Question

The value of $\cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ }$ is

Answer 1

Generally, trigonometric identities are equalities that involve trigonometric functions and are useful whenever trigonometric functions are involved in an expression or an equation. The six basic trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent and all the fundamental trigonometric identities are derived from the six trigonometric ratios. We need to analyze the given information so that we are able to solve the problem. Here we are asked to calculate the value of$\cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ }$.
We need to apply the appropriate trigonometric identities to obtain the required answer.
Formula to be used:
The trigonometric identities that are used to solve the given problem are as follows.
a) $cos\left( {A + B} \right) + \cos \left( {A-B} \right) = 2cosAcosB$
b)$\cos \left( {A-B} \right) = cosAcosB + \sin A\sin B$

Complete step by step answer:
The given expression is$\cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ }$.
To find: The value of $\cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ }$
For our convenience, we are rewriting the following terms.
Here, $\cos {52^ \circ }$can be written as$\cos \left( {{{60}^ \circ } - {8^\circ }} \right)$
Similarly, $\cos {68^ \circ }$can be written as$\cos \left( {{{60}^ \circ } + {8^\circ }} \right)$
Also, $\cos {172^ \circ }$ can be written as$\cos \left( {{{180}^ \circ } - {8^\circ }} \right)$
Now, we shall apply the formula$\cos \left( {A-B} \right) = cosAcosB + \sin A\sin B$
That is $\cos \left( {{{180}^ \circ } - {8^\circ }} \right) = cos{180^ \circ }cos{8^\circ } + \sin {180^ \circ }\sin {8^\circ }$
$= - 1 \times \cos {8^\circ } + 0 \times \sin {8^\circ }$
(We know that$\cos {180^\circ } = - 1$ and$\sin {180^\circ } = 0$ )
$= - \cos {8^\circ }$
Hence, $\cos {172^ \circ }$ $= - \cos {8^\circ }$
Now, we shall substitute the obtained results in the given expression.
$\cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ } = \cos \left( {{{60}^ \circ } - {8^\circ }} \right) + \cos \left( {{{60}^ \circ } + {8^\circ }} \right) - \cos {8^\circ }$
Now, we shall apply the formula $cos\left( {A + B} \right) + \cos \left( {A-B} \right) = 2cosAcosB$on the first two terms
$\Rightarrow \cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ } = 2\cos {60^ \circ }\cos {8^\circ } - \cos {8^\circ }$
We know that$\cos {60^\circ } = \dfrac{1}{2}$
$\Rightarrow \cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ } = 2\dfrac{1}{2}\cos {8^\circ } - \cos {8^\circ }$
$\Rightarrow \cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ } = \cos {8^\circ } - \cos {8^\circ }$
Hence we get$\Rightarrow \cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ } = 0$

Note: Generally, the trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation and these identities are useful whenever expressions involving trigonometric functions need to be simplified. Here we are asked to calculate the value of $\cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ }$. In this question, we have applied the appropriate trigonometric identities to obtain the desired answer. Hence, the value $\cos {52^ \circ } + \cos {68^ \circ } + \cos {172^ \circ }$ is zero.