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Evaluate \[\sin \left( {\dfrac{{2\pi }}{3}} \right)\] ?

Last updated date: 16th May 2024
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Hint: Here in this question, we have to find the exact value of a given trigonometric function by using the sine sum or difference identity. First rewrite the given angle in the form of addition or difference, then the standard trigonometric formula sine sum i.e., \[\sin (A + B)\] or sine difference i.e., \[sin(A - B)\] identity defined as \[\sin A.cosB + cosA.sinB\] and \[sinA.cosB - cosA.sinB\] using one of these we get the required value.

Complete step-by-step answer:
To evaluate the given question by using a formula of cosine addition defined as the sine addition formula calculates the sine of an angle that is either the sum or difference of two other angles. It arises from the law of sines and the distance formula. By using the sine addition formula, the cosine of both the sum and difference of two angles can be found with the two angles' sines and cosines.
Consider the given function
 \[\sin \left( {\dfrac{{2\pi }}{3}} \right)\] -------(1)
The angle \[\dfrac{{2\pi }}{3}\] can be written as \[\pi - \dfrac{\pi }{3}\] , then
Equation (1) becomes
 \[ \Rightarrow \sin \left( {\pi - \dfrac{\pi }{3}} \right)\] ------(2)
Apply the trigonometric cosine identity of difference \[\sin (A - B) = \] \[sinA.cosB - cosA.sinB\] .
Here \[A = \pi \] and \[B = \dfrac{\pi }{3}\]
Substitute A and B in formula then
 \[ \Rightarrow \sin \left( {\pi - \dfrac{\pi }{3}} \right) = sin\pi .cos\dfrac{\pi }{3} - cos\pi .sin\dfrac{\pi }{3}\]
By using specified cosine and sine angle i.e., \[cos\dfrac{\pi }{3} = \dfrac{1}{2}\] , \[cos\pi = - 1\] , \[sin\dfrac{\pi }{3} = \dfrac{{\sqrt 3 }}{2}\] and \[sin\pi = 0\]
On, Substituting the values, we have
 \[ \Rightarrow \sin \left( {\pi - \dfrac{\pi }{3}} \right) = (0).\left( {\dfrac{1}{2}} \right) - ( - 1).\dfrac{{\sqrt 3 }}{2}\]
On simplification we get
 \[ \Rightarrow \sin \left( {\pi - \dfrac{\pi }{3}} \right) = 0 + \dfrac{{\sqrt 3 }}{2}\]
 \[ \Rightarrow \sin \left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{{\sqrt 3 }}{2}\]
Hence, the exact functional value of \[\sin \left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{{\sqrt 3 }}{2}\] .
So, the correct answer is “ \[ \dfrac{{\sqrt 3 }}{2}\] ”.

Note: Simply this can also be solve by using a ASTC rule i.e.,
 \[ \Rightarrow \sin \left( {\dfrac{{2\pi }}{3}} \right) = \sin \left( {\pi - \dfrac{\pi }{3}} \right)\]
By using the ASTC rule of trigonometry, the angle \[\pi - \dfrac{\pi }{3}\] or angle \[180 - \theta \] lies in the second quadrant. sine function is positive here, hence the angle must in positive, then
 \[ \Rightarrow \sin \left( {\dfrac{{2\pi }}{3}} \right) = \sin \left( {\dfrac{\pi }{3}} \right)\]
 \[ \Rightarrow \sin \left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{{\sqrt 3 }}{2}\]
While solving this type of question, we must know about the ASTC rule.
And also know the cosine sum or difference identity, for this we have a standard formula. To find the value for the trigonometry function we need the table of trigonometry ratios for standard angles.