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/√(1-cos²(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 x-axis.
- The cosec of 3pi/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 cosec 3pi/4 = 1/y = 1.4142 (approx)
☛ Also Check:
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 x-axis, 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 y-coordinate (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/√(1-cos²(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.