\cos $ on both sides, we get
$ \Rightarrow \cos \theta = \cos {45^\circ }$
Now by using (1) in the half – angle formula, we get
$ \Rightarrow \cos {22.5^\circ } = + \sqrt {\dfrac{{1 + \cos {{45}^\circ }}}{2}} - - - - - \left( 2 \right)$
We know that $\cos {45^\circ } = \dfrac{1}{{\sqrt 2 }}$
Now equation (2) becomes,
$ \Rightarrow \cos {22.5^\circ } = + \sqrt {\dfrac{{1 - \dfrac{1}{{\sqrt 2 }}}}{2}} $
$ \Rightarrow \cos {22.5^\circ } = \sqrt {\dfrac{{\dfrac{{\sqrt 2 - 1}}{{\sqrt 2 }}}}{2}} $
$ \Rightarrow \cos {22.5^\circ } = \sqrt {\dfrac{{\sqrt 2 - 1}}{{2\sqrt 2 }}} $,
On calculating this we get,
$\cos {22.5^\circ } \approx + 0.3827$

Therefore, the exact value of $\cos 22.5$ by using half angle formula is $ + 0.3827$

Note: Steps to evaluate a trigonometric function using half – angle formulas are first rewrite the trigonometric function and the angle as half of a unit circle value and then we have to determine the sign of the trigonometric function and then we substitute the angle value into the right identity. After this we replace $\cos x$ with its actual value and then we simplify the half – angle formula to solve.
In this problem we used the half – angle formula of cosine to find the exact value of $\cos 22.5$. The most important step in this problem we have done is taking $\dfrac{\theta }{2} = 22.5$. From this we found the value of $\cos \theta $ and then we substituted these values in the half – angle formula for cosine and by this way we got our required answer.