Question

How do you evaluate $\cos 870^\circ $?

Answer 1

Hint: Here, in the given question, we are given a trigonometric ratio $\cos 870^\circ $ and we need to find the value of it. As we know the function $y = \cos x$ has a period of $2\pi $ or $360^\circ $, i.e. the value of $\cos x$ repeats after an interval of $2\pi $ or $360^\circ $. For any positive integer $n$, angle $\left( {360^\circ \times n + \theta } \right)$ is coterminal to angle $\theta $. Coterminal angles are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side. Therefore, for any positive integer $n$, we have $\cos \left( {360^\circ \times n + \theta } \right) = \cos \theta $. Therefore, we will write $\cos 870^\circ $ as $\cos \left( {360^\circ \times 2 + 150^\circ } \right)$ and proceed.

Complete step-by-step answer:
We know that the function $y = \cos 870^\circ $ has a period of $2\pi $ or $360^\circ $.

Given, $\cos 870^\circ $
As we know $\cos \left( {360^\circ \times n + \theta } \right) = \cos \theta $. Therefore, we can write the above written statement as,
$ = \cos \left( {360^\circ \times 2 + 150^\circ } \right)$
$ = \cos 150^\circ $
As we know $\cos \left( {180^\circ - \theta } \right) = - \cos \theta $, because in second quadrant $\cos ine$ function is negative. Therefore, we get
$ \Rightarrow \cos \left( {180^\circ - 30^\circ } \right) = - \cos 30^\circ $
As we know value of $\cos 30^\circ $ is $\dfrac{{\sqrt 3 }}{2}$
Hence, the value of $\cos 870^\circ $ is $ - \dfrac{{\sqrt 3 }}{2}$.
So, the correct answer is “$ - \dfrac{{\sqrt 3 }}{2}$”.

Note: To solve these type of questions we should know all the required values of standard angles say, $0^\circ ,30^\circ ,60^\circ ,90^\circ ,180^\circ ,270^\circ ,360^\circ $ respectively for each trigonometric term such as $\sin ,\cos ,\tan ,\cos ec,\sec ,\cot $. Remember that $\sin e$ and $\cos ine$ functions and their reciprocals $\cos ecant$ and $\sec ant$ functions are periodic functions with period $2\pi $ or $360^\circ $. $\operatorname{Tan} gent$ and $cotangent$ functions are periodic with period $\pi $ or $180^\circ $.