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 x-axis.
• The sin of 2pi/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 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:

• ± √(1-cos²(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:

• tan 3pi/2
• sin pi/3
• cos 17pi/12
• tan 2pi/3
• sec pi/3
• sin 3pi/2

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:

• ± √(1-cos²(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 x-axis, 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 y-coordinate (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.