Find Value Of Sin Pi/3 - Cuemath

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