Arccosecant Calculator
Enter a value of 1 or more, or −1 or less, to find its arccosecant, the angle θ = csc⁻¹(x), in degrees or radians.
Updated
Arccosecant Calculator
Our Arccosecant Calculator finds the angle whose cosecant is a given value, arccsc(x) or csc⁻¹(x). Type a number of 1 or more (or −1 or less), a fraction such as 5/4 or a root such as √2 and the angle appears straight away in degrees or radians, to 10 decimal places. Common values give exact answers (arccsc(2) = 30° = π/6), and the calculator also checks the result, lists the other angles with the same cosecant, and explains why values between −1 and 1 have no arccosecant.
Quick answer
The arccosecant of x is the angle whose cosecant is x, between −90° and 90°. For example, arccsc(2) = 30° because csc 30° = 1 ÷ sin 30° = 2. Enter any value of 1 or more, or −1 or less, above and the calculator shows the angle instantly.
- Formula: θ = arccsc(x) = arcsin(1 ÷ x)
- Key values: arccsc(1) = 90°, arccsc(√2) = 45°, arccsc(2) = 30°, arccsc(−1) = −90°
- Domain: x ≤ −1 or x ≥ 1; values between −1 and 1 have no arccosecant
- Range: −90° to 90° (−π/2 to π/2), except 0°
What Is Arccosecant?
Arccosecant is the inverse of the cosecant function. Cosecant takes an angle and gives a ratio; arccosecant takes the ratio back to the angle. In a right triangle, if you know the hypotenuse and the side opposite an angle, arccosecant gives the angle:
θ = arccsc(hyp ÷ opposite)
It is written arccsc x, arccosec x, csc⁻¹ x or acsc x. The ⁻¹ means "inverse function", not a power: csc⁻¹ x is not 1 ÷ csc x (that is the sine).
Arccosecant Table of Common Values
| x | Degrees | Radians |
|---|---|---|
| 1 | 90° | π/2 |
| 2√3/3 ≈ 1.1547 | 60° | π/3 |
| √2 ≈ 1.4142 | 45° | π/4 |
| 2 | 30° | π/6 |
| √5 + 1 ≈ 3.2361 | 18° | π/10 |
| −2 | −30° | −π/6 |
| −√2 ≈ −1.4142 | −45° | −π/4 |
| −2√3/3 ≈ −1.1547 | −60° | −π/3 |
| −1 | −90° | −π/2 |
2√3/3 is the same number as 2/√3. Other values have no simple exact angle, for example arccsc(1.5) ≈ 41.8103° and arccsc(3) ≈ 19.4712°. Large values give angles close to 0°, but never 0° itself.
The Arccosecant Graph
- Two branches: one for x ≥ 1 (from 90° at x = 1 down toward 0°) and one for x ≤ −1 (from −90° at x = −1 up toward 0°).
- Gap: nothing between x = −1 and x = 1, because no cosecant lies there.
- Horizontal asymptote: y = 0°, approached from both sides but never reached.
- End points: (1, 90°) and (−1, −90°) are on the graph.
- Odd function: arccsc(−x) = −arccsc x, symmetric about the origin.
Finding Arccsc With Arcsin
Calculators, Excel, Python and JavaScript have no arccosecant key or function. Because csc θ = 1 ÷ sin θ, take the reciprocal first and then use arcsin:
arccsc(1.5) = arcsin(0.6667)
≈ 41.81°
If x is between −1 and 1, 1 ÷ x is bigger than 1 and arcsin gives an error, which is the same as saying the arccosecant does not exist.
Why Arccosecant Gives One Angle
Many angles share the same cosecant: csc 30° = csc 150° = csc 390° = 2. A function must give one answer, so arccosecant returns the principal value, the one between −90° and 90°. To find every angle with csc θ = x, use both of these and add whole turns:
θ = 180° − arccsc(x) + 360° × k
For csc θ = 2 between 0° and 360° that gives 30° and 150°. The calculator shows the second angle in its "Other angles" row.
Arccosecant Rules and Identities
- Inverse: csc(arccsc x) = x for x ≤ −1 or x ≥ 1.
- With arcsin: arccsc x = arcsin(1 ÷ x).
- Negative values: arccsc(−x) = −arccsc x.
- With arcsec: arcsec x + arccsc x = 90° (π/2).
- Calculus: the derivative of arccsc x is −1 ÷ (|x|√(x2 − 1)).
- Other convention: some calculus books use 0° to 90° and 180° to 270°, which gives arccsc(−2) = 210° instead of −30°.
Worked Examples
Angle of a ladder. A 5 m ladder reaches 4 m up a wall. The ladder is the hypotenuse and the height is opposite the angle with the ground:
= arcsin(0.8)
≈ 53.13°
A fraction. arccsc(5/3) = arcsin(3/5) ≈ 36.8699°, the smaller acute angle of a 3-4-5 triangle.
