Welcome to Calcugo's cosecant calculator, where we study the mysterious csc trig function. Arguably, even if you've had a course in trigonometry, you may have missed this one, since we rarely use it. Nevertheless, the csc in math is a well-defined function that can still appear in some textbooks. Type an angle in degrees, radians, or gradians and the calculator returns csc(α) instantly — or switch modes and work backwards from a cosecant value to the angle (the arccosecant).
Since time is money, let's jump right into the topic! Below you'll find the cosecant formula, the cosecant graph, alternative ways to find csc, and a worked example using the calculator.
What Is csc? The Cosecant Formula
Geometry begins with the simplest object — the triangle. The idea behind trigonometry is to relate the size of a triangle's inside angles to its sides. To get a firmer grip on how it's done, we focus on right triangles (the ones the Pythagorean theorem is about). There, if we take one of the acute angles and make it larger, then the opposite side has to be longer too. We define the trigonometric functions as ratios of a triangle's sides.
So what is the cosecant? The cosecant of an angle is the hypotenuse divided by the side (leg) opposite the angle. In other words, it is the reciprocal of the sine:
csc(α) = hypotenuse / opposite = c / a = 1 / sin(α)
Note that this definition does not mention how big the triangle is — that's a crucial strength of trigonometry. Even if we scale the triangle to twice its size, the value of csc(α) does not change.
From the triangle picture, sin, cos, and csc have a meaning only for angles between 0 and 90 degrees (0 and π/2 in radians), simply because it's a right triangle. But we don't have to pay that price. We can translate the whole reasoning to the two-dimensional plane. Let A = (x, y) be a point, and let α be the (directed) angle measured counterclockwise from the positive horizontal axis to the segment from the origin to A. Substituting a → y, b → x, and c → √(x² + y²) (the distance from the origin to A) gives the general cosecant formula:
csc(α) = √(x² + y²) / y
This extends the cosecant to any real angle — even negative ones and angles beyond a full turn of 360°. Since sine appears in the denominator, csc(α) is undefined whenever y = 0, i.e., at α = 0°, 180°, 360°, … (every multiple of 180°), where the sine equals zero.
Cosecant Graph
Because csc(α) = 1/sin(α), the cosecant graph is the reciprocal of the sine wave. Wherever sin(α) = 0, the cosecant shoots off to ±∞, producing vertical asymptotes at 0, ±180°, ±360°, and so on. Between the asymptotes the curve forms a series of U-shaped branches. The cosecant never takes a value strictly between −1 and 1: its range is csc(α) ≤ −1 or csc(α) ≥ 1, and each branch has a minimum of 1 (at 90°, 90°±360°, …) or a maximum of −1 (at 270°, 270°±360°, …). Like sine, the cosecant is periodic with period 2π and is an odd function: csc(−α) = −csc(α).
How to Find csc: Alternative csc Formulas
Depending on what you already know, there is more than one way to reach the cosecant:
- From sine (reciprocal): csc(α) = 1 / sin(α). This is the definition used by the calculator.
- From a right triangle (SOHCAHTOA): csc(α) = hypotenuse / opposite.
- From coordinates: csc(α) = √(x² + y²) / y for a point (x, y) on the terminal side of the angle.
- From the Pythagorean identity: csc²(α) = 1 + cot²(α), so csc(α) = ±√(1 + cot²(α)) (the sign follows the quadrant).
- Cofunction identity: csc(α) = sec(90° − α) — the cosecant of an angle equals the secant of its complement.
Cosecant Values Table (csc 0, csc 30°, csc 45°…)
The exact cosecant value is easy to remember for a handful of standard angles. Note the “undef” entries at 0° and 180°, where the sine is zero.
| α (degrees) | α (radians) | csc(α) — exact | csc(α) — decimal |
|---|---|---|---|
| 0° | 0 | undefined | ±∞ |
| 30° | π/6 | 2 | 2 |
| 45° | π/4 | √2 | 1.4142135624 |
| 60° | π/3 | 2√3/3 | 1.1547005384 |
| 90° | π/2 | 1 | 1 |
| 120° | 2π/3 | 2√3/3 | 1.1547005384 |
| 135° | 3π/4 | √2 | 1.4142135624 |
| 150° | 5π/6 | 2 | 2 |
| 180° | π | undefined | ±∞ |
| 210° | 7π/6 | −2 | −2 |
| 270° | 3π/2 | −1 | −1 |
| 330° | 11π/6 | −2 | −2 |
Remember the periodicity of the cosecant — csc(α + 360°) = csc(α) — if your angle is not shown in the table above.
Example: Using the Cosecant Calculator
- Keep the mode on Angle → cosecant.
- Enter the angle in the input field. For example, type 30.
- Choose the angle unit — degrees, radians, or gradians.
- Click Calculate. The calculator instantly shows csc(30°) = 2.
- It also shows the same angle converted to every unit, the related trig values (sin, cos, sec, tan, cot), and the quadrant the angle lands in.
Want to go the other way? Switch the mode to Cosecant value → angle, type a number with absolute value at least 1 (for example 2), and the calculator returns the angle in the [−90°, 90°] range, excluding 0° (this is the arccosecant, or inverse cosecant).
FAQs
What is the cosecant of 30, 45, 60, and 90 degrees?
csc 30° = 2, csc 45° = √2 ≈ 1.4142, csc 60° = 2√3/3 ≈ 1.1547, and csc 90° = 1. The cosecant of 0° and 180° is undefined because the sine of those angles is zero.
Is cosecant an even or odd function?
Cosecant is an odd function, meaning csc(−α) = −csc(α). Its graph is symmetric about the origin, just like the sine it is built from.
What is the range of the cosecant function?
The cosecant never lies strictly between −1 and 1. Its range is csc(α) ≤ −1 or csc(α) ≥ 1. That is why the inverse mode only accepts values with |x| ≥ 1 — there is no angle whose cosecant is, say, 0.5.
How are cosecant and sine related?
They are reciprocals: csc(α) = 1/sin(α), and equivalently sin(α) = 1/csc(α). Wherever sine is zero, the cosecant is undefined, and where sine reaches ±1 the cosecant also reaches ±1.
How do I convert between degrees, radians, and gradians?
A full circle is 360° = 2π rad = 400 grad. To convert degrees to radians, multiply by π/180; to convert degrees to gradians, multiply by 10/9. This calculator shows all three units automatically.