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Sin pi

The value of sin pi is 0. Sin pi radians in degrees is written as sin ((π) × 180°/π), i.e., sin (180°). In this article, we will discuss the methods to find the value of sin pi with examples.

Sin pi: 0
Sin (-pi): 0
Sin pi in degrees: sin (180°)

What is the Value of Sin pi?

The value of sin pi is 0. Sin pi can also be expressed using the equivalent of the given angle (pi) in degrees (180°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi radians = pi × (180°/pi) = 180° or 180 degrees
∴ sin pi = sin π = sin(180°) = 0

Explanation:

For sin pi, the angle pi lies on the negative x-axis. Thus, sin pi value = 0
Since the sine function is a periodic function, we can represent sin pi as, sin pi = sin(pi + n × 2pi), n ∈ Z.
⇒ sin pi = sin 3pi = sin 5pi , and so on.
Note: Since, sine is an odd function, the value of sin(-pi) = -sin(pi) = 0.

Methods to Find Value of Sin pi

The value of sin pi is given as 0. We can find the value of sin pi by:

Using Unit Circle
Using Trigonometric Functions

Sin pi Using Unit Circle

To find the value of sin π using the unit circle:

Rotate ‘r’ anticlockwise to form pi angle with the positive x-axis.
The sin of pi equals the y-coordinate(0) of the point of intersection (-1, 0) of unit circle and r.

Hence the value of sin pi = y = 0

Sin pi in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin pi as:

± √(1-cos²(pi))
± tan(pi)/√(1 + tan²(pi))
± 1/√(1 + cot²(pi))
± √(sec²(pi) - 1)/sec(pi)
1/cosec(pi)

Note: Since pi lies on the negative x-axis, the final value of sin pi is 0.

We can use trigonometric identities to represent sin pi as,

sin(pi - pi) = sin 0
-sin(pi + pi) = -sin 2pi
cos(pi/2 - pi) = cos(-pi/2)
-cos(pi/2 + pi) = -cos 3pi/2

☛ Also Check:

Examples Using Sin pi

Example 1: Using the value of sin pi, solve: (1-cos²(pi)).

Solution:
We know, (1-cos²(pi)) = (sin²(pi)) = 0
⇒ (1-cos²(pi)) = 0
Example 2: Find the value of sin(pi) if cos(pi) is -1 and tan(pi) is 0.

Solution:
Since, tan pi = sin pi/cos (pi)
⇒ sin pi = 0
Example 3: Find the value of 2 × (sin(pi/2) cos(pi/2)). [Hint: Use sin pi = 0]

Solution:

Using the sin 2a formula,
2 sin(pi/2) cos(pi/2) = sin(2 × pi/2) = sin pi
∵ sin pi = 0
⇒ 2 × (sin(pi/2) cos(pi/2)) = 0

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FAQs on Sin pi

What is Sin pi?

Sin pi is the value of sine trigonometric function for an angle equal to pi radians. The value of sin pi is 0.

How to Find Sin pi in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin π can be given in terms of other trigonometric functions as:

± √(1-cos²(pi))
± tan(pi)/√(1 + tan²(pi))
± 1/√(1 + cot²(pi))
± √(sec²(pi) - 1)/sec(pi)
1/cosec(pi)

☛ Also check: trigonometric table

How to Find the Value of Sin pi?

The value of sin pi can be calculated by constructing an angle of π radians with the x-axis, and then finding the coordinates of the corresponding point (-1, 0) on the unit circle. The value of sin pi is equal to the y-coordinate (0). ∴ sin pi = 0.

What is the Value of Sin pi in Terms of Cos pi?

Using trigonometric identities, we can write sin pi in terms of cos pi as, sin(pi) = √(1-cos²(pi)). Here, the value of cos pi is equal to -1.

What is the Value of Sin pi in Terms of Sec pi?

Since the sine function can be represented using the secant function, we can write sin pi as -√(sec²(pi) - 1)/sec pi. The value of sec pi is equal to -1.

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