Cosecant Calculator - csc(x)
Enter an angle in degrees, radians, or as a multiple of pi to find its cosecant value. The result updates as you type, with a worked step-by-step solution, a graph of csc(x), and a reference table of common angles. You can also reverse-solve: enter a known csc value and get the principal angle back.
Formula
Worked example
To find csc(30 deg): sin(30 deg) = 0.5, so csc(30 deg) = 1 / 0.5 = 2. For a right triangle with hypotenuse 10 and opposite side 5, csc = 10 / 5 = 2, confirming the angle is 30 degrees.
What is the cosecant function?
The cosecant of an angle x, written csc(x), is defined as the reciprocal of the sine: csc(x) = 1 / sin(x). In a right triangle, it equals the hypotenuse divided by the side opposite the angle. Because sine ranges from -1 to 1, cosecant is always either >= 1 or <= -1, never between those values. The function is undefined wherever sin(x) = 0, which happens at every integer multiple of 180 degrees (or pi radians), creating vertical asymptotes at those points. Cosecant is a periodic function with period 360 degrees (2pi radians), matching the period of sine.
How to calculate cosecant step by step
Step 1: Convert your angle to radians if working in radians (degrees x pi / 180). Step 2: Compute sin(x) using a calculator or the unit circle. Step 3: Take the reciprocal: csc(x) = 1 / sin(x). If sin(x) = 0 the result is undefined. For example, csc(45 deg): sin(45 deg) = sqrt(2)/2 ≈ 0.7071, so csc(45 deg) = 1 / 0.7071 ≈ 1.4142 = sqrt(2). On a right triangle, cosecant is even simpler: measure the hypotenuse and the side opposite the angle, then divide hypotenuse by opposite.
Cosecant vs. arccosecant (inverse cosecant)
Cosecant takes an angle and returns a ratio. Its inverse, arccosecant (arccsc or csc-1), goes the other way: given a ratio y with |y| >= 1, it returns the principal angle whose cosecant equals y. The principal value is defined on [-90, -0) union (0, 90] degrees, or equivalently [-pi/2, 0) union (0, pi/2] in radians. The formula is arccsc(y) = arcsin(1 / y). Importantly, arccosecant is NOT the same as 1 / cos(x); that would be the secant function. Toggle "Reverse solve" in this calculator to use the inverse.
The cosecant graph and its properties
The graph of y = csc(x) consists of U-shaped curves separated by vertical asymptotes at each multiple of 180 degrees. On the interval (0, 180 deg), csc(x) starts at positive infinity, falls to 1 at 90 deg, then rises back to positive infinity. On (180 deg, 360 deg), csc(x) mirrors this shape below the x-axis, reaching a minimum of -1 at 270 deg. Key properties: period = 360 deg (2pi), range = (-infinity, -1] union [1, +infinity), no x-intercepts, not defined at x = n x 180 deg for any integer n, and the function is odd: csc(-x) = -csc(x).
Common angles and their cosecant values
| Angle (deg) | Angle (rad) | Exact csc | Decimal csc |
|---|---|---|---|
| 0 | 0 | undefined | undefined |
| 15 | pi/12 | sqrt(6) + sqrt(2) | 3.8637 |
| 30 | pi/6 | 2 | 2.0000 |
| 45 | pi/4 | sqrt(2) | 1.4142 |
| 60 | pi/3 | 2sqrt(3)/3 | 1.1547 |
| 90 | pi/2 | 1 | 1.0000 |
| 120 | 2pi/3 | 2sqrt(3)/3 | 1.1547 |
| 135 | 3pi/4 | sqrt(2) | 1.4142 |
| 150 | 5pi/6 | 2 | 2.0000 |
| 180 | pi | undefined | undefined |
| 210 | 7pi/6 | -2 | -2.0000 |
| 225 | 5pi/4 | -sqrt(2) | -1.4142 |
| 240 | 4pi/3 | -2sqrt(3)/3 | -1.1547 |
| 270 | 3pi/2 | -1 | -1.0000 |
| 300 | 5pi/3 | -2sqrt(3)/3 | -1.1547 |
| 315 | 7pi/4 | -sqrt(2) | -1.4142 |
| 330 | 11pi/6 | -2 | -2.0000 |
| 360 | 2pi | undefined | undefined |
Exact and decimal csc values for the most frequently used angles in trigonometry.
Frequently asked questions
What is csc in trigonometry?
csc stands for cosecant, one of the six standard trigonometric functions. It is the reciprocal of sine: csc(x) = 1 / sin(x). In a right triangle, it equals the length of the hypotenuse divided by the length of the side opposite the angle.
Why is cosecant undefined at 0 and 180 degrees?
Because csc(x) = 1 / sin(x), and sin(0) = sin(180 deg) = 0. Division by zero is undefined, so csc has vertical asymptotes at every multiple of 180 degrees (equivalently, every multiple of pi radians). Near these angles, the cosecant value approaches positive or negative infinity.
Is csc the same as arcsin?
No. csc(x) is 1 / sin(x), the reciprocal of sine. arcsin (or sin-1) is the inverse function of sine, returning the angle whose sine equals a given value. The inverse of cosecant is arccosecant (arccsc), calculated as arcsin(1 / y).
What values can cosecant take?
The range of cosecant is all real numbers with |csc(x)| >= 1. In interval notation: (-infinity, -1] union [1, +infinity). Values between -1 and 1 are impossible for any real angle.
How do I find cosecant without a calculator?
For common angles, use the unit circle. sin(30 deg) = 1/2, so csc(30 deg) = 2. sin(45 deg) = sqrt(2)/2, so csc(45 deg) = sqrt(2) ≈ 1.414. sin(60 deg) = sqrt(3)/2, so csc(60 deg) = 2/sqrt(3) = 2sqrt(3)/3 ≈ 1.155. sin(90 deg) = 1, so csc(90 deg) = 1. For other angles, compute sin first, then take the reciprocal.
How does cosecant behave with negative angles?
Cosecant is an odd function: csc(-x) = -csc(x). This means csc(-30 deg) = -2, csc(-90 deg) = -1, and so on. A negative angle simply reflects the result across zero.