Negative values. arccsc(−3) ≈ −19.4712° and arccsc(−7/3) ≈ −25.3769°. A negative cosecant always gives a negative angle.
Arccosecant in Excel and Code
| Tool | arccsc(2) in degrees |
|---|---|
| Excel / Google Sheets | =DEGREES(ASIN(1/2)) |
| Python | math.degrees(math.asin(1 / 2)) |
| JavaScript | Math.asin(1 / 2) * 180 / Math.PI |
| Scientific calculator | DEG mode, sin⁻¹(1 ÷ 2) |
Floating-point maths gives 30.000000000000004 here. For x between −1 and 1, Excel returns #NUM!, Python raises "math domain error" and JavaScript returns NaN.
Common Mistakes
- Reading csc⁻¹ as 1 ÷ csc. csc⁻¹ x is an angle; 1 ÷ csc θ is just sin θ.
- Skipping the reciprocal. arccsc(2) is arcsin(1/2) = 30°, not arcsin(2), which is an error.
- Values between −1 and 1. arccsc(0.5) does not exist.
- Expecting 0°. No finite number has an arccosecant of 0°.
- Wrong angle mode. arccsc(2) is 30 in degrees but 0.5236 in radians.
Key Terms
- Inverse
- A function that undoes another; arccosecant undoes cosecant.
- Cosecant
- 1 ÷ sin θ, or hypotenuse ÷ opposite.
- Domain
- The inputs allowed; for arccosecant, x ≤ −1 or x ≥ 1.
- Range
- The possible outputs; here −90° to 90°, except 0°.
- Arcsin
- The inverse of sine; arccsc x = arcsin(1 ÷ x).
- Arcsec
- The inverse of secant; arcsec x + arccsc x = 90°.
How to Use the Arccosecant Calculator
- Enter the value. Type the cosecant value x, which must be 1 or more, or −1 or less, or press one of the buttons (2, √2, 2√3/3 and −2). You can type a decimal such as 1.5, a fraction such as 5/4, or a root such as √2 or 2/√3.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad) for the answer. The table below the result always shows both.
- Read and copy the result. The angle θ = arccsc(x) appears as you type, between −90° and 90°, to 10 decimal places, with the exact angle for common values (for example arccsc(√2) = 45° = π/4 rad). The table also checks the answer and gives the other angle with the same cosecant. Values between −1 and 1 have no arccosecant and the calculator says so. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter −2 and the result is −30°, or −π/6 rad. The table shows that csc(−30°) = −2 and that 210° has the same cosecant.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is arccosecant?
Arccosecant is the inverse of cosecant. arccsc(x), also written csc−1(x) or arccosec(x), is the angle between −90° and 90° whose cosecant is x. For example arccsc(2) = 30°, because csc 30° = 1 ÷ sin 30° = 2.
What is arccsc 1, arccsc √2, arccsc 2 and arccsc(−1)?
arccsc(1) = 90° (π/2), arccsc(2√3/3) = 60° (π/3), arccsc(√2) = 45° (π/4), arccsc(2) = 30° (π/6), arccsc(−2) = −30° (−π/6) and arccsc(−1) = −90° (−π/2).
Why is arccsc(0.5) undefined?
Cosecant is 1 ÷ sin θ, and sine is never bigger than 1 or smaller than −1, so a cosecant is always 1 or more, or −1 or less. No angle has a cosecant of 0.5, so arccsc(0.5) does not exist.
What are the domain and range of arccosecant?
The domain is x ≤ −1 or x ≥ 1. The range is −90° to 90° (−π/2 to π/2) without 0°, because csc 0° is undefined. Large positive or negative x gives angles close to 0°.
Is csc⁻¹(x) the same as 1/csc(x)?
No. csc−1(x) is the inverse function and returns an angle: csc−1(2) = 30°. 1 ÷ csc(x) is simply sin(x).
How do you calculate arccsc on a calculator?
Calculators have no arccsc key. Use arccsc(x) = arcsin(1 ÷ x): in DEG mode work out 1 ÷ x, then press SHIFT sin (sin−1). For x = 2 that is sin−1(0.5) = 30°. In Excel use =DEGREES(ASIN(1/2)).
How do I find all angles with a given cosecant?
Use θ = arccsc(x) + 360° × k and θ = 180° − arccsc(x) + 360° × k for any whole number k. For csc θ = 2 between 0° and 360° that gives 30° and 150°.
What is arccsc(1.5) in degrees?
arccsc(1.5) = arcsin(2/3) ≈ 41.8103°, or about 0.7297 radians. It has no simple exact form. The other angle from 0° to 360° with the same cosecant is 138.1897°.
Are arccsc and arccosec the same?
Yes. arccsc, arccosec, acsc and csc−1 are all names for the inverse cosecant. arccosec and cosec are common in British textbooks, arccsc and csc in American ones.
Is the Arccosecant Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Arccosecant Calculator to find the angle for any cosecant value in degrees or radians instantly.