Inverse Secant Calculator

Inverse secant is the function that returns the angle whose secant equals a given number. Inverse secant is written as sec⁻¹(x), arcsec(x) or asec(x), and it reverses the secant function: sec(60°) = 2, so sec⁻¹(2) = 60°.

The secant is the reciprocal of the cosine, sec θ = 1/cos θ, so the inverse secant is the arccosine of the reciprocal: sec⁻¹(x) = cos⁻¹(1/x). Arcsecant accepts numbers with |x| ≥ 1 and returns an angle from 0° to 180°, except 90°.

In a right triangle, the secant of an angle θ equals the hypotenuse divided by the adjacent side. The inverse secant takes that hypotenuse/adjacent ratio and returns θ. A 13 m cable anchored 12 m from the base of a mast makes θ = sec⁻¹(13/12) = 22.62° with the ground.

• Input: x ≤ −1 or x ≥ 1.
• Output: an angle in [0°, 90°) for x ≥ 1, or (90°, 180°] for x ≤ −1.
• Special values: sec⁻¹(1) = 0°, sec⁻¹(2) = 60°, sec⁻¹(−1) = 180°.

The −1 in sec⁻¹(x) marks the inverse function. The reciprocal 1/sec(x) is the cosine cos(x).

To calculate inverse secant, take the reciprocal of x and apply the inverse cosine: sec⁻¹(x) = cos⁻¹(1/x). Follow these 4 steps:

1. Find the secant value x. For a right triangle, x = hypotenuse ÷ adjacent. Example: 5 ÷ 4 = 1.25.
2. Check that |x| ≥ 1. Values between −1 and 1 have no real arcsecant.
3. Take the reciprocal and apply arccos: cos⁻¹(1/1.25) = cos⁻¹(0.8) = 36.87°.
4. Convert the unit if needed: 36.87° × π/180 = 0.6435 rad.

Check the answer: sec(36.87°) = 1/cos(36.87°) = 1/0.8 = 1.25, which matches x.

Exact angles: sec⁻¹(2/√3) = 30° = π/6, sec⁻¹(√2) = 45° = π/4, sec⁻¹(2) = 60° = π/3 and sec⁻¹(−2) = 120° = 2π/3.

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

1. Pick an input method: Decimal / x for a secant value, or Hyp / Adj for 2 sides of a right triangle.
2. Type the value. The field accepts decimals (1.25), fractions (5/4), 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 cosine 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 / Adj mode, the hypotenuse must be at least as long as the adjacent side.

Horizon dip mode returns how far below eye level the horizon sits: dip = arcsec((R + h)/R), with R the planet radius and h your eye height. From 100 m on Earth the dip is 0.321° (19.3 arcminutes) and the horizon is 35.7 km away.

The inverse secant formula is θ = sec⁻¹(hypotenuse / adjacent) = cos⁻¹(adjacent / hypotenuse). θ is the angle, and the adjacent side touches θ.

Related forms of the formula:

• Identity: sec(sec⁻¹ x) = x for |x| ≥ 1, and sec⁻¹(sec θ) = θ for θ in [0°, 180°], θ ≠ 90°.
• Complement rule: sec⁻¹ x + csc⁻¹ x = 90° for |x| ≥ 1.
• Arctangent form: sec⁻¹ x = tan⁻¹(√(x² − 1)) for x ≥ 1.
• Derivative: d/dx sec⁻¹ x = 1/(|x|·√(x² − 1)) for |x| > 1.
• Integral: ∫ dx / (x√(x² − 1)) = sec⁻¹ x + C for x > 1, the form that appears in calculus tables.

Example: a 10 m rope reaches from a dock post to a boat 6 m away along the dock. sec θ = 10/6 = 1.667, so θ = sec⁻¹(1.667) = 53.13° between the rope and the dock.

Inverse secant in degrees returns an angle from 0° to 180°, never 90°. sec⁻¹(1) = 0°, sec⁻¹(2) = 60° and sec⁻¹(−2) = 120°.

Inputs x ≥ 1 give angles from 0° up to 90°; inputs x ≤ −1 give angles from 90° up to 180°. The angle 90° is missing: cos 90° = 0, so sec 90° is undefined.

Large inputs approach 90°: sec⁻¹(10) = 84.26° and sec⁻¹(100) = 89.43°. In degrees, minutes and seconds, sec⁻¹(3) = 70.5288° = 70° 31′ 43.6″.

Common values in degrees:

• sec⁻¹(1) = 0°
• sec⁻¹(2/√3) = 30°
• sec⁻¹(√2) = 45°
• sec⁻¹(2) = 60°
• sec⁻¹(−2) = 120°
• sec⁻¹(−√2) = 135°
• sec⁻¹(−1) = 180°

Inverse secant in radians returns an angle from 0 to π, except π/2. sec⁻¹(2) = π/3 = 1.0472 rad and sec⁻¹(−1) = π = 3.1416 rad.

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

To convert degrees to radians, multiply by π/180: 135° × π/180 = 3π/4 = sec⁻¹(−√2).

• sec⁻¹(2/√3) = π/6 = 0.5236 rad
• sec⁻¹(√2) = π/4 = 0.7854 rad
• sec⁻¹(2) = π/3 = 1.0472 rad
• sec⁻¹(−2) = 2π/3 = 2.0944 rad
• sec⁻¹(−√2) = 3π/4 = 2.3562 rad

The domain of inverse secant is x ≤ −1 or x ≥ 1, and the range is 0° ≤ θ ≤ 180° with θ ≠ 90