Inverse Cosecant Calculator

Inverse cosecant is the function that returns the angle whose cosecant equals a given number. Inverse cosecant is written as csc⁻¹(x), arccsc(x), acsc(x) or cosec⁻¹(x), and it reverses the cosecant function: csc(30°) = 2, so csc⁻¹(2) = 30°.

The cosecant is the reciprocal of the sine, csc θ = 1/sin θ, so the inverse cosecant is the arcsine of the reciprocal: csc⁻¹(x) = sin⁻¹(1/x). Arccosecant accepts numbers with |x| ≥ 1 and returns an angle from −90° to 90°, except 0°.

In a right triangle, the cosecant of an angle θ equals the hypotenuse divided by the opposite side. The inverse cosecant takes that hypotenuse/opposite ratio and returns θ. A 13 m ladder that reaches 5 m up a wall stands at θ = csc⁻¹(13/5) = 22.62°.

• Input: x ≤ −1 or x ≥ 1.
• Output: an angle in (0°, 90°] for x ≥ 1, or [−90°, 0°) for x ≤ −1.
• Special values: csc⁻¹(1) = 90°, csc⁻¹(2) = 30°, csc⁻¹(−1) = −90°.

The −1 in csc⁻¹(x) marks the inverse function. The reciprocal 1/csc(x) is the sine sin(x).

To calculate inverse cosecant, take the reciprocal of x and apply the inverse sine: csc⁻¹(x) = sin⁻¹(1/x). Follow these 4 steps:

1. Find the cosecant value x. For a right triangle, x = hypotenuse ÷ opposite. Example: 5 ÷ 3 = 1.6667.
2. Check that |x| ≥ 1. Values between −1 and 1 have no real arccosecant.
3. Take the reciprocal and apply arcsin: sin⁻¹(1/1.6667) = sin⁻¹(0.6) = 36.87°.
4. Convert the unit if needed: 36.87° × π/180 = 0.6435 rad.

Check the answer: csc(36.87°) = 1/sin(36.87°) = 1/0.6 = 1.6667, which matches x.

Exact angles: csc⁻¹(2) = 30° = π/6, csc⁻¹(√2) = 45° = π/4, csc⁻¹(2/√3) = 60° = π/3 and csc⁻¹(1) = 90° = π/2.

Enter a value with |x| ≥ 1, or the hypotenuse and opposite side, and the calculator returns csc⁻¹(x) as you type. The steps:

1. Pick an input method: Decimal / x for a cosecant value, or Hyp / Opp for 2 sides of a right triangle.
2. Type the value. The field accepts decimals (1.25), fractions (5/3), roots (√2), π and expressions.
3. Choose the result unit: degrees, radians, gradians, turns, arcminutes, arcseconds, milliradians, microradians or multiples of π.
4. Choose the rounding, from whole numbers to 15 decimal places.
5. Read the angle in all 9 units, the reciprocal sine value, the triangle, the unit circle, the graph and the worked steps.

The calculator rejects values between −1 and 1 and states the reason. In Hyp / Opp mode, the hypotenuse must be at least as long as the opposite side.

Mach cone mode returns the half-angle of the shock cone behind a supersonic object: μ = arccsc(M), with M = speed ÷ speed of sound. At 686 m/s in 343 m/s air, M = 2 and μ = 30°. Below Mach 1 there is no cone, and the calculator says so.

The inverse cosecant formula is θ = csc⁻¹(hypotenuse / opposite) = sin⁻¹(opposite / hypotenuse). θ is the angle, and the opposite side faces θ.

Related forms of the formula:

• Identity: csc(csc⁻¹ x) = x for |x| ≥ 1, and csc⁻¹(csc θ) = θ for θ in [−90°, 90°], θ ≠ 0°.
• Complement rule: csc⁻¹ x + sec⁻¹ x = 90° for |x| ≥ 1.
• Arctangent form: csc⁻¹ x = tan⁻¹(1/√(x² − 1)) for x > 1.
• Derivative: d/dx csc⁻¹ x = −1/(|x|·√(x² − 1)) for |x| > 1.
• Integral: ∫ csc⁻¹ x dx = x·csc⁻¹ x + ln(x + √(x² − 1)) + C for x > 1.

Example: a kite string 50 m long holds the kite 25 m above the ground. csc θ = 50/25 = 2, so θ = csc⁻¹(2) = 30° above the horizon.

Inverse cosecant in degrees returns an angle from −90° to 90°, never 0°. csc⁻¹(1) = 90°, csc⁻¹(2) = 30° and csc⁻¹(−2) = −30°.

