Question

Find the value of \[\cos \dfrac{\pi }{8}\]

Answer 1

Hint: This question belongs to the topic trigonometry. For this question, we can not put the numerical value of the above trigonometric function. So, if we want to solve this question, we have to use one of the trigonometric identities related to angles. After using the identity, we can easily find the value of the above trigonometric function.
The identity which we are going to use in this question is,
\[\cos 2\theta =2{{\cos }^{2}}\theta -1\]

Complete step-by-step answer:
Now, we will see the complete solution.
First of all, we must write the trigonometric function given in the question,
The trigonometric function is,
\[\cos \dfrac{\pi }{8}\]
Now if we compare our trigonometric function with the identity which we are going to use to solve this question, then,
\[\begin{align}
  &\Rightarrow \theta =\dfrac{\pi }{8} \\
 &\Rightarrow 2\theta =\dfrac{\pi }{4} \\
\end{align}\]
Now, putting the values which we got after comparing into the trigonometric identity stated above, we get,
\[\begin{align}
  &\Rightarrow \cos 2\theta =2{{\cos }^{2}}\theta -1 \\
 &\Rightarrow \cos \dfrac{\pi }{4}=2{{\cos }^{2}}\dfrac{\pi }{8}-1 \\
 &\Rightarrow 2{{\cos }^{2}}\dfrac{\pi }{8}=\cos \dfrac{\pi }{4}+1 \\
\end{align}\]
As we know that, \[\cos \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}\]
Therefore,
\[\begin{align}
  &\Rightarrow 2{{\cos }^{2}}\dfrac{\pi }{8}=\dfrac{1}{\sqrt{2}}+1 \\
 &\Rightarrow 2{{\cos }^{2}}\dfrac{\pi }{8}=0.707+1 \\
 &\Rightarrow {{\cos }^{2}}\dfrac{\pi }{8}=\dfrac{1.707}{2} \\
 &\Rightarrow {{\cos }^{2}}\dfrac{\pi }{8}=0.8535 \\
\end{align}\]
If we take the square root of both the sides, then we will get the final answer.
So, the change in equation after taking the square root will be,
\[\begin{align}
  &\Rightarrow {{\cos }^{2}}\dfrac{\pi }{8}=0.8535 \\
 &\Rightarrow \cos \dfrac{\pi }{8}=0.9238 \\
\end{align}\]
Therefore, the value of \[\cos \dfrac{\pi }{8}=0.9238\]

Note: There are many identities in trigonometry, but we need to choose any one of them which will be useful for us in solving this question, so select the identity carefully. The chances of error in this question can be in the interchange in the value of \[\dfrac{\pi }{4}\]and \[\dfrac{\pi }{8}\]. We also need to be careful in the mathematical calculations, as we are finding the answer in decimals rather than fraction.