Question

How do you find the exact value of \[\dfrac{{\cos 5\pi }}{6}\]?

Answer 1

Hint: We have to calculate the value for \[\dfrac{{\cos 5\pi }}{6}\]. So, first of all, we will split the $\cos 5\pi $ and find the value for $\cos 5\pi $ by using the identity - $\cos \left( {\pi + x} \right) = - \cos x$. After getting the value of $\cos 5\pi $ divide it by 6.

Complete step by step solution:
In the question, we have to find the exact value for \[\dfrac{{\cos 5\pi }}{6}\]. We can see that the whole $\cos 5\pi $ is to be divided by 6. So, first we have to find out the value for $\cos 5\pi $. Therefore, $\cos 5\pi $ should be split so that we can find the value of $\cos 5\pi $ in an easy way. $\cos 5\pi $ can be rewritten as –
$ \Rightarrow \cos 5\pi = \cos \left( {4\pi + \pi } \right)$
We know that, -
$\cos \left( {\pi + x} \right) = - \cos x$
To prove above identity, we will use cosine addition formula –
$\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$
So, now using the formula of cosine add formula, let $A = \pi $ and $B = x$, we get –
$\cos \left( {\pi + x} \right) = \cos \pi \cos x - \sin x\sin \pi $
Since, we know that, $\cos \pi = - 1$ and $\sin \pi = 0$, we get –
$\cos \left( {\pi + x} \right) = - \cos x$
Hence, proved
Therefore, using the above identity, we get –
$ \Rightarrow \cos \left( {4\pi + \pi } \right) = \cos \pi $
Hence, we expressed $\cos 5\pi $ in a simple way, that is, $\cos \pi $, which means that the value of $\cos 5\pi $ will be equal to $\cos \pi $.
Now, we know that $\cos \pi = - 1$
Therefore, to find the value of \[\dfrac{{\cos 5\pi }}{6}\] we have to put $\cos \pi $ in the place of $\cos 5\pi $. Then, it can be rewritten as –
$\therefore \dfrac{{\cos \pi }}{6}$
Putting $\cos \pi = - 1$ in the above, we get –
$\therefore \dfrac{{\cos \pi }}{6} = \dfrac{{ - 1}}{6}$

Hence, the required value of \[\dfrac{{\cos 5\pi }}{6}\] is $ - \dfrac{1}{6}$

Note:
Many students can make mistakes and misunderstand the question as $\cos \left( {\dfrac{{5\pi }}{6}} \right)$ which will make the solution different from the above question and gives the answer as $ - \dfrac{{\sqrt 3 }}{2}$. So, see the question carefully.
If the function is written as $\cos \left( { - x} \right)$ then, we can write it as positive $\cos x$ as it is the even function and its graph is symmetric to y – axis.