Cosecant Calculator
Cosecant is the trig function most calculators refuse to give you a button for. Homework, wave formulas, and integration tables all use csc(x), and every time you have to build it yourself out of sine and a reciprocal, hoping the angle mode was right.
This calculator takes an angle in degrees or radians and returns the cosecant directly, with the exact value recognized at the standard angles, a proper undefined result where sine is zero, and a table of all six trig functions at your angle so you can cross-check the reciprocal by eye.
Cosecant is undefined wherever sine is zero: at 0, 180, 360 degrees and every other multiple of 180 degrees (multiples of pi in radians).
Quick Answer
Key takeaways
- Cosecant is exactly one thing: the reciprocal of sine. Every property it has, its range, its asymptotes, its sign, comes from that one definition.
- It is undefined at multiples of 180 degrees, where sine is zero, and it never takes a value strictly between -1 and 1.
- At the standard angles the values are exact: csc(30 degrees) = 2, csc(45 degrees) = √2, csc(90 degrees) = 1. This calculator names them instead of leaving you with a long decimal.
- Degrees and radians are the whole ballgame: csc(30 radians) and csc(30 degrees) are different numbers, so the unit selector matters more than the arithmetic.
- Quick Answer
- How to use the cosecant calculator
- Cosecant from the unit circle: 1 over sine
- Worked example, checked against this exact tool: csc(30 degrees)
- Where cosecant is undefined, and why the graph explodes
- Exact values worth memorizing
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to use the cosecant calculator
Type the angle, choose degrees or radians to match the problem in front of you, and press Calculate Cosecant. The headline result is the cosecant to six decimal places, with the exact value in brackets at the standard angles. Underneath, the working shows the sine value the answer came from, so you can verify the reciprocal step yourself: cosecant times sine must come back to 1.
The six-function table under the result is deliberate. Cosecant belongs to a family, and seeing sin, cos, and tan next to csc, sec, and cot exposes sign errors instantly. Its closest sibling on this site works the adjacent-over-hypotenuse ratio instead: the cosine calculator takes the same kind of angle input and returns cos(x), and cosine is the cofunction here, since sin(x) = cos(90 degrees – x), which makes csc(x) = sec(90 degrees – x).
Cosecant from the unit circle: 1 over sine
The definition fits in one line, and everything else on this page is a consequence of it.
OpenStax Precalculus states the same definition in unit-circle terms: for a point (x, y) where the terminal side of the angle meets the unit circle, csc(t) = 1/y whenever y is not zero, which is the hypotenuse-over-opposite ratio with a hypotenuse of 1. The NIST Digital Library of Mathematical Functions treats cosecant as one of the six basic functions with the same reciprocal relationship, and its shift identities (csc(x) = sec(x + pi/2) with sign conventions) follow from it. Because cosecant is built from sine, it inherits sine’s period of 360 degrees and its sign pattern, flipped in size: small sine, enormous cosecant.
Worked example, checked against this exact tool: csc(30 degrees)
The calculator is pre-filled with 30 degrees, and every figure below is what it returns.
| Function | Value at 30 degrees | Exact form |
|---|---|---|
| sin | 0.500000 | 1/2 |
| cos | 0.866025 | √3/2 |
| tan | 0.577350 | 1/√3 |
| csc | 2.000000 | 2 |
| sec | 1.154701 | 2/√3 |
| cot | 1.732051 | √3 |
The formula line under the result reads: sin(30 degrees) = 0.500000000, so csc = 1 / 0.500000000 = 2.000000. That one line is the entire computation, printed so a teacher, a grader, or your own doubt can audit it. Angles measured from a vertex come up constantly in this family of problems, and the central angle calculator is the companion when the angle itself is what you are solving for from an arc and a radius.
Where cosecant is undefined, and why the graph explodes
Cosecant fails exactly where sine succeeds in hitting zero: 0, 180, 360 degrees, and every other multiple of 180 degrees (n x pi in radians, for any whole number n). Approach one of those angles and the values do not drift, they blow up: csc(1 degree) is about 57.30, csc(0.1 degree) is about 572.96, and at the angle itself there is no value at all, just a vertical asymptote with the graph fleeing to positive infinity on one side and negative infinity on the other.
This calculator handles that case honestly. Enter 0 or 180 degrees and it tells you cosecant is undefined because sin = 0 at multiples of 180 degrees, and 1/0 has no value, while still printing the other five functions that do exist at your angle. Tools that return a giant number like 1.6e16 at these angles are showing you floating-point noise dressed up as an answer; a multiple of 180 degrees deserves the word undefined, not a sixteen-digit coincidence.
Exact values worth memorizing
Ten angles cover nearly every textbook problem, and this calculator’s exact-value detection is built on them: 30, 45, 90, 135, 150, 210, 225, 270, 315, and 330 degrees. The pattern is easier than the list. In the first quadrant, csc runs 2, √2, 1 as the angle runs 30, 45, 90. The second quadrant repeats the first with the supplements (150 is 2, 135 is √2). The third and fourth quadrants repeat the sizes with negative signs, because sine is negative there: csc(210 degrees) = -2, csc(270 degrees) = -1.
