Cosecant Calculator
Calculate cosecant (csc) values for angles in degrees and radians with exact values for special angles
Calculate Cosecant (csc)
Enter any real number for the angle
Quick Reference
Common angles:
30° → csc = 2
45° → csc = √2 ≈ 1.414
60° → csc = 2√3/3 ≈ 1.155
90° → csc = 1
Cosecant Results
Domain Analysis
Domain: All real numbers except x ≠ nπ (where n is any integer) in radians, or x ≠ n×180° in degrees.
Range: (-∞, -1] ∪ [1, ∞)
Special Angles
Cosecant Properties
csc(x) = 1/sin(x)
Period: 360° (2π radians)
Range: (-∞, -1] ∪ [1, ∞)
Odd function: csc(-x) = -csc(x)
Undefined when sin(x) = 0
Understanding the Cosecant Function
What is Cosecant?
The cosecant function (csc) is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function: csc(x) = 1/sin(x).
Right Triangle Definition
In a right triangle, cosecant of an angle is the ratio of the hypotenuse to the opposite side: csc(θ) = hypotenuse/opposite.
Domain and Range
- •Domain: All real numbers except multiples of π (or 180°)
- •Range: (-∞, -1] ∪ [1, ∞)
Key Properties
Reciprocal Identity
csc(x) = 1/sin(x)
Pythagorean Identity
1 + cot²(x) = csc²(x)
Periodicity
csc(x + 2π) = csc(x)
Odd Function
csc(-x) = -csc(x)