Question

How do you evaluate $\sin \left( \dfrac{5\pi }{3} \right)$ ?
(a) Using trigonometric angle identities
(b) Using linear formulas
(c) a and b both
(d) none of the above

Answer 1

Hint: In this problem we are to find the value of $\sin \left( \dfrac{5\pi }{3} \right)$. We will try to use the trigonometric double angle identities of $\tan 2\theta $ to find and simplify the value of our needed problem. We can start with the fact that $\sin \left( 2\pi -\theta \right)=\sin \left( -\theta \right)$ and consider $\theta =\dfrac{\pi }{3}$ to get ahead with the problem and evaluate the value.

Complete step by step solution:
According to the question, we start with,
$\sin \left( \dfrac{5\pi }{3} \right)$
Again, we can write, $\dfrac{5\pi }{3}$ with the terms, $2\pi -\dfrac{\pi }{3}$ ,
$\Rightarrow \sin \left( 2\pi -\dfrac{\pi }{3} \right)$
As, $\sin \left( 2\pi -\theta \right)=\sin \left( -\theta \right)$ , because the angle is in the fourth quadrant where the value of sin function is negative, we get,
So, we get to know, $\sin \left( -\dfrac{\pi }{3} \right)$ .
Now, say, $\theta =\dfrac{\pi }{3}$ ,
So, adding sin function on both sides, $\sin \theta =\sin \dfrac{\pi }{3}$
And as we can also see, $\dfrac{\pi }{3}$ lies in the first quadrant where the value of all functions are always positive.
Also, we know, $\sin \left( -x \right)=-\sin x$ , from the use of trigonometric identities.
Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Identity inequalities which are true for every value occurring on both sides of an equation. Geometrically, these identities involve certain functions of one or more angles. There are various distinct identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.
So,
$\sin \left( -\dfrac{\pi }{3} \right)=-\sin \dfrac{\pi }{3}$
As, $\sin \dfrac{\pi }{3}=\dfrac{\sqrt{3}}{2}$ ,
We can conclude, $\sin \left( \dfrac{5\pi }{3} \right)=-\dfrac{\sqrt{3}}{2}$

So, the correct answer is “Option a”.

Note: To understand how the values of trigonometric ratios like $\sin \left( \dfrac{5\pi }{3} \right)$ change in different quadrants, first we have to understand ASTC rule. The ASTC rule is nothing but the "all sin tan cos" rule in trigonometry. The angles which lie between 0° and 90° are said to lie in the first quadrant. The angles between 90° and 180° are in the second quadrant, angles between 180° and 270° are in the third quadrant and angles between 270° and 360° are in the fourth quadrant. In the first quadrant, the values for sin, cos and tan are positive. In the second quadrant, the values for sin are positive only. In the third quadrant, the values for tan are positive only. In the fourth quadrant, the values for cos are positive only.