Inverse questions run the other way: given a cosecant value, which angle produced it? That is arccsc territory, and the same direction-flip applies to the better-known functions: the acos calculator returns the angle from a cosine value, and the arctan calculator does it from a tangent value. Keeping the forward tools and the inverse tools separate is what stops the classic mix-up of feeding a ratio into a function that expects an angle.
What competing calculators miss
Cosecant tools in the search results mostly evaluate the reciprocal and stop. Three things separate a lookup from an answer you can trust.
Miss 1: undefined points shown as garbage numbers. At 0 or 180 degrees, several tools display an enormous rounded figure produced by dividing by a near-zero floating-point sine. That number is an artifact, and quoting it in homework or a report is simply wrong. Here the result says undefined and explains why.
Miss 2: no exact values. A result of 1.414214 invites the question of whether the true answer was √2. At the ten standard angles this calculator states the exact form next to the decimal, which is the form exams and symbolic work actually want.
Miss 3: cosecant in isolation. A lone csc value cannot be sanity-checked. The six-function table lets you verify the reciprocal against the sine sitting one row away, and it is how most errors (wrong unit, wrong quadrant sign) get caught before they travel.
One honest limit: this is a forward calculator. It does not evaluate arccsc, solve triangles, or graph the function, and its exact-value recognition covers the ten standard angles rather than arbitrary surds.
Frequently Asked Questions
Cosecant is the reciprocal of sine: csc(x) = 1 / sin(x). On a right triangle it is the hypotenuse divided by the opposite side, and on the unit circle it is 1 divided by the y-coordinate. It is one of the six standard trigonometric functions, alongside sine, cosine, tangent, secant, and cotangent.
Exactly 2. Sin(30 degrees) is 1/2, and the reciprocal of 1/2 is 2. This calculator recognizes the standard angles and reports the exact value next to the decimal, so csc(30 degrees) shows as 2.000000 with exact value 2.
Undefined. Sin(0) is 0, and 1 divided by 0 has no value. The same happens at every multiple of 180 degrees (pi radians), where the cosecant graph has vertical asymptotes. This calculator detects those angles and explains the undefined result instead of showing an error or a huge number.
Most scientific calculators have no cosecant key. Compute sin(x) first, then press the reciprocal key (1/x or x⁻¹), or simply divide 1 by the sine value. Check the angle mode before you start: the degrees-versus-radians mix-up is the most common way this goes wrong.
They are reciprocals, so their values multiply to 1 (where both are defined) and their graphs are mirror images in size: where sine peaks at 1, cosecant bottoms out at 1; where sine approaches 0, cosecant runs off to infinity. Sine stays between -1 and 1; cosecant never lands strictly between -1 and 1.
1, because sin(pi/2) is 1 and 1/1 is 1. It is the smallest positive value cosecant ever takes, reached at 90 degrees and every full turn after it.
Yes, wherever sine is negative: in the third and fourth quadrants, between 180 and 360 degrees. Cosecant is an odd function, so csc(-x) = -csc(x); csc(-30 degrees) is -2, the negative of csc(30 degrees).
All real numbers except those strictly between -1 and 1: in interval form, (-infinity, -1] union [1, infinity). Its period is 360 degrees (2pi radians), the same as sine, so csc(x + 360 degrees) = csc(x).
Arccsc, also written csc⁻¹, is the inverse cosecant: it takes a value with absolute size at least 1 and returns the angle whose cosecant it is. This calculator runs the forward direction, angle to cosecant; inverse work belongs to the dedicated inverse-function tools.
In physics and engineering wherever sine appears in a denominator: wave and optics formulas, certain integration techniques in calculus (the integral of csc is a standard result), antenna and signal work, and anywhere a hypotenuse-over-opposite ratio shows up in triangle problems.
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References
- OpenStax, Precalculus: the other trigonometric functions, defining cosecant as the reciprocal of sine on the unit circle (csc t = 1/y). OpenStax Precalculus: The Other Trigonometric Functions. Accessed October 2026.
- National Institute of Standards and Technology, Digital Library of Mathematical Functions, Section 4.15: trigonometric function graphics and shift identities involving cosecant. NIST DLMF: Trigonometric Functions, Section 4.15. Accessed October 2026.
- Auburn Educational Association Network: how to calculate cosecant on a calculator, including the reciprocal method and angle-mode checks. AEANET: How Do I Calculate Cosecant on a Calculator?. Accessed October 2026.
- Union Test Prep, CLT Quantitative Reasoning study guide: unit circle values and the reciprocal definitions of cosecant, secant, and cotangent. Union Test Prep: Quantitative Reasoning Study Guide. Accessed October 2026.
- University of Misan, Engineering Mathematics lecture notes: the six basic trigonometric functions and their ratio definitions, including cosecant as c/a. Engineering Mathematics Lecture Notes: Trigonometric Functions. Accessed October 2026.