Question

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

Answer 1

Hint: In this question, we have to find the exact value of a trigonometric function whose angle is given in the form of a radian. Thus, we apply the trigonometric radian formula to get our required solution. First of all, we change the angle $ \dfrac{2\pi }{3} $ into $ \dfrac{\pi }{3} $ by subtracting $ \pi $ from $ \dfrac{\pi }{3} $ . Then, we will apply the trigonometric formula $ \sin (\pi -a)=\sin a $ , and make necessary calculations, to get the exact value of the trigonometric function.

Complete step by step answer:
According to the question, we have to get the exact value of $ \sin \dfrac{2\pi }{3} $ .
So, we will apply the trigonometric radian formula, that is we have to find such radian angles that lie in the first quadrant.
The trigonometric function given to us: $ \sin \dfrac{2\pi }{3} $ -------- (1)
Now, we will subtract $ \pi $ from $ \dfrac{\pi }{3} $ in equation (1), because we know that $ \dfrac{\pi }{3} $ lie in the first quadrant, and $ \pi -\dfrac{\pi }{3}=\dfrac{2\pi }{3} $ , therefore we get
 $ \Rightarrow \sin \dfrac{2\pi }{3}=\sin \left( \pi -\dfrac{\pi }{3} \right) $
Now, we will apply the trigonometric formula $ \sin (\pi -a)=\sin a $ , where a is some angle. We know that the first quadrant lies between $ \text{0 to }\dfrac{\text{ }\!\!\pi\!\!\text{ }}{2} $ and the second quadrant lies between $ \dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}\text{ to }\!\!\pi\!\!\text{ } $ , also trigonometric sine is positive in the first quadrant as well as in the second quadrant, therefore we get
 $ \Rightarrow \sin \left( \pi -\dfrac{\pi }{3} \right)=\sin \dfrac{\pi }{3} $
Now, from the angle-radian table, we know that
 $ \Rightarrow \sin \dfrac{\pi }{3}=\dfrac{\sqrt{3}}{2} $
Therefore, we get our required solution.
Thus, the exact value of $ \sin \dfrac{2\pi }{3} $ is $ \dfrac{\sqrt{3}}{2}=\dfrac{1.732}{2}=0.866 $ approximately.

Note:
 Always remember the formula and make the necessary calculations to avoid mathematical errors and confusion. You can also find the exact value of $ \sin \dfrac{2\pi }{3} $ , by using a graphical method. In this method, you can draw the angle $ \sin \dfrac{2\pi }{3} $ with the help of $ \dfrac{\text{opposite side}}{\text{hypotenuse}} $ formula of sine. After that, you can get the exact value of $ \sin \dfrac{2\pi }{3} $ as $ \dfrac{\sqrt{3}}{2} $ which is equal to 0.866 approximately.