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:

• ± √(1-cos²(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 x-axis.
• The sin of pi/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 pi/3 = y = 0.866 (approx)

☛ Also Check:

• cos 2pi
• cos 3pi/2
• cos 5pi/3
• sec pi/4
• cos 7pi/4
• sin 5pi/8

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:

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