Inputs x ≥ 1 give positive angles from 90° down toward 0°; inputs x ≤ −1 give negative angles. The angle 0° is missing: sin 0° = 0, so csc 0° is undefined.

Large inputs approach 0°: csc⁻¹(10) = 5.74° and csc⁻¹(100) = 0.57°. In degrees, minutes and seconds, csc⁻¹(3) = 19.4712° = 19° 28′ 16.4″.

Common values in degrees:

• csc⁻¹(1) = 90°
• csc⁻¹(2/√3) = 60°
• csc⁻¹(√2) = 45°
• csc⁻¹(2) = 30°
• csc⁻¹(−2) = −30°
• csc⁻¹(−√2) = −45°
• csc⁻¹(−1) = −90°

Inverse cosecant in radians returns an angle from −π/2 to π/2, except 0. csc⁻¹(2) = π/6 = 0.5236 rad and csc⁻¹(1) = π/2 = 1.5708 rad.

On the unit circle, the cosecant is the y-intercept of the tangent line at the point for θ. The tangent line at angle θ meets the y-axis at y = csc θ, which lies at or beyond ±1.

To convert degrees to radians, multiply by π/180: −45° × π/180 = −π/4 = csc⁻¹(−√2).

• csc⁻¹(2) = π/6 = 0.5236 rad
• csc⁻¹(√2) = π/4 = 0.7854 rad
• csc⁻¹(2/√3) = π/3 = 1.0472 rad
• csc⁻¹(1) = π/2 = 1.5708 rad
• csc⁻¹(−2) = −π/6 = −0.5236 rad

The domain of inverse cosecant is x ≤ −1 or x ≥ 1, and the range is −90° ≤ θ ≤ 90° with θ ≠ 0°. In interval notation: domain (−∞, −1] ∪ [1, ∞), range [−π/2, 0) ∪ (0, π/2].

The cosecant equals 1/sin θ, and |sin θ| ≤ 1, so |csc θ| ≥ 1 for every angle. Values such as csc⁻¹(0.5) have no real answer.

• Domain: (−∞, −1] ∪ [1, ∞)
• Range: [−π/2, 0) ∪ (0, π/2]
• Monotonic: decreasing on each piece of the domain
• Odd function: csc⁻¹(−x) = −csc⁻¹(x)
• Horizontal asymptote: y = 0 as x → ±∞

Scientific calculators have no csc⁻¹ key, so enter sin⁻¹(1 ÷ x). On a Casio fx-991, the keys for csc⁻¹(2) are SHIFT, sin, 1 ÷ 2, ), = and the screen shows 30.

1. Set the angle mode: D for degrees, R for radians.
2. Press SHIFT sin (Casio) or 2nd SIN (TI) for sin⁻¹.
3. Type 1 ÷ x, close the bracket and press =.

The x⁻¹ key works too: type 2, press x⁻¹ to get 0.5, then press SHIFT sin, Ans, =.

Math ERROR means |x| < 1. Negative inputs need no extra step: sin⁻¹(1 ÷ −2) = −30° directly.

On the Android (Google) Calculator, open the scientific keys, tap INV, tap sin⁻¹, type 1 ÷ x and tap =. csc⁻¹(2) = sin⁻¹(1 ÷ 2) returns 30 in DEG mode.

1. Tap the expand arrow to show the scientific keys.
2. Check the mode label: DEG for degrees.
3. Tap INV, then tap sin⁻¹.
4. Type 1 ÷ 2 and tap =.

For csc⁻¹(−2), type 1 ÷ −2 inside the bracket; the result is −30.

On the iPhone Calculator, switch to Scientific, compute 1 ÷ x, then tap 2nd and sin⁻¹. For csc⁻¹(2), type 1 ÷ 2 =, tap 2nd, tap sin⁻¹: the display shows 30.

1. Turn the iPhone sideways, or choose Scientific from the calculator icon menu.
2. Check that Rad is not shown on the display (degrees).
3. Type 1 ÷ 2 and tap =. The display shows 0.5.
4. Tap 2nd, then tap sin⁻¹.

The 1/x key in Scientific mode replaces the division: type 2, tap 1/x, then 2nd and sin⁻¹.

Windows Calculator has a csc⁻¹ key: choose Scientific, type the value, open Trigonometry, select 2nd, then select csc⁻¹. The walkthrough below uses 1 ÷ x with sin⁻¹, the method that works on every calculator.

1. Open the menu (☰) and choose Scientific.
2. Check the angle button: DEG.
3. Type 1 ÷ 2 and press =.
4. Open Trigonometry, select 2nd, then select sin⁻¹.

