Question

How do you use the double angle or half angle formulas to simplify $\cos 22.5$ ?

Answer 1

Hint : Here in this question, we have to find the exact value of $\cos 22.5$ . To find this exact value by using the half angle formula of cos i.e., $\cos \left( {\dfrac{\theta }{2}} \right) = \pm \sqrt {\dfrac{{1 + \cos \left( \theta \right)}}{2}}$ con simplification the angle $\theta$ and by the table of value of standard angle of trigonometric ratios and on further simplification we get the required solution.

Complete step by step solution:
Consider the given trigonometric ratio
$\Rightarrow \,\cos 22.5$
Now find the exact value of this.
Here angle $\theta = 22.5$ can be also written as $\theta = \dfrac{{45}}{2}$ .
This can be find, out by using the half angle formula.
Half-angle formulas allow the expression of trigonometric functions of angles equal to $\dfrac{\alpha }{2}$ in terms of $\alpha$ , which can simplify the functions and make it easier to perform more complex calculations, such as integration, on them. Half-angle formulas are especially useful in finding the values of unknown trigonometric functions.
Now, consider the half angle formula of cosine is
$\Rightarrow \,\cos \left( {\dfrac{\theta }{2}} \right) = \pm \sqrt {\dfrac{{1 + \cos \left( \theta \right)}}{2}}$
(we will ignore the negative version since for this example our angle ${22.5^0}$ falls in Quadrant I where cos is positive)
Where $\theta = 45$ , on substituting the value $\theta$ in formula, then
$\Rightarrow \,\cos \left( {\dfrac{{45}}{2}} \right) = \sqrt {\dfrac{{1 + \cos \left( {45} \right)}}{2}}$
On Table of standard angle of trigonometric ratios the value of $\tan {45^0} = \dfrac{1}{{\sqrt 2 }}$ , then
$\Rightarrow \,\cos \left( {\dfrac{{45}}{2}} \right) = \sqrt {\dfrac{{1 + \dfrac{1}{{\sqrt 2 }}}}{2}}$
$\Rightarrow \,\cos \left( {\dfrac{{45}}{2}} \right) = \sqrt {\dfrac{{\dfrac{{\sqrt 2 + 1}}{{\sqrt 2 }}}}{2}}$
$\Rightarrow \,\cos \left( {22.5} \right) = \sqrt {\dfrac{{\sqrt 2 + 1}}{{2\sqrt 2 }}}$
On simplification, we get
$\Rightarrow \,\cos \left( {22.5} \right) = 0.92388$
Hence, the exact value of $\cos 22.5$ is $0.92388$ .
So, the correct answer is “ $0.92388$ ”.

Note : Here the given question belongs to the topic trigonometry. In the question we have the
word cos which means it is cosine trigonometry ratio. Here we must know the trigonometric standard formula of Half angle and double angle and by the table of trigonometric ratios for the standard angles we simplify the given trigonometric function and hence we obtain the required result for the given question.