Question

How to use the half-angle formula to find $\cos \left( {\dfrac{\pi }{8}} \right)$?

Answer 1

Hint: In order to solve this question, we will have to use the half-angle formula. Along with that, we have to use trigonometric identities to find the solution. Then, after using both half angle formula and trigonometric identities we will have to transform it up until we get the exact value.

Complete step by step solution:
Before starting to solve the given question, we should first know the half-angle formula. Half angle formulas allow expressions of trigonometric functions of angles to be equal to $\dfrac{\alpha }{2}$ in terms of $\alpha $, which helps to easily simplify and solve the functions. The half-angle formula of cosine is:
$\cos \left( {\dfrac{\alpha }{2}} \right) = \sqrt {\dfrac{{1 + \cos \alpha }}{2}} $-----(1)
Here in this question, we are given $\cos \left( {\dfrac{\pi }{8}} \right)$ and have to make use of a half-angle formula in order to find its value.
Let us put $\dfrac{\pi }{8}$ instead of $\dfrac{\alpha }{2}$ in equation (1) and we get,
$\cos \left( {\dfrac{\alpha }{2}} \right) = \cos \left( {\dfrac{\pi }{8}} \right)$
$\cos \left( {\dfrac{\pi }{8}} \right)$ can be also be written as $\cos \left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right)$. We write $\left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right)$ instead of $\left( {\dfrac{\pi }{8}} \right)$ in equation (1) as we want the denominator to be$2$as the formula requires it.
$\cos \left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right) = \sqrt {\dfrac{{1 + \cos \left( {\dfrac{\pi }{4}} \right)}}{2}} \\
\Rightarrow \cos \left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right) = \sqrt {\dfrac{{1 + \dfrac{{\sqrt 2 }}{2}}}{2}} \\
\Rightarrow \cos \left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right) = \sqrt {\dfrac{{\dfrac{{2 + \sqrt 2 }}{2}}}{2}} \\ $
$
\Rightarrow \cos \left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right) = \sqrt {\dfrac{{2 + \sqrt 2 }}{4}} \\
\therefore\cos \left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right) = \dfrac{{\sqrt {2 + \sqrt 2 } }}{2} \\ $
Hence, the answer is $\dfrac{{\sqrt {2 + \sqrt 2 } }}{2}$.

Note: Since, we are not getting the exact value in these types of questions so, we need to have a particular logic of every question in order to have a solution to the improper angled questions. Questions whose exact value can be easily calculated are only asked in exams. Each step should be performed carefully so that there are no calculation mistakes and confusion. Students should keep the nature of the values of the trigonometric functions in different quadrants in their mind while solving such questions.