Cosecant Calculator
Enter an angle in degrees or radians to find its cosecant, csc θ.
Updated
Cosecant Calculator
Our Cosecant Calculator finds csc θ for any angle in degrees or radians. Type the angle and the answer appears straight away to 10 decimal places, with the exact value for special angles (such as √2 for 45°), the angle in the other unit, the quadrant, and sin θ and cos θ. It tells you clearly when the cosecant is undefined, at 0°, 180° and similar angles, and gives csc 30° as exactly 2 rather than the 2.0000000000000004 many programs show.
Quick answer
The cosecant of an angle is the reciprocal of the sine: 1 ÷ sin θ, or the hypotenuse divided by the opposite side. For example, csc 30° = 1 ÷ 0.5 = 2. Enter any angle above and the calculator shows csc θ instantly.
- Formula: csc θ = 1 ÷ sin θ = hypotenuse ÷ opposite
- Key values: csc 30° = 2, csc 45° ≈ 1.4142, csc 60° ≈ 1.1547, csc 90° = 1
- Undefined: at 0°, 180°, 360° and every k × 180°, where sin θ = 0
- Range and period: csc θ ≤ −1 or csc θ ≥ 1, repeating every 360° (2π)
What Is Cosecant?
Cosecant is one of the three reciprocal trigonometric functions, with secant and cotangent. It flips the sine upside down, so in a right triangle it compares the hypotenuse with the side opposite the angle:
csc θ = 1 ÷ sin θ
sin θ = opposite ÷ hypotenuse
Because sine never goes beyond −1 or 1, the cosecant never goes between them: dividing 1 by a number smaller than 1 always gives something larger than 1. Where sin θ is 0, the cosecant has no value at all. It is often written csc, and sometimes cosec.
Cosecant Table of Common Angles
| Angle | Exact | Decimal |
|---|---|---|
| 0° (0) | undefined | undefined |
| 30° (π/6) | 2 | 2 |
| 45° (π/4) | √2 | 1.4142 |
| 60° (π/3) | 2√3/3 | 1.1547 |
| 90° (π/2) | 1 | 1 |
| 120° (2π/3) | 2√3/3 | 1.1547 |
| 135° (3π/4) | √2 | 1.4142 |
| 150° (5π/6) | 2 | 2 |
| 180° (π) | undefined | undefined |
| 270° (3π/2) | −1 | −1 |
| 360° (2π) | undefined | undefined |
2√3/3 is the same number as 2/√3. Each cosecant is simply 1 divided by the sine in the same row of a sine table.
The Cosecant Graph
- Shape: U-shaped branches that open up where sin θ > 0 and down where sin θ < 0.
- Touches the sine: csc θ and sin θ meet where sin θ = ±1, at 90° and 270°.
- Asymptotes: vertical lines at 0°, 180°, 360° and every k × 180°.
- Gap: no part of the graph lies between −1 and 1.
- Odd function: csc(−θ) = −csc θ, symmetric about the origin.
Cosecant Rules and Identities
- Reciprocal: csc θ = 1 ÷ sin θ.
- Pythagorean: 1 + cot2θ = csc2θ.
- Complement: csc(90° − θ) = sec θ.
- Supplement: csc(180° − θ) = csc θ.
- Negative angle: csc(−θ) = −csc θ.
- Full turn: csc(θ + 360°) = csc θ.
- Calculus: the derivative of cot θ is −csc2θ, and the derivative of csc θ is −csc θ cot θ.
Worked Examples
Exact value of csc 60°.
= 1 ÷ (√3/2)
= 2/√3 = 2√3/3
≈ 1.1547
Length of a ladder. A ladder makes 60° with the ground and reaches 3 m up a wall. The wall height is the side opposite the angle and the ladder is the hypotenuse:
= 3 × 1.1547
≈ 3.46 m
Radian inputs. csc(1) ≈ 1.1884, csc(2) ≈ 1.0998, csc(3) ≈ 7.0862, csc(4) ≈ −1.3213 and csc(5) ≈ −1.0428. 4 and 5 radians (about 229° and 286°) lie between 180° and 360°, where sin θ is negative, so their cosecants are negative.
Cosecant in Excel and Code
| Tool | csc 30° |
|---|---|
| Excel / Google Sheets | =CSC(RADIANS(30)) |
| Older Excel | =1/SIN(RADIANS(30)) |
| Python | 1 / math.sin(math.radians(30)) |
| JavaScript | 1 / Math.sin(30 * Math.PI / 180) |
| Scientific calculator | DEG mode, sin 30, then 1/x |
Python and JavaScript have no built-in cosecant, and all of these expect radians. Floating-point maths gives 2.0000000000000004 for csc 30° and about 8.17 × 1015 at 180° instead of an error.
