Quick answer
The cosecant is the reciprocal trigonometric ratio csc(θ) = 1 ÷ sin θ. csc(30°) = 2.
Enter an angle and choose degrees or radians. The calculator returns csc(θ).
Related tools: Trigonometry Calculator: All six trigonometric functions of an angle. SOHCAHTOA Calculator: Find the other two sides of a right triangle from one side and an acute angle. Reference Angle Calculator: Acute angle between the terminal side and the x-axis.
How to use the Cosecant Calculator
- Enter angle.
- Choose unit: Degrees and Radians.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Angle | A number | — |
| Unit | Choose one: Degrees; Radians | Starts at Degrees |
Formula and method
Formula
csc(θ)
IEEE-754 double precision; shown to at most 6 decimal places.
Angle is converted to radians and evaluated with IEEE-754 math functions. Values where the function is undefined are rejected.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Angle = 30; Unit = Degrees.
| Result | Value |
|---|---|
| csc(30°) | 2 |
Example 2
Inputs: Angle = 90; Unit = Degrees.
| Result | Value |
|---|---|
| csc(90°) | 1 |
The first line is the headline result (csc(30°)). IEEE-754 double precision; shown to at most 6 decimal places.
Relationship to the basic ratios
csc(θ) is the reciprocal of sin θ, so it is large wherever sin θ is close to zero. In a right triangle it equals the hypotenuse divided by the opposite side.
Where it is undefined
csc(θ) is undefined at 0°, 180° and 360° (where sin θ = 0), and the calculator reports this instead of returning a huge number. Values of csc θ are never between −1 and 1; they are ≥ 1 or ≤ −1.
Worked example
For θ = 30°, 1 ÷ sin θ gives 2. Checking with the basic ratios confirms the reciprocal relationship, and the result matches the value from the Trigonometry Calculator.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Angle: entering "0" shows "csc is undefined at this angle." beside that field and no result.
Choosing between the Cosecant Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Trigonometry Calculator](/math/trigonometry-calculator/) | angle and unit | sin |
| [SOHCAHTOA Calculator](/math/sohcahtoa-calculator/) | acute angle, known side length and known side is the | Opposite |
| [Reference Angle Calculator](/math/reference-angle/) | angle | Reference angle |
| [Inverse Sine Calculator](/math/inverse-sine/) | value | arcsin(0.5) |
| [Coterminal Angle Calculator](/math/coterminal-angle/) | angle | Smallest non-negative coterminal angle |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- Undefined at some angles (for example tan 90°)
- Angles may be negative or larger than 360°
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for angle to confirm this.
Frequently asked questions
What is csc(30°) exactly?
2, found from the reciprocal rule: 1 ÷ sin θ at 30°.
How is csc related to sin θ?
They are reciprocals, so csc(θ) × sin θ = 1.
Do I need degrees or radians?
Either: choose the unit next to the angle; the same angle gives the same ratio in both.
Can the angle be negative or large?
Yes. Angles repeat every 360°, so 420° gives the same result as 60°.
What does the Cosecant Calculator calculate?
csc of an angle in degrees or radians. It returns csc(30°).
What formula does the Cosecant Calculator use?
csc(θ). Angle is converted to radians and evaluated with IEEE-754 math functions.
What does the “Unit” option change?
It sets how the cosecant is worked out. The choices are Degrees and Radians.
What is an example of a cosecant result?
With angle 30 and unit Degrees, the csc(30°) is 2.
Related calculators
- Trigonometry Calculator — All six trigonometric functions of an angle.
- SOHCAHTOA Calculator — Find the other two sides of a right triangle from one side and an acute angle.
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
- Inverse Sine Calculator — arcsin (inverse trigonometric function) with the result in degrees and radians.
- Coterminal Angle Calculator — Angles sharing the same terminal side.
- Cosine Calculator — cos of an angle in degrees or radians.
- Secant Calculator — sec of an angle in degrees or radians.
- Cotangent Calculator — cot of an angle in degrees or radians.
- Law of Sines Calculator — Solve a triangle from two angles and one side (law of sines).
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.