Question

How do you use the angle sum identity to find the exact value of $\cos 165?$

Answer 1

As we know that there are three basic trigonometric identities that involve the sums of angles, the functions which are involved in these identities are sine, cosine and tangent. A trigonometric identity is an equation based on trigonometry which is always true. The angle sum identity like $\cos (A + B) = \cos A\cos B - \sin A\sin B$. Whether it is sum or difference of angles in trigonometric functions, they are used to find out the most functional values or exact value of any angles. Some of the most familiar values of all trigonometric ratios are ${30^ \circ },{45^ \circ },{90^ \circ },{60^ \circ }$ and so on.

Complete step by step answer:
Here we have $\cos 165$, we can write $165$ as a sum of $120$ and $45$. So we have $\cos (165) = \cos (120 + 45)$, Now by applying the angle sum identity of cosine which is $\cos (A + B) = \cos A\cos B - \sin A\sin B$, we get :

$\cos (165) = \cos 120\times\cos 45 - \sin 120\times\sin 45$
Also we know the values:
$\cos 120$ can be written as $\cos (180 - 60)$. Since we know the value that if there is $\cos (180 - x)$ then it equals $- \cos x$, Similarly $\cos (180 - 60) = - \cos 60$. We know that $\cos 60 = \dfrac{1}{2}$. So $\cos 120 = - \dfrac{1}{2}$.

We now calculate for $\sin 120$. WE can write $\sin 120 = \sin (180 - 60)$. If there is $\sin (180 - x) = \sin x$, we can apply this and we get $\sin (180 = 60) = \sin 60$ and we know that the value of $\sin 60 = \dfrac{{\sqrt 3 }}{2}$.

We know that $\sin 45 = \cos 45 = \dfrac{1}{{\sqrt 2 }}$ also written as $\dfrac{{\sqrt 2 }}{2}$. Now substituting all the values in the formula we get:
$\cos (165) = (\dfrac{{ - 1}}{2})\times(\dfrac{{\sqrt 2 }}{2}) - \dfrac{{\sqrt 3 }}{2}\times\dfrac{{\sqrt 2 }}{2} \\\ \Rightarrow \cos (165) = - \dfrac{{\sqrt 2 }}{2}(\dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2})$
So we have $\cos 165 = - \dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}$.

Hence the solution of $\cos 165$ is $- \dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}$.

Note: We should always remember the angle sum identity of every trigonometric function before solving it. Trigonometric functions are also called circular functions and these basic functions are also known as trigonometric ratios. There are multiple trigonometric formulas and identities which represent the relation between the functions and enable to find the value of unknown angles. We should always remember to determine in which quadrant the angle will lie as it will say about the positive and negative value of cosine. There is both angle sum identity and angle difference formula to calculate the values of angles.