With the csc⁻¹ key: type 2, open Trigonometry, select 2nd and select csc⁻¹. The result is 30.

Most software has no arccosecant function, so use asin(1/x). Results come out in radians.

• Excel and Google Sheets: =ASIN(1/2) returns 0.5236; =DEGREES(ASIN(1/2)) returns 30. Excel has CSC but no ACSC.
• Python: math.asin(1/2) returns 0.5236. NumPy: np.arcsin(1/x).
• JavaScript: Math.asin(1 / 2) * 180 / Math.PI returns 30.000000000000004.
• MATLAB and Octave: acsc(2) for radians and acscd(2) for degrees.
• Wolfram Alpha: ArcCsc[2] returns π/6.

Inputs between −1 and 1 make 1/x larger than 1 in size, so ASIN returns #NUM! in Excel and NaN in JavaScript.

Without a calculator, use exact values for special angles, or the arcsine series of 1/x.

1. Special values: csc 30° = 2, csc 45° = √2 and csc 60° = 2/√3, so csc⁻¹(2) = 30°, csc⁻¹(√2) = 45° and csc⁻¹(2/√3) = 60°.
2. Series: with u = 1/x, csc⁻¹ x = u + u³/6 + 3u⁵/40 + 5u⁷/112 + …
3. Negative inputs: csc⁻¹(−x) = −csc⁻¹(x).

Example: csc⁻¹(3), with u = 1/3: 0.333333 + 0.006173 + 0.000309 + 0.000019 = 0.339835 rad = 19.471°. The true value is 0.339837 rad.

For large x, the first term alone is close: csc⁻¹(10) ≈ 0.1 rad = 5.73°, against the true 5.74°.

The graph of y = csc⁻¹(x) has 2 branches: one falls from (1, π/2) toward y = 0, and the other falls from y = 0 to (−1, −π/2).

• Right branch: starts at (1, 90°) and approaches 0° from above as x grows.
• Left branch: approaches 0° from below as x → −∞ and ends at (−1, −90°).
• Horizontal asymptote y = 0 (the x-axis); no part of the graph lies between x = −1 and x = 1.
• Vertical tangents at x = ±1, where the slope −1/(|x|√(x² − 1)) falls without bound.
• Point symmetry about the origin: the function is odd.

The inverse cosecant table lists csc⁻¹(x) for common values of x, in degrees and radians. Click a row to load it into the calculator.

Exact values to know: csc⁻¹(1) = π/2, csc⁻¹(2/√3) = π/3, csc⁻¹(√2) = π/4 and csc⁻¹(2) = π/6. Negative inputs give the same angles with a minus sign.

The inverse cosecant of a negative number is a negative angle: csc⁻¹(−x) = −csc⁻¹(x). csc⁻¹(−2) = −30°, csc⁻¹(−√2) = −45° and csc⁻¹(−1) = −90°.

Arccosecant is an odd function, like arcsine. Negative cosecants come from negative sines, so the angle sits in quadrant IV, between −90° and 0°.

What is csc⁻¹(−2)?

csc⁻¹(−2) = −30° = −π/6 rad. sin(−30°) = −0.5, so csc(−30°) = 1/(−0.5) = −2. Through arcsine: sin⁻¹(1/−2) = sin⁻¹(−0.5) = −30°.

arccsc x equals arcsin(1/x), and arccsc x + arcsec x = 90° for every |x| ≥ 1. Arccosecant takes hypotenuse/opposite, the reciprocal of the sine ratio.

For x = 2: csc⁻¹(2) = 30°, sin⁻¹(1/2) = 30° and sec⁻¹(2) = 60°, and 30° + 60° = 90°.

Inverse cosecant finds an angle from a slanted length and the height it reaches, the setup of ladders, ramps and lines of sight.

• Ladders: a 6 m ladder that reaches 5.8 m up a wall leans at csc⁻¹(6/5.8) = 75.16° with the ground, close to the 75° safety rule.
• Kites and drones: a 100 m line holding a kite 60 m up gives an elevation of csc⁻¹(100/60) = 36.87°.
• Optics: the critical angle for total internal reflection is θc = csc⁻¹(n₁/n₂). From glass (n = 1.5) to air, θc = csc⁻¹(1.5) = 41.81°.
• Ramps: a 10 m ramp that climbs 0.5 m has a slope of csc⁻¹(20) = 2.87°.
• Calculus: csc⁻¹ appears in antiderivatives of 1/(x√(x² − 1)), with a minus sign compared with sec⁻¹.