Sin pi/3
The value of sin pi/3 is 0.8660254. . .. Sin pi/3 radians in degrees is written as sin ((π/3) × 180°/π), i.e., sin (60°). In this article, we will discuss the methods to find the value of sin pi/3 with examples.
 Sin pi/3: √3/2
 Sin pi/3 in decimal: 0.8660254. . .
 Sin (pi/3): 0.8660254. . . or (√3/2)
 Sin pi/3 in degrees: sin (60°)
What is the Value of Sin pi/3?
The value of sin pi/3 in decimal is 0.866025403. . .. Sin pi/3 can also be expressed using the equivalent of the given angle (pi/3) in degrees (60°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/3 radians = pi/3 × (180°/pi) = 60° or 60 degrees
∴ sin pi/3 = sin π/3 = sin(60°) = √3/2 or 0.8660254. . .
Explanation:
For sin pi/3, the angle pi/3 lies between 0 and pi/2 (First Quadrant). Since sine function is positive in the first quadrant, thus sin pi/3 value = √3/2 or 0.8660254. . .
Since the sine function is a periodic function, we can represent sin pi/3 as, sin pi/3 = sin(pi/3 + n × 2pi), n ∈ Z.
⇒ sin pi/3 = sin 7pi/3 = sin 13pi/3 , and so on.
Note: Since, sine is an odd function, the value of sin(pi/3) = sin(pi/3).
Methods to Find Value of Sin pi/3
The sine function is positive in the 1st quadrant. The value of sin pi/3 is given as 0.86602. . .. We can find the value of sin pi/3 by:
 Using Trigonometric Functions
 Using Unit Circle
Sin pi/3 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin pi/3 as:
 ± √(1cos²(pi/3))
 ± tan(pi/3)/√(1 + tan²(pi/3))
 ± 1/√(1 + cot²(pi/3))
 ± √(sec²(pi/3)  1)/sec(pi/3)
 1/cosec(pi/3)
Note: Since pi/3 lies in the 1st Quadrant, the final value of sin pi/3 will be positive.
We can use trigonometric identities to represent sin pi/3 as,
 sin(pi  pi/3) = sin 2pi/3
 sin(pi + pi/3) = sin 4pi/3
 cos(pi/2  pi/3) = cos pi/6
 cos(pi/2 + pi/3) = cos 5pi/6
Sin pi/3 Using Unit Circle
To find the value of sin π/3 using the unit circle:
 Rotate ‘r’ anticlockwise to form pi/3 angle with the positive xaxis.
 The sin of pi/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 pi/3 = y = 0.866 (approx)
☛ Also Check:
Examples Using Sin pi/3

Example 1: Using the value of sin pi/3, solve: (1cos²(pi/3)).
Solution:
We know, (1cos²(pi/3)) = (sin²(pi/3)) = 0.75
⇒ (1cos²(pi/3)) = 0.75 
Example 2: Find the value of 5 sin(pi/3)/7 cos(pi/6).
Solution:
Using trigonometric identities, we know, sin(pi/3) = cos(pi/2  pi/3) = cos(pi/6).
⇒ sin(pi/3) = cos(pi/6)
⇒ Value of 5 sin(pi/3)/7 cos(pi/6) = 5/7 
Example 3: Find the value of sin(pi/3) if cosec(pi/3) is 1.1547.
Solution:
Since, sin pi/3 = 1/csc(pi/3)
⇒ sin pi/3 = 1/1.1547 = 0.866
FAQs on Sin pi/3
What is Sin pi/3?
Sin pi/3 is the value of sine trigonometric function for an angle equal to pi/3 radians. The value of sin pi/3 is √3/2 or 0.866 (approx).
What is the Value of Sin pi/3 in Terms of Cot pi/3?
We can represent the sine function in terms of the cotangent function using trig identities, sin pi/3 can be written as 1/√(1 + cot²(pi/3)). Here, the value of cot pi/3 is equal to 0.5774.
How to Find Sin pi/3 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin π/3 can be given in terms of other trigonometric functions as:
 ± √(1cos²(pi/3))
 ± tan(pi/3)/√(1 + tan²(pi/3))
 ± 1/√(1 + cot²(pi/3))
 ± √(sec²(pi/3)  1)/sec(pi/3)
 1/cosec(pi/3)
☛ Also check: trigonometry table
How to Find the Value of Sin pi/3?
The value of sin pi/3 can be calculated by constructing an angle of π/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 pi/3 is equal to the ycoordinate (0.866). ∴ sin pi/3 = 0.866.
What is the Exact Value of sin pi/3?
The exact value of sin pi/3 can be given accurately up to 8 decimal places as 0.86602540 and √3/2 in fraction.
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