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 x-axis.
- The sin of 4pi/3 equals the y-coordinate(-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:
- ± √(1-cos²(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:
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) = -√(1-cos²(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:
- ± √(1-cos²(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 x-axis, 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 y-coordinate (-0.866). ∴ sin 4pi/3 = -0.866.