Sin 4pi/3
The value of sin 4pi/3 is 0.8660254. . .. Sin 4pi/3 radians in degrees is written as sin ((4π/3) × 180°/π), i.e., sin (240°). In this article, we will discuss the methods to find the value of sin 4pi/3 with examples.
 Sin 4pi/3: (√3/2)
 Sin 4pi/3 in decimal: 0.8660254. . .
 Sin (4pi/3): 0.8660254. . . or √3/2
 Sin 4pi/3 in degrees: sin (240°)
What is the Value of Sin 4pi/3?
The value of sin 4pi/3 in decimal is 0.866025403. . .. Sin 4pi/3 can also be expressed using the equivalent of the given angle (4pi/3) in degrees (240°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 4pi/3 radians = 4pi/3 × (180°/pi) = 240° or 240 degrees
∴ sin 4pi/3 = sin 4π/3 = sin(240°) = (√3/2) or 0.8660254. . .
Explanation:
For sin 4pi/3, the angle 4pi/3 lies between pi and 3pi/2 (Third Quadrant). Since sine function is negative in the third quadrant, thus sin 4pi/3 value = (√3/2) or 0.8660254. . .
Since the sine function is a periodic function, we can represent sin 4pi/3 as, sin 4pi/3 = sin(4pi/3 + n × 2pi), n ∈ Z.
⇒ sin 4pi/3 = sin 10pi/3 = sin 16pi/3 , and so on.
Note: Since, sine is an odd function, the value of sin(4pi/3) = sin(4pi/3).
Methods to Find Value of Sin 4pi/3
The sine function is negative in the 3rd quadrant. The value of sin 4pi/3 is given as 0.86602. . .. We can find the value of sin 4pi/3 by:
 Using Unit Circle
 Using Trigonometric Functions
Sin 4pi/3 Using Unit Circle
To find the value of sin 4π/3 using the unit circle:
 Rotate ‘r’ anticlockwise to form 4pi/3 angle with the positive xaxis.
 The sin of 4pi/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 4pi/3 = y = 0.866 (approx)
Sin 4pi/3 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 4pi/3 as:
 ± √(1cos²(4pi/3))
 ± tan(4pi/3)/√(1 + tan²(4pi/3))
 ± 1/√(1 + cot²(4pi/3))
 ± √(sec²(4pi/3)  1)/sec(4pi/3)
 1/cosec(4pi/3)
Note: Since 4pi/3 lies in the 3rd Quadrant, the final value of sin 4pi/3 will be negative.
We can use trigonometric identities to represent sin 4pi/3 as,
 sin(pi  4pi/3) = sin(pi/3)
 sin(pi + 4pi/3) = sin 7pi/3
 cos(pi/2  4pi/3) = cos(5pi/6)
 cos(pi/2 + 4pi/3) = cos 11pi/6
☛ Also Check:
Examples Using Sin 4pi/3

Example 1: Find the value of 5 sin(4pi/3)/7 cos(5pi/6).
Solution:
Using trigonometric identities, we know, sin(4pi/3) = cos(pi/2  4pi/3) = cos(5pi/6).
⇒ sin(4pi/3) = cos(5pi/6)
⇒ Value of 5 sin(4pi/3)/7 cos(5pi/6) = 5/7 
Example 2: Find the value of 2 × (sin(2pi/3) cos(2pi/3)). [Hint: Use sin 4pi/3 = 0.866]
Solution:
Using the sin 2a formula,
2 sin(2pi/3) cos(2pi/3) = sin(2 × 2pi/3) = sin 4pi/3
∵ sin 4pi/3 = 0.866
⇒ 2 × (sin(2pi/3) cos(2pi/3)) = 0.866 
Example 3: Using the value of sin 4pi/3, solve: (1cos²(4pi/3)).
Solution:
We know, (1cos²(4pi/3)) = (sin²(4pi/3)) = 0.75
⇒ (1cos²(4pi/3)) = 0.75
FAQs on Sin 4pi/3
What is Sin 4pi/3?
Sin 4pi/3 is the value of sine trigonometric function for an angle equal to 4pi/3 radians. The value of sin 4pi/3 is (√3/2) or 0.866 (approx).
What is the Value of Sin 4pi/3 in Terms of Cos 4pi/3?
Using trigonometric identities, we can write sin 4pi/3 in terms of cos 4pi/3 as, sin(4pi/3) = √(1cos²(4pi/3)). Here, the value of cos 4pi/3 is equal to (1/2).
How to Find Sin 4pi/3 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 4π/3 can be given in terms of other trigonometric functions as:
 ± √(1cos²(4pi/3))
 ± tan(4pi/3)/√(1 + tan²(4pi/3))
 ± 1/√(1 + cot²(4pi/3))
 ± √(sec²(4pi/3)  1)/sec(4pi/3)
 1/cosec(4pi/3)
☛ Also check: trigonometry table
What is the Value of Sin 4pi/3 in Terms of Sec 4pi/3?
Since the sine function can be represented using the secant function, we can write sin 4pi/3 as √(sec²(4pi/3)  1)/sec 4pi/3. The value of sec 4pi/3 is equal to 2.
How to Find the Value of Sin 4pi/3?
The value of sin 4pi/3 can be calculated by constructing an angle of 4π/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 4pi/3 is equal to the ycoordinate (0.866). ∴ sin 4pi/3 = 0.866.
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