Cosec 3pi/2

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

• Cosec 3pi/2: -1
• Cosec (-3pi/2): 1
• Cosec 3pi/2 in degrees: csc (270°)

What is the Value of Cosec 3pi/2?

The value of cosec 3pi/2 is -1. Cosec 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
∴ csc 3pi/2 = csc 3π/2 = cosec(270°) = -1

Explanation:

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

Methods to Find Value of Cosec 3pi/2

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

• Using Trigonometric Functions
• Using Unit Circle

Cosec 3pi/2 in Terms of Trigonometric Functions

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

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

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

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

• csc(pi - 3pi/2) = csc(-pi/2)
• -cosec(pi + 3pi/2) = -cosec 5pi/2
• sec(pi/2 - 3pi/2) = sec(-pi)
• -sec(pi/2 + 3pi/2) = -sec 2pi

Cosec 3pi/2 Using Unit Circle

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

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

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

☛ Also Check:

• cot 3pi/2
• tan 2pi
• cot 5pi/3
• sin 9pi/4
• tan 11pi/12
• tan pi/3

FAQs on Cosec 3pi/2

What is Cosec 3pi/2?

Cosec 3pi/2 is the value of cosecant trigonometric function for an angle equal to 3π/2. The value of cosec 3pi/2 is -1.

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

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

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

☛ Also check: trigonometric table

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

We know, using trig identities, we can write csc 3pi/2 as -√(1 + tan²(3pi/2))/tan 3pi/2. Here, the value of tan 3pi/2 is equal to not defined.

What is the Exact Value of Cosec 3pi/2?

The exact value of csc 3pi/2 is -1.

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

The value of csc 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 csc 3pi/2 is equal to the reciprocal of the y-coordinate (-1). ∴ csc 3pi/2 = -1.