Question

How do you find the exact value of \[\cos \left( -\dfrac{2\pi }{3} \right)\]?

Answer 1

Hint: In the given question, we have been asked to find the exact value of the given trigonometric function. In order to find the exact value, first by using the trigonometric unit circle and the trigonometric table of special arcs we will remove the negative sign from the given angle of the given function. Later we know that the given angle is negative in the second quadrant and the value is just opposite the value of cos 60 degrees. In this way we will get the exact value of the given trigonometric function.

Complete step by step solution:
We have given that,
\[\Rightarrow \cos \left( -\dfrac{2\pi }{3} \right)\]
Using the trigonometric unit circle, we have
\[\Rightarrow \cos \left( -\dfrac{2\pi }{3} \right)=\cos \left( \dfrac{2\pi }{3} \right)\]
Now solving,
\[\Rightarrow \cos \left( \dfrac{2\pi }{3} \right)\]
We know that,
\[\pi ={{180}^{0}}\ then\ 2\pi ={{360}^{0\ }}\]
Thus, the value of given function is,
\[\Rightarrow \dfrac{2\pi }{3}=\dfrac{360}{3}={{120}^{0}}\]
Therefore, we have
\[\Rightarrow \cos \left( \dfrac{2\pi }{3} \right)=\cos \left( {{120}^{0}} \right)\]
By trigonometric unit circle;
The value of \[\cos \left( {{60}^{0}} \right)\]is opposite to the value of \[\cos \left( {{120}^{0}} \right)\]as they both are in different quadrants.
Using the trigonometric ratios table,
\[\cos \left( {{60}^{0}} \right)=\dfrac{1}{2}\]Then,
\[\cos \left( {{120}^{0}} \right)=-\dfrac{1}{2}\]

Therefore,
\[\Rightarrow \cos \left( \dfrac{2\pi }{3} \right)=-\dfrac{1}{2}\]
Hence, it is the required exact value.

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.