Cosec pi/4

The value of cosec pi/4 is 1.4142135. . .. Cosec pi/4 radians in degrees is written as cosec ((π/4) × 180°/π), i.e., csc (45°). In this article, we will discuss the methods to find the value of csc pi/4 with examples.

• Cosec pi/4: √2
• Cosec pi/4 in decimal: 1.4142135. . .
• Cosec (-pi/4): -1.4142135. . . or -√2
• Cosec pi/4 in degrees: csc (45°)

What is the Value of Cosec pi/4?

The value of cosec pi/4 in decimal is 1.414213562. . .. Cosec pi/4 can also be expressed using the equivalent of the given angle (pi/4) in degrees (45°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/4 radians = pi/4 × (180°/pi) = 45° or 45 degrees
∴ cosec pi/4 = csc π/4 = cosec(45°) = √2 or 1.4142135. . .

Explanation:

For csc pi/4, the angle pi/4 lies between 0 and pi/2 (First Quadrant). Since cosecant function is positive in the first quadrant, thus csc pi/4 value = √2 or 1.4142135. . .
Since the cosecant function is a periodic function, we can represent cosec pi/4 as, cosec pi/4 = cosec(pi/4 + n × 2pi), n ∈ Z.
⇒ cosec pi/4 = cosec 9pi/4 = csc 17pi/4 , and so on.
Note: Since, cosecant is an odd function, the value of cosec(-pi/4) = -cosec(pi/4).

Methods to Find Value of Cosec pi/4

The cosecant function is positive in the 1st quadrant. The value of cosec pi/4 is given as 1.41421. . .. We can find the value of csc pi/4 by:

• Using Unit Circle
• Using Trigonometric Functions

Cosec pi/4 Using Unit Circle

To find the value of csc π/4 using the unit circle:

• Rotate ‘r’ anticlockwise to form pi/4 angle with the positive x-axis.
• The csc of pi/4 equals the reciprocal of the y-coordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.

Hence the value of csc pi/4 = 1/y = 1.4142 (approx)

Cosec pi/4 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the csc pi/4 as:

• ± 1/√(1-cos²(pi/4))
• ± √(1 + tan²(pi/4))/tan(pi/4)
• ± √(1 + cot²(pi/4))
• ± sec(pi/4)/√(sec²(pi/4) - 1))
• 1/sin(pi/4)

Note: Since pi/4 lies in the 1st Quadrant, the final value of csc pi/4 will be positive.

We can use trigonometric identities to represent cosec pi/4 as,

• csc(pi - pi/4) = csc 3pi/4
• -cosec(pi + pi/4) = -cosec 5pi/4
• sec(pi/2 - pi/4) = sec pi/4
• -sec(pi/2 + pi/4) = -sec 3pi/4

☛ Also Check:

• tan 5pi/12
• sin pi/6
• cos 7pi/3
• tan 19pi/3
• cot pi/6
• tan 5pi/4

FAQs on Cosec pi/4

What is Cosec pi/4?

Cosec pi/4 is the value of cosecant trigonometric function for an angle equal to π/4. The value of csc pi/4 is √2 or 1.4142 (approx).

What is the Value of Csc pi/4 in Terms of Cos pi/4?

Using trigonometric identities, we can write csc pi/4 in terms of cos pi/4 as, csc(pi/4) = 1/√(1 - cos²(pi/4)). Here, the value of cos pi/4 is equal to 0.7071.

How to Find Cosec pi/4 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of csc pi/4 can be given in terms of other trigonometric functions as:

• ± 1/√(1-cos²(pi/4))
• ± √(1 + tan²(pi/4))/tan(pi/4)
• ± √(1 + cot²(pi/4))
• ± sec(pi/4)/√(sec²(pi/4) - 1))
• 1/sin(pi/4)

☛ Also check: trigonometric table

How to Find the Value of Cosec pi/4?

The value of csc pi/4 can be calculated by constructing an angle of π/4 radians with the x-axis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of cosec pi/4 is equal to the reciprocal of the y-coordinate (0.7071). ∴ csc pi/4 = 1.4142.

What is the Exact Value of Cosec pi/4?

The exact value of cosec pi/4 can be given accurately up to 8 decimal places as 1.41421356 or as √2.