Question

How do you evaluate $\cos \left( \dfrac{4\pi }{3} \right)$ ?

Answer 1

Hint: To evaluate the given trigonometric expression i.e. $\cos \left( \dfrac{4\pi }{3} \right)$, we will first convert this angle given in the cosine into the angle of which we know the value of cosine. To achieve this, we are going to write $\pi +\dfrac{\pi }{3}$ in place of $\dfrac{4\pi }{3}$. And then we will apply the property of the trigonometry angle conversion i.e. $\cos \left( \pi +\theta \right)=-\cos \theta $.

Complete step by step answer:
The trigonometric expression which we have to evaluate is as follows:
$\cos \left( \dfrac{4\pi }{3} \right)$
We are going to replace $\dfrac{4\pi }{3}$ by $\pi +\dfrac{\pi }{3}$ in the above cosine angle we get,
$\Rightarrow \cos \left( \pi +\dfrac{\pi }{3} \right)$
As the angle given in the above cosine is in third quadrant and we know that in the third quadrant cosine is negative so we can open this above cosine as:
$\cos \left( \pi +\theta \right)=-\cos \theta $
$\Rightarrow -\cos \dfrac{\pi }{3}$
We know that the value of cosine of $\dfrac{\pi }{3}$ is equal to half. Now, writing what we just stated in the mathematical form we get:
$\cos \dfrac{\pi }{3}=\dfrac{1}{2}$
Using the above relation in $-\cos \dfrac{\pi }{3}$ we get,
$\Rightarrow -\dfrac{1}{2}$

Hence, we have evaluated the given trigonometric expression to $-\dfrac{1}{2}$.

Note: In the above solution, you have seen the following trigonometric conversion:
$\cos \left( \pi +\theta \right)=-\cos \theta $
From this, we can infer that whenever we add $\pi $ into any angle and then we take cosine, sine, tangent then no change in the trigonometric functions are observed the only change we will observe in the sign of the trigonometric functions.
$\begin{align}
  & \cos \left( \pi +\theta \right)=-\cos \theta \\
 & \sin \left( \pi +\theta \right)=-\sin \theta \\
 & \tan \left( \pi +\theta \right)=\tan \theta \\
\end{align}$
As you can see that all the angles given above lie in the third quadrant and we know that only tangent and cotangent are positive and all the trigonometric functions like cosine, sine, secant and cosecant are negative.
This is the important information that we have shown above which will be helpful in solving the similar problems where you require sine, tangent angles simplification.