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Sin 2pi/3
The value of sin 2pi/3 is 0.8660254. . .. Sin 2pi/3 radians in degrees is written as sin ((2π/3) × 180°/π), i.e., sin (120°). In this article, we will discuss the methods to find the value of sin 2pi/3 with examples.
 Sin 2pi/3: √3/2
 Sin 2pi/3 in decimal: 0.8660254. . .
 Sin (2pi/3): 0.8660254. . . or (√3/2)
 Sin 2pi/3 in degrees: sin (120°)
What is the Value of Sin 2pi/3?
The value of sin 2pi/3 in decimal is 0.866025403. . .. Sin 2pi/3 can also be expressed using the equivalent of the given angle (2pi/3) in degrees (120°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 2pi/3 radians = 2pi/3 × (180°/pi) = 120° or 120 degrees
∴ sin 2pi/3 = sin 2π/3 = sin(120°) = √3/2 or 0.8660254. . .
Explanation:
For sin 2pi/3, the angle 2pi/3 lies between pi/2 and pi (Second Quadrant). Since sine function is positive in the second quadrant, thus sin 2pi/3 value = √3/2 or 0.8660254. . .
Since the sine function is a periodic function, we can represent sin 2pi/3 as, sin 2pi/3 = sin(2pi/3 + n × 2pi), n ∈ Z.
⇒ sin 2pi/3 = sin 8pi/3 = sin 14pi/3 , and so on.
Note: Since, sine is an odd function, the value of sin(2pi/3) = sin(2pi/3).
Methods to Find Value of Sin 2pi/3
The sine function is positive in the 2nd quadrant. The value of sin 2pi/3 is given as 0.86602. . .. We can find the value of sin 2pi/3 by:
 Using Unit Circle
 Using Trigonometric Functions
Sin 2pi/3 Using Unit Circle
To find the value of sin 2π/3 using the unit circle:
 Rotate ‘r’ anticlockwise to form 2pi/3 angle with the positive xaxis.
 The sin of 2pi/3 equals the ycoordinate(0.866) of the point of intersection (0.5, 0.866) of unit circle and r.
Hence the value of sin 2pi/3 = y = 0.866 (approx)
Sin 2pi/3 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 2pi/3 as:
 ± √(1cos²(2pi/3))
 ± tan(2pi/3)/√(1 + tan²(2pi/3))
 ± 1/√(1 + cot²(2pi/3))
 ± √(sec²(2pi/3)  1)/sec(2pi/3)
 1/cosec(2pi/3)
Note: Since 2pi/3 lies in the 2nd Quadrant, the final value of sin 2pi/3 will be positive.
We can use trigonometric identities to represent sin 2pi/3 as,
 sin(pi  2pi/3) = sin pi/3
 sin(pi + 2pi/3) = sin 5pi/3
 cos(pi/2  2pi/3) = cos(pi/6)
 cos(pi/2 + 2pi/3) = cos 7pi/6
☛ Also Check:
Examples Using Sin 2pi/3

Example 1: Using the value of sin 2pi/3, solve: (1cos²(2pi/3)).
Solution:
We know, (1cos²(2pi/3)) = (sin²(2pi/3)) = 0.75
⇒ (1cos²(2pi/3)) = 0.75 
Example 2: Simplify: 7 (sin(2pi/3)/sin(8pi/3))
Solution:
We know sin 2pi/3 = sin 8pi/3
⇒ 7 sin(2pi/3)/sin(8pi/3) = 7(sin(2pi/3)/sin(2pi/3))
= 7(1) = 7 
Example 3: Find the value of 5 sin(2pi/3)/7 cos(pi/6).
Solution:
Using trigonometric identities, we know, sin(2pi/3) = cos(pi/2  2pi/3) = cos(pi/6).
⇒ sin(2pi/3) = cos(pi/6)
⇒ Value of 5 sin(2pi/3)/7 cos(pi/6) = 5/7
FAQs on Sin 2pi/3
What is Sin 2pi/3?
Sin 2pi/3 is the value of sine trigonometric function for an angle equal to 2pi/3 radians. The value of sin 2pi/3 is √3/2 or 0.866 (approx).
What is the Value of Sin 2pi/3 in Terms of Cosec 2pi/3?
Since the cosecant function is the reciprocal of the sine function, we can write sin 2pi/3 as 1/cosec(2pi/3). The value of cosec 2pi/3 is equal to 1.15470.
How to Find Sin 2pi/3 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 2π/3 can be given in terms of other trigonometric functions as:
 ± √(1cos²(2pi/3))
 ± tan(2pi/3)/√(1 + tan²(2pi/3))
 ± 1/√(1 + cot²(2pi/3))
 ± √(sec²(2pi/3)  1)/sec(2pi/3)
 1/cosec(2pi/3)
☛ Also check: trigonometric table
How to Find the Value of Sin 2pi/3?
The value of sin 2pi/3 can be calculated by constructing an angle of 2π/3 radians with the xaxis, and then finding the coordinates of the corresponding point (0.5, 0.866) on the unit circle. The value of sin 2pi/3 is equal to the ycoordinate (0.866). ∴ sin 2pi/3 = 0.866.
What is the Value of Sin 2pi/3 in Terms of Tan 2pi/3?
We know, using trig identities, we can write sin 2pi/3 as tan(2pi/3)/√(1 + tan²(2pi/3)). Here, the value of tan 2pi/3 is equal to 1.732050.
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