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Sin 3pi/2

The value of sin 3pi/2 is -1. Sin 3pi/2 radians in degrees is written as sin ((3π/2) × 180°/π), i.e., sin (270°). In this article, we will discuss the methods to find the value of sin 3pi/2 with examples.

Sin 3pi/2: -1
Sin (-3pi/2): 1
Sin 3pi/2 in degrees: sin (270°)

What is the Value of Sin 3pi/2?

The value of sin 3pi/2 is -1. Sin 3pi/2 can also be expressed using the equivalent of the given angle (3pi/2) in degrees (270°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/2 radians = 3pi/2 × (180°/pi) = 270° or 270 degrees
∴ sin 3pi/2 = sin 3π/2 = sin(270°) = -1

Explanation:

For sin 3pi/2, the angle 3pi/2 lies on the negative y-axis. Thus, sin 3pi/2 value = -1
Since the sine function is a periodic function, we can represent sin 3pi/2 as, sin 3pi/2 = sin(3pi/2 + n × 2pi), n ∈ Z.
⇒ sin 3pi/2 = sin 7pi/2 = sin 11pi/2 , and so on.
Note: Since, sine is an odd function, the value of sin(-3pi/2) = -sin(3pi/2).

Methods to Find Value of Sin 3pi/2

The value of sin 3pi/2 is given as -1. We can find the value of sin 3pi/2 by:

Using Unit Circle
Using Trigonometric Functions

Sin 3pi/2 Using Unit Circle

To find the value of sin 3π/2 using the unit circle:

Rotate ‘r’ anticlockwise to form 3pi/2 angle with the positive x-axis.
The sin of 3pi/2 equals the y-coordinate(-1) of the point of intersection (0, -1) of unit circle and r.

Hence the value of sin 3pi/2 = y = -1

Sin 3pi/2 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 3pi/2 as:

± √(1-cos²(3pi/2))
± tan(3pi/2)/√(1 + tan²(3pi/2))
± 1/√(1 + cot²(3pi/2))
± √(sec²(3pi/2) - 1)/sec(3pi/2)
1/cosec(3pi/2)

Note: Since 3pi/2 lies on the negative y-axis, the final value of sin 3pi/2 is -1.

We can use trigonometric identities to represent sin 3pi/2 as,

sin(pi - 3pi/2) = sin(-pi/2)
-sin(pi + 3pi/2) = -sin 5pi/2
cos(pi/2 - 3pi/2) = cos(-pi)
-cos(pi/2 + 3pi/2) = -cos 2pi

☛ Also Check:

Examples Using Sin 3pi/2

Example 1: Find the value of 2 × (sin(3pi/4) cos(3pi/4)). [Hint: Use sin 3pi/2 = -1]

Solution:

Using the sin 2a formula,
2 sin(3pi/4) cos(3pi/4) = sin(2 × 3pi/4) = sin 3pi/2
∵ sin 3pi/2 = -1
⇒ 2 × (sin(3pi/4) cos(3pi/4)) = -1
Example 2: Simplify: 2 (sin(3pi/2)/sin(7pi/2))

Solution:

We know sin 3pi/2 = sin 7pi/2
⇒ 2 sin(3pi/2)/sin(7pi/2) = 2(sin(3pi/2)/sin(3pi/2))
= 2(1) = 2
Example 3: Using the value of sin 3pi/2, solve: (1-cos²(3pi/2)).

Solution:
We know, (1-cos²(3pi/2)) = (sin²(3pi/2)) = 1
⇒ (1-cos²(3pi/2)) = 1

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FAQs on Sin 3pi/2

What is Sin 3pi/2?

Sin 3pi/2 is the value of sine trigonometric function for an angle equal to 3pi/2 radians. The value of sin 3pi/2 is -1.

What is the Value of Sin 3pi/2 in Terms of Cos 3pi/2?

Using trigonometric identities, we can write sin 3pi/2 in terms of cos 3pi/2 as, sin(3pi/2) = -√(1-cos²(3pi/2)). Here, the value of cos 3pi/2 is equal to 0.

What is the Value of Sin 3pi/2 in Terms of Cosec 3pi/2?

Since the cosecant function is the reciprocal of the sine function, we can write sin 3pi/2 as 1/cosec(3pi/2). The value of cosec 3pi/2 is equal to -1.

How to Find the Value of Sin 3pi/2?

The value of sin 3pi/2 can be calculated by constructing an angle of 3π/2 radians with the x-axis, and then finding the coordinates of the corresponding point (0, -1) on the unit circle. The value of sin 3pi/2 is equal to the y-coordinate (-1). ∴ sin 3pi/2 = -1.

How to Find Sin 3pi/2 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 3π/2 can be given in terms of other trigonometric functions as:

± √(1-cos²(3pi/2))
± tan(3pi/2)/√(1 + tan²(3pi/2))
± 1/√(1 + cot²(3pi/2))
± √(sec²(3pi/2) - 1)/sec(3pi/2)
1/cosec(3pi/2)

☛ Also check: trigonometric table

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