Question

How do you find the value of $\cos {300^ \circ }$ ?

Answer 1

Hint : In the given question, we are required to find the value of $\cos {300^ \circ }$ . We will use the trigonometric formulae and identities such as $\cos \left( {{{360}^ \circ } - \theta } \right) = \cos \theta$ to find the value of the trigonometric function at the particular angle. We should be clear with the signs of all trigonometric functions in the four quadrants.

Complete step-by-step answer :
So, we have to find the value of the trigonometric function $\cos {300^ \circ }$ .
Now, we know that the trigonometric function cosine is positive in the fourth quadrant. Also, the value of the cosine function gets repeated after a regular interval of $2\pi$ radians.
We know that the angle ${300^ \circ }$ lies in the fourth quadrant. So, the cosine of the angle will be a positive value.
So, we have, $\cos {300^ \circ }$
$\Rightarrow \cos {300^ \circ } = \cos \left( {{{360}^ \circ } - {{60}^ \circ }} \right)$
Now, we know that the value of $\cos \left( {{{360}^ \circ } - \theta } \right)$ is equal to $\cos \theta$ . So, we get,
$\Rightarrow \cos {300^ \circ } = \cos \left( {{{60}^ \circ }} \right)$
Now, we know that cosine and sine are complementary ratios of each other. So, we have, $\sin \left( {{{90}^ \circ } - \theta } \right) = \cos \theta$ . Hence, we get,
$\Rightarrow \cos {300^ \circ } = \sin \left( {{{90}^ \circ } - {{60}^ \circ }} \right)$
Simplifying the expression,
$\Rightarrow \cos {300^ \circ } = \sin \left( {{{30}^ \circ }} \right)$
Now, we also know that the value of $\sin \left( {{{30}^ \circ }} \right)$ is $\left( {\dfrac{1}{2}} \right)$ .
Hence, we get,
$\Rightarrow \cos {300^ \circ } = \dfrac{1}{2}$
Therefore, the value of $\cos {300^ \circ }$ is $\left( {\dfrac{1}{2}} \right)$ .
So, the correct answer is “$\left( {\dfrac{1}{2}} \right)$ ”.

Note : We can also solve the given problem using the periodicity of the cosine and sine functions. Periodic Function is a function that repeats its value after a certain interval. For a real number $T > 0$ , $f\left( {x + T} \right) = f\left( x \right)$ for all x. If T is the smallest positive real number such that $f\left( {x + T} \right) = f\left( x \right)$ for all x, then T is called the fundamental period. The fundamental period of cosine is $2\pi$ radians. So, $\cos \left( \theta \right) = \cos \left( {\theta - 2\pi } \right)$ . Hence, we get, $\cos {300^ \circ } = \cos \left( {{{300}^ \circ } - {{360}^ \circ }} \right) = \cos \left( { - {{60}^ \circ }} \right) = \dfrac{1}{2}$ as we know that cosine is also positive in fourth quadrant of Cartesian plane.