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Sin 3pi
The value of sin 3pi is 0. Sin 3pi radians in degrees is written as sin ((3π) × 180°/π), i.e., sin (540°). In this article, we will discuss the methods to find the value of sin 3pi with examples.
 Sin 3pi: 0
 Sin (3pi): 0
 Sin 3pi in degrees: sin (540°)
What is the Value of Sin 3pi?
The value of sin 3pi is 0. Sin 3pi can also be expressed using the equivalent of the given angle (3pi) in degrees (540°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi radians = 3pi × (180°/pi) = 540° or 540 degrees
∴ sin 3pi = sin 3π = sin(540°) = 0
Explanation:
For sin 3pi, the angle 3pi > 2pi. We can represent sin 3pi as, sin(3pi mod 2pi) = sin(pi). For sin 3pi, the angle 3pi lies on the negative xaxis. Thus, sin 3pi value = 0
Since the sine function is a periodic function, we can represent sin 3pi as, sin 3pi = sin(3pi + n × 2pi), n ∈ Z.
⇒ sin 3pi = sin 5pi = sin 7pi , and so on.
Note: Since, sine is an odd function, the value of sin(3pi) = sin(3pi) = 0.
Methods to Find Value of Sin 3pi
The value of sin 3pi is given as 0. We can find the value of sin 3pi by:
 Using Unit Circle
 Using Trigonometric Functions
Sin 3pi Using Unit Circle
To find the value of sin 3π using the unit circle, represent 3pi in the form (1 × 2pi) + pi [∵ 3pi>2pi] ∵ sine is a periodic function, sin 3pi = sin pi.
 Rotate ‘r’ anticlockwise to form pi or 3pi angle with the positive xaxis.
 The sin of 3pi equals the ycoordinate(0) of the point of intersection (1, 0) of unit circle and r.
Hence the value of sin 3pi = y = 0
Sin 3pi in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 3pi as:
 ± √(1cos²(3pi))
 ± tan(3pi)/√(1 + tan²(3pi))
 ± 1/√(1 + cot²(3pi))
 ± √(sec²(3pi)  1)/sec(3pi)
 1/cosec(3pi)
Note: Since 3pi lies on the negative xaxis, the final value of sin 3pi is 0.
We can use trigonometric identities to represent sin 3pi as,
 sin(pi  3pi) = sin(2pi)
 sin(pi + 3pi) = sin 4pi
 cos(pi/2  3pi) = cos(5pi/2)
 cos(pi/2 + 3pi) = cos 7pi/2
☛ Also Check:
Examples Using Sin 3pi

Example 1: Using the value of sin 3pi, solve: (1cos²(3pi)).
Solution:
We know, (1cos²(3pi)) = (sin²(3pi)) = 0
⇒ (1cos²(3pi)) = 0 
Example 2: Find the value of sin(3pi) if cos(3pi) is 1 and tan 3pi = 0.
Solution:
Since, tan 3pi = sin 3pi/cos 3pi
⇒ sin 3pi = 0 
Example 3: Find the value of 2 × (sin(3pi/2) cos(3pi/2)). [Hint: Use sin 3pi = 0]
Solution:
Using the sin 2a formula,
2 sin(3pi/2) cos(3pi/2) = sin(2 × 3pi/2) = sin 3pi
∵ sin 3pi = 0
⇒ 2 × (sin(3pi/2) cos(3pi/2)) = 0
FAQs on Sin 3pi
What is Sin 3pi?
Sin 3pi is the value of sine trigonometric function for an angle equal to 3pi radians. The value of sin 3pi is 0.
What is the Value of Sin 3pi in Terms of Cos 3pi?
Using trigonometric identities, we can write sin 3pi in terms of cos 3pi as, sin(3pi) = √(1cos²(3pi)). Here, the value of cos 3pi is equal to 1.
How to Find Sin 3pi in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 3π can be given in terms of other trigonometric functions as:
 ± √(1cos²(3pi))
 ± tan(3pi)/√(1 + tan²(3pi))
 ± 1/√(1 + cot²(3pi))
 ± √(sec²(3pi)  1)/sec(3pi)
 1/cosec(3pi)
☛ Also check: trigonometry table
What is the Exact Value of sin 3pi?
The exact value of sin 3pi is 0.
How to Find the Value of Sin 3pi?
The value of sin 3pi can be calculated by constructing an angle of 3π radians with the xaxis, and then finding the coordinates of the corresponding point (1, 0) on the unit circle. The value of sin 3pi is equal to the ycoordinate (0). ∴ sin 3pi = 0.
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