Question

What is \[\sin \dfrac{2\pi}{3} \] degrees ?

Answer 1

Hint: In this question, we need to find the value of \[\sin \dfrac{2\pi}{3} \] .Mathematically, \[Pi (\pi)\] is Greek letter and is a mathematical constant . In trigonometry, the value of \[π\] is \[180^{o}\] . We can find the value of \[\sin \dfrac{2\pi}{3} \] by using trigonometric identities and values. Sine is nothing but a ratio of the opposite side of a right angle to the hypotenuse of the right angle. The basic trigonometric functions are sine , cosine and tangent. The value of \[\sin 60^{o}\] is used to find the value. With the help of the Trigonometric functions , we can find the value of \[\sin \dfrac{2\pi}{3} \] .
Trigonometric Table :

Angles	\[0^{o}\]	\[30^{o}\]	\[45^{o}\]	\[60^{o}\]	\[90^{o}\]
Sine	\[0\]	\[\dfrac{1}{2}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{\sqrt{3}}{2}\]	\[1\]
Cosine	\[1\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{1}{2}\]	\[0\]

Complete step-by-step solution:
Here need to find the value of \[\sin \dfrac{2\pi}{3} \] ,
We know that the value of \[\pi\] is \[180^{o} \], by substituting the value of \[\pi\] in \[\sin \dfrac{2\pi}{3} \]
We get,
\[\sin \dfrac{2\pi}{3}=\sin \dfrac{2\times(180^{o})}{3} \]
By simplifying,
We get,
\[\sin \dfrac{2\pi}{3}=sin 120^{o}\]
Now, We need to find the value of \[sin 120^{o}\] .
We can find the value of \[\sin 120^{o} \] by using other angles of sine functions.
We can rewrite \[120^{o}\] as \[180^{o} – 60^{o}\]
Thus we get,
\[\sin 120^{o} = \sin(180^{o} – 60^{o})\]
We know that \[\sin(180^{o} – x)\ = \sin x\] ,
Therefore we get,
\[\sin 120^{o} = \sin 60^{o}\]
We know that the value of \[\sin 60^{o} \] is equal to \[\dfrac{\sqrt{3}}{2}\] .
Hence we get the value of \[\sin 120^{o}\] as \[\dfrac{\sqrt{3}}{2}\].
Thus we can say that the value of \[\sin \ 60^{o}\] is equal to the value of \[\sin 120^{o}\]
Therefore the value of \[\sin \dfrac{2\pi}{3} \] is \[\dfrac{\sqrt{3}}{2}\]
Final answer :
The value of \[\sin \dfrac{2\pi}{3} \] is \[\dfrac{\sqrt{3}}{2}\] .


Note: The concept used in this problem is trigonometric identities and ratios. Trigonometric identities are nothing but they involve trigonometric functions including variables and constants. The common technique used in this problem is the use of trigonometric functions. Geometrically, \[\sin \dfrac{2\pi}{3} \] lies in the second quadrant. Hence the value of \[\sin \dfrac{2\pi}{3} \] is non-negative.
Alternative solution :
Given,
\[\sin \dfrac{2\pi}{3} \]
By substituting the value of \[\pi\]
We get,
\[\sin \dfrac{2\pi}{3}=\sin \dfrac{2\times(180^{o})}{3} \]
By simplifying,
We get,
\[\sin \dfrac{2\pi}{3}=sin 120^{o}\]
Now, We need to find the value of \[\sin 120^{o}\] .
We can find the value of \[\sin 120^{o}\] by using the values of cosine functions .
We can also rewrite \[120^{o} \] as \[(90^{o} + 30^{o})\]
Thus we get,
\[\sin 120^{o} = \sin (90^{o} + 30^{o})\]
We know that \[\sin (90^{o} + x)\ = \cos x\]
Therefore we get,
\[\sin 120^{o}= \cos 30^{o}\]
We know that the value of \[\cos 30^{o}\] is \[\dfrac{\sqrt{3}}{2}\] .
Thus we get, the value of \[\sin \dfrac{2\pi}{3} = \dfrac{\sqrt{3}}{2}\ \]