Common Mistakes
- Confusing csc with sin⁻¹. csc θ = 1 ÷ sin θ; sin−1 (arcsin) returns an angle.
- Mixing up cosecant and secant. Cosecant pairs with sine, secant with cosine.
- Expecting values below 1. A cosecant between −1 and 1 is always a mistake.
- Wrong angle mode. csc 30 is 2 in degrees but about −1.0121 in radians.
- Missing the undefined angles. csc 0°, csc 180° and csc 360° have no value.
Key Terms
- Hypotenuse
- The longest side of a right triangle, opposite the right angle.
- Opposite
- The side across from θ in a right triangle.
- Reciprocal
- 1 divided by a value; cosecant is the reciprocal of sine.
- Asymptote
- A line the graph approaches but never reaches; cosecant has them at k × 180°.
- Secant
- The reciprocal of cosine, sec θ = 1 ÷ cos θ.
- Arccsc
- The inverse of cosecant, written csc⁻¹ or acsc.
How to Use the Cosecant Calculator
- Enter the angle. Type the angle θ, for example 30, or press one of the angle buttons (30°, 45°, 60° and 90° in degrees, π/6, π/4, π/3 and π/2 in radians). In radians you can type a decimal such as 1 or a multiple of π such as pi/6 or 5π/6.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad). Typing 30° or 1 rad also sets the unit for that angle.
- Read and copy the result. csc θ appears as you type, to 10 decimal places, with the exact value where there is one (for example csc(45°) = √2 ≈ 1.4142135624). The table below shows the angle in the other unit, the quadrant, sin θ and cos θ. At 0°, 180°, 360° and similar angles the cosecant is undefined and the calculator says so. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter 210 in degrees and the result is −2, with the note that 210° is in quadrant III with a reference angle of 30°, and 210° = 7π/6 rad.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is the cosecant of an angle?
The cosecant is the reciprocal of the sine: csc θ = 1 ÷ sin θ. In a right triangle it is the hypotenuse divided by the side opposite the angle: csc θ = hypotenuse ÷ opposite. For example csc 30° = 1 ÷ 0.5 = 2.
What is csc 0°, csc 30°, csc 45°, csc 60° and csc 90°?
csc 0° is undefined, because sin 0° = 0. csc 30° = 2, csc 45° = √2 ≈ 1.4142, csc 60° = 2√3/3 ≈ 1.1547 and csc 90° = 1. Also csc 270° = −1.
Why is csc 0° undefined?
csc θ = 1 ÷ sin θ and sin 0° = 0, so csc 0° would mean dividing by zero. The same happens at 180°, 360° and every k × 180°. Close to those angles the cosecant becomes huge (csc 0.1° ≈ 573), and the graph has vertical asymptotes there.
Can cosecant be between −1 and 1?
No. Sine is always between −1 and 1, so its reciprocal is always 1 or more, or −1 or less: |csc θ| ≥ 1. The smallest positive value is csc 90° = 1 and the largest negative value is csc 270° = −1.
Is cosecant the same as sin⁻¹?
No. csc θ = 1 ÷ sin θ is the reciprocal of sine. sin−1(x), or arcsin, is the inverse function that returns an angle: sin−1(0.5) = 30°, while csc 30° = 2.
When is cosecant negative?
Cosecant has the same sign as sine: positive in quadrants I and II, negative in quadrants III and IV. For example csc 210° = −2 and csc 225° = −√2 ≈ −1.4142.
What is the period of cosecant?
360°, or 2π radians, the same as sine. csc(θ + 360°) = csc θ, so csc 390° = csc 30° = 2. Cosecant is an odd function: csc(−θ) = −csc θ, so csc(−30°) = −2.
How do you calculate csc on a calculator or in Excel?
Most calculators have no csc key, so work out 1 ÷ sin θ. In Excel use =CSC(RADIANS(30)) or =1/SIN(RADIANS(30)), both 2. In code use 1 / Math.sin(x) in JavaScript or 1 / math.sin(x) in Python, with x in radians; these give 2.0000000000000004 for 30° and a huge number instead of an error at 180°.
Where is the cosecant used?
The identity 1 + cot2θ = csc2θ is used to simplify expressions, and in calculus the derivative of cot θ is −csc2θ. In geometry, csc θ gives the length of a slanted line for each unit of height, such as a ladder: a ladder at 60° to the ground needs 1.1547 m of length for every metre it reaches up the wall.
Is the Cosecant Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Cosecant Calculator to find csc θ for any angle in degrees or radians instantly.