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 xaxis.
 The csc of pi/4 equals the reciprocal of the ycoordinate(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/√(1cos²(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:
Examples Using Cosec pi/4

Example 1: Simplify: 8 (csc(pi/4)/csc(9pi/4))
Solution:
We know csc pi/4 = csc 9pi/4
⇒ 8 csc(pi/4)/csc(9pi/4) = 8(csc(pi/4)/csc(pi/4))
= 8(1) = 8 
Example 2: Find the value of 6 cosec(pi/4)/8 sec(pi/4).
Solution:
Using trigonometric identities, we know, cosec(pi/4) = sec(pi/2  pi/4) = sec pi/4.
⇒ csc(pi/4) = sec(pi/4)
⇒ Value of 6 cosec(pi/4)/8 sec(pi/4) = 3/4 
Example 3: Find the value of csc pi/4 if sin pi/4 is 0.7071.
Solution:
Since, csc pi/4 = 1/sin(pi/4)
⇒ csc pi/4 = 1/0.7071 = 1.4142
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/√(1cos²(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 xaxis, 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 ycoordinate (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.
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