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Sin 5pi

The value of sin 5pi is 0. Sin 5pi radians in degrees is written as sin ((5π) × 180°/π), i.e., sin (900°). In this article, we will discuss the methods to find the value of sin 5pi with examples.

Sin 5pi: 0
Sin (-5pi): 0
Sin 5pi in degrees: sin (900°)

What is the Value of Sin 5pi?

The value of sin 5pi is 0. Sin 5pi can also be expressed using the equivalent of the given angle (5pi) in degrees (900°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi radians = 5pi × (180°/pi) = 900° or 900 degrees
∴ sin 5pi = sin 5π = sin(900°) = 0

Explanation:

For sin 5pi, the angle 5pi > 2pi. We can represent sin 5pi as, sin(5pi mod 2pi) = sin(pi). For sin 5pi, the angle 5pi lies on the negative x-axis. Thus, sin 5pi value = 0
Since the sine function is a periodic function, we can represent sin 5pi as, sin 5pi = sin(5pi + n × 2pi), n ∈ Z.
⇒ sin 5pi = sin 7pi = sin 9pi , and so on.
Note: Since, sine is an odd function, the value of sin(-5pi) = -sin(5pi) = 0.

Methods to Find Value of Sin 5pi

The value of sin 5pi is given as 0. We can find the value of sin 5pi by:

Using Unit Circle
Using Trigonometric Functions

Sin 5pi Using Unit Circle

To find the value of sin 5π using the unit circle, represent 5pi in the form (2 × 2pi) + pi [∵ 5pi>2pi] ∵ sine is a periodic function, sin 5pi = sin pi.

Rotate ‘r’ anticlockwise to form pi or 5pi angle with the positive x-axis.
The sin of 5pi equals the y-coordinate(0) of the point of intersection (-1, 0) of unit circle and r.

Hence the value of sin 5pi = y = 0

Sin 5pi in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 5pi as:

± √(1-cos²(5pi))
± tan(5pi)/√(1 + tan²(5pi))
± 1/√(1 + cot²(5pi))
± √(sec²(5pi) - 1)/sec(5pi)
1/cosec(5pi)

Note: Since 5pi lies on the negative x-axis, the final value of sin 5pi is 0.

We can use trigonometric identities to represent sin 5pi as,

sin(pi - 5pi) = sin(-4pi)
-sin(pi + 5pi) = -sin 6pi
cos(pi/2 - 5pi) = cos(-9pi/2)
-cos(pi/2 + 5pi) = -cos 11pi/2

☛ Also Check:

Examples Using Sin 5pi

Example 1: Using the value of sin 5pi, solve: (1-cos²(5pi)).

Solution:
We know, (1-cos²(5pi)) = (sin²(5pi)) = 0
⇒ (1-cos²(5pi)) = 0
Example 2: Find the value of sin(5pi) if cos(5pi) = -1 and tan 5pi = 0.

Solution:
Since, tan 5pi = sin 5pi/cos 5pi
⇒ sin 5pi = 0
Example 3: Find the value of 5 sin(5pi)/7 sin(pi/2).

Solution:

Using trigonometric values, we know, sin(5pi) = 0 and sin pi/2 = 1.
⇒ Value of 5 sin(5pi)/7 sin(pi/2) = 0

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FAQs on Sin 5pi

What is Sin 5pi?

Sin 5pi is the value of sine trigonometric function for an angle equal to 5pi radians. The value of sin 5pi is 0.

How to Find Sin 5pi in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 5π can be given in terms of other trigonometric functions as:

± √(1-cos²(5pi))
± tan(5pi)/√(1 + tan²(5pi))
± 1/√(1 + cot²(5pi))
± √(sec²(5pi) - 1)/sec(5pi)
1/cosec(5pi)

☛ Also check: trigonometric table

What is the Exact Value of sin 5pi?

The exact value of sin 5pi is 0.

How to Find the Value of Sin 5pi?

The value of sin 5pi can be calculated by constructing an angle of 5π radians with the x-axis, and then finding the coordinates of the corresponding point (-1, 0) on the unit circle. The value of sin 5pi is equal to the y-coordinate (0). ∴ sin 5pi = 0.

What is the Value of Sin 5pi in Terms of Cos 5pi?

Using trigonometric identities, we can write sin 5pi in terms of cos 5pi as, sin(5pi) = √(1-cos²(5pi)). Here, the value of cos 5pi is equal to -1.

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