Question

How do you use the angle sum or difference identity to find the exact value of $\cos \dfrac{{5\pi }}{{12}}$ ?

Answer 1

Hint : Trigonometry is used to study the relation between the sides of a right-angled triangle that is the base, the perpendicular and the hypotenuse. The trigonometric ratios have great importance in daily life and various other branches of mathematics too, so there are various trigonometric identities used for simplifying the function and making the calculations easier, so all the trigonometric ratios are interrelated. To find the trigonometric value of large angles, there are many ways. In the given question, we have to find the cosine of $\dfrac{{5\pi }}{{12}}$ degrees by using the sum or difference identity, so first, we will write $\dfrac{{5\pi }}{{12}}$ degrees as the sum of two angles such that their value is already known or can be found easily.

Complete step-by-step answer :
As the trigonometric values of $\dfrac{\pi }{6}$ and $\dfrac{\pi }{4}$ are easy to find, $\dfrac{{5\pi }}{{12}}$ degrees can be written as a sum of $\dfrac{{2\pi }}{{12}}$ and $\dfrac{{3\pi }}{{12}}$ –
$\cos (\dfrac{{5\pi }}{{12}}) = \cos (\dfrac{{2\pi }}{{12}} + \dfrac{{3\pi }}{{12}})$
We know that –
$\cos (A + B) = \cos A\cos B - \sin A\sin B \\\ \Rightarrow \cos (\dfrac{{2\pi }}{{12}} + \dfrac{{3\pi }}{{12}}) = \cos \dfrac{{2\pi }}{{12}}\cos \dfrac{{3\pi }}{{12}} - \sin \dfrac{{2\pi }}{{12}}\sin \dfrac{{3\pi }}{{12}} \\\ \Rightarrow \cos \dfrac{{5\pi }}{{12}} = \cos \dfrac{\pi }{6}\cos \dfrac{\pi }{4} - \sin \dfrac{\pi }{6}\sin \dfrac{\pi }{4} \\\ \Rightarrow \cos \dfrac{{5\pi }}{{12}} = \dfrac{{\sqrt 3 }}{2} \times \dfrac{1}{{\sqrt 2 }} - \dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }} \\\ \Rightarrow \cos \dfrac{{5\pi }}{{12}} = \dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }} \\\$
Hence, the exact functional value of $\cos \dfrac{{5\pi }}{{12}}$ is $\dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}$
So, the correct answer is “ $\dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}$ ”.

Note : We know the value of the cosine function when the angle lies between 0 and $\dfrac{\pi }{2}$ . So we wrote the angle as the sum of $\dfrac{\pi }{6}$ and $\dfrac{\pi }{4}$ . We have applied the identity that the cosine of the sum of two angles a and b is equal to the difference of the product of the cosine of angle a and cosine of angle b and the product of the sine of angle a and the sine of angle b that is $\cos (a + b) = \cos a\cos b - \sin a\sin b$ . Similarly, $\cos (a - b) = \cos a\cos b + \sin a\sin b$ . There are many such identities in trigonometry that can be used to solve similar questions.