Question

How do you find the exact value of $\sin \left( \dfrac{2\pi }{3} \right)$ ?

Answer 1

Hint : To solve the given question, first we need to apply the reference angle by finding the angle equivalent trigonometric values in the first quadrant. Then we need to know about the fact that $\pi$ radians is equals to ${{180}^{0}}$ . Converting the angle into degrees and using the trigonometric ratios table, we will get the required exact value of $\sin \left( \dfrac{2\pi }{3} \right)$ .
Formula used:
$\pi$ radian = ${{180}^{0}}$
$\sin \theta =\dfrac{perpendicular}{hypotenuse}$

Trigonometric ratio table used to find the sine and cosine of the angle:

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Angles(in degrees)	$\sin \theta$	$\cos \theta$
${{0}^{0}}$	0	1
${{30}^{0}}$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$
${{45}^{0}}$	$\dfrac{1}{\sqrt{2}}$	$\dfrac{1}{\sqrt{2}}$
${{60}^{0}}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$
${{90}^{0}}$	1	0

Complete step-by-step answer :
We have given that,
$\sin \left( \dfrac{2\pi }{3} \right)$
Applying the reference angle by finding the angle equivalent trigonometric values in the first quadrant.
We will obtain,
$\Rightarrow \sin \left( \dfrac{2\pi }{3} \right)=\sin \left( \pi -\left( \dfrac{2\pi }{3} \right) \right)$
Simplifying the RHS of the above expression,
$\Rightarrow \sin \left( \dfrac{2\pi }{3} \right)=\sin \left( \left( \dfrac{3\pi -2\pi }{3} \right) \right)$
$\Rightarrow \sin \left( \dfrac{2\pi }{3} \right)=\sin \dfrac{\pi }{3}$
As we know that,
The value of $\pi ={{180}^{0}}$
So, $\dfrac{\pi }{3}=\dfrac{{{180}^{0}}}{3}={{60}^{0}}$
Therefore,
Substituting this value, we will get
$\Rightarrow \sin \left( \dfrac{2\pi }{3} \right)=\sin {{60}^{0}}$
Using the trigonometric ratios table,
$\sin {{60}^{0}}=\dfrac{\sqrt{3}}{2}$
Therefore,
$\Rightarrow \sin \left( \dfrac{2\pi }{3} \right)=\dfrac{\sqrt{3}}{2}$
Thus,
The exact value of $\sin \left( \dfrac{2\pi }{3} \right)$ is equal to $\dfrac{\sqrt{3}}{2}$ .
So, the correct answer is “ $\dfrac{\sqrt{3}}{2}$ ”.

Note : One must be careful while noted down the values from the trigonometric table to avoid any error in the answer. We must know the basic value of sine and cosine of the angles like ${{0}^{0}}$ , ${{30}^{0}}$ , ${{60}^{0}}$ , ${{90}^{0}}$ etc. Whenever we get this type of problem, first convert the radians to degrees to make the process of solving the question easier. The sine, cosine and the tangent are the three basic functions in introduction to trigonometry which shows the relation between all the sides of the triangles.