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Cosec pi/2

The value of cosec pi/2 is 1. Cosec pi/2 radians in degrees is written as cosec ((π/2) × 180°/π), i.e., csc (90°). In this article, we will discuss the methods to find the value of csc pi/2 with examples.

Cosec pi/2: 1
Cosec (-pi/2): -1
Cosec pi/2 in degrees: cosec (90°)

What is the Value of Cosec pi/2?

The value of cosec pi/2 is 1. Cosec pi/2 can also be expressed using the equivalent of the given angle (pi/2) in degrees (90°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/2 radians = pi/2 × (180°/pi) = 90° or 90 degrees
∴ csc pi/2 = cosec π/2 = csc(90°) = 1

Explanation:

For csc pi/2, the angle pi/2 lies on the positive y-axis. Thus, cosec pi/2 value = 1
Since the cosecant function is a periodic function, we can represent csc pi/2 as, cosec pi/2 = csc(pi/2 + n × 2pi), n ∈ Z.
⇒ csc pi/2 = cosec 5pi/2 = cosec 9pi/2 , and so on.
Note: Since, cosecant is an odd function, the value of cosec(-pi/2) = -cosec(pi/2).

Methods to Find Value of Cosec pi/2

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

Using Trigonometric Functions
Using Unit Circle

Cosec pi/2 in Terms of Trigonometric Functions

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

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

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

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

cosec(pi - pi/2) = csc pi/2
-cosec(pi + pi/2) = -cosec 3pi/2
sec(pi/2 - pi/2) = sec 0
-sec(pi/2 + pi/2) = -sec pi

Cosec pi/2 Using Unit Circle

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

Rotate ‘r’ anticlockwise to form pi/2 angle with the positive x-axis.
The cosec of pi/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 cosec pi/2 = 1/y = 1

☛ Also Check:

Examples Using Cosec pi/2

Example 1: Simplify: 12 (csc(pi/2)/csc(5pi/2))

Solution:

We know csc pi/2 = csc 5pi/2
⇒ 12 csc(pi/2)/csc(5pi/2) = 12(csc(pi/2)/csc(pi/2))
= 12(1) = 12
Example 2: Using the value of csc pi/2, solve: (1 + cot²(pi/2)).

Solution:
We know, (1 + cot²(pi/2)) = (csc²(pi/2)) = 1
⇒ (1 + cot²(pi/2)) = 1
Example 3: Find the value of csc pi/2 if sin pi/2 is 1.

Solution:
Since, csc pi/2 = 1/sin(pi/2)
⇒ csc pi/2 = 1/1 = 1

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FAQs on Cosec pi/2

What is Cosec pi/2?

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

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

We can represent the csc function in terms of the cotangent function using trig identities, cosec pi/2 can be written as √(1 + cot²(pi/2)). Here, the value of cot pi/2 is equal to 0.

How to Find the Value of Cosec pi/2?

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

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

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

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

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

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

☛ Also check: trigonometry table

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