Evaluate \[\sin \left( {\dfrac{{2\pi }}{3}} \right)\] ?
Last updated date: 27th Mar 2023
•
Total views: 206.1k
•
Views today: 4.83k
Answer
206.1k+ views
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.
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.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
