Cosec 3pi/4
The value of cosec 3pi/4 is 1.4142135. . .. Cosec 3pi/4 radians in degrees is written as cosec ((3π/4) × 180°/π), i.e., csc (135°). In this article, we will discuss the methods to find the value of cosec 3pi/4 with examples.
 Cosec 3pi/4: √2
 Cosec 3pi/4 in decimal: 1.4142135. . .
 Cosec (3pi/4): 1.4142135. . . or √2
 Cosec 3pi/4 in degrees: csc (135°)
What is the Value of Cosec 3pi/4?
The value of csc 3pi/4 in decimal is 1.414213562. . .. Cosec 3pi/4 can also be expressed using the equivalent of the given angle (3pi/4) in degrees (135°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/4 radians = 3pi/4 × (180°/pi) = 135° or 135 degrees
∴ csc 3pi/4 = csc 3π/4 = cosec(135°) = √2 or 1.4142135. . .
Explanation:
For cosec 3pi/4, the angle 3pi/4 lies between pi/2 and pi (Second Quadrant). Since cosecant function is positive in the second quadrant, thus csc 3pi/4 value = √2 or 1.4142135. . .
Since the cosecant function is a periodic function, we can represent cosec 3pi/4 as, csc 3pi/4 = cosec(3pi/4 + n × 2pi), n ∈ Z.
⇒ csc 3pi/4 = cosec 11pi/4 = cosec 19pi/4 , and so on.
Note: Since, cosecant is an odd function, the value of csc(3pi/4) = cosec(3pi/4).
Methods to Find Value of Cosec 3pi/4
The cosecant function is positive in the 2nd quadrant. The value of cosec 3pi/4 is given as 1.41421. . .. We can find the value of csc 3pi/4 by:
 Using Trigonometric Functions
 Using Unit Circle
Cosec 3pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the csc 3pi/4 as:
 ± 1/√(1cos²(3pi/4))
 ± √(1 + tan²(3pi/4))/tan(3pi/4)
 ± √(1 + cot²(3pi/4))
 ± sec(3pi/4)/√(sec²(3pi/4)  1))
 1/sin(3pi/4)
Note: Since 3pi/4 lies in the 2nd Quadrant, the final value of csc 3pi/4 will be positive.
We can use trigonometric identities to represent csc 3pi/4 as,
 csc(pi  3pi/4) = csc pi/4
 cosec(pi + 3pi/4) = cosec 7pi/4
 sec(pi/2  3pi/4) = sec(pi/4)
 sec(pi/2 + 3pi/4) = sec 5pi/4
Cosec 3pi/4 Using Unit Circle
To find the value of cosec 3π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form 3pi/4 angle with the positive xaxis.
 The cosec of 3pi/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 cosec 3pi/4 = 1/y = 1.4142 (approx)
☛ Also Check:
Examples Using Cosec 3pi/4

Example 1: Simplify: 13 (csc(3pi/4)/csc(11pi/4))
Solution:
We know csc 3pi/4 = csc 11pi/4
⇒ 13 csc(3pi/4)/csc(11pi/4) = 13(csc(3pi/4)/csc(3pi/4))
= 13(1) = 13 
Example 2: Using the value of csc 3pi/4, solve: (1 + cot²(3pi/4)).
Solution:
We know, (1 + cot²(3pi/4)) = (csc²(3pi/4)) = 2
⇒ (1 + cot²(3pi/4)) = 2 
Example 3: Find the value of (sec(3pi/8) csc(3pi/8))/2. [Hint: Use csc 3pi/4 = 1.4142]
Solution:
Using the sin 2a formula,
(sec(3pi/8) csc(3pi/8))/2 = 1/(2 × cos(3pi/8) sin(3pi/8)) = 1/sin(3pi/4)
= 1/sin(3pi/4) = csc 3pi/4 = 1.4142
⇒ (sec(3pi/8) csc(3pi/8))/2 = 1.4142
FAQs on Cosec 3pi/4
What is Cosec 3pi/4?
Cosec 3pi/4 is the value of cosecant trigonometric function for an angle equal to 3π/4. The value of cosec 3pi/4 is √2 or 1.4142 (approx).
How to Find the Value of Cosec 3pi/4?
The value of csc 3pi/4 can be calculated by constructing an angle of 3π/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 csc 3pi/4 is equal to the reciprocal of the ycoordinate (0.7071). ∴ csc 3pi/4 = 1.4142.
What is the Value of Cosec 3pi/4 in Terms of Sec 3pi/4?
Since the cosec function can be represented using the secant function, we can write csc 3pi/4 as sec(3pi/4)/√(sec²(3pi/4)  1). The value of sec 3pi/4 is equal to 1.4142.
How to Find Cosec 3pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of csc 3pi/4 can be given in terms of other trigonometric functions as:
 ± 1/√(1cos²(3pi/4))
 ± √(1 + tan²(3pi/4))/tan(3pi/4)
 ± √(1 + cot²(3pi/4))
 ± sec(3pi/4)/√(sec²(3pi/4)  1))
 1/sin(3pi/4)
☛ Also check: trigonometric table
What is the Value of Csc 3pi/4 in Terms of Cos 3pi/4?
Using trigonometric identities, we can write csc 3pi/4 in terms of cos 3pi/4 as, csc(3pi/4) = 1/√(1  cos²(3pi/4)). Here, the value of cos 3pi/4 is equal to 0.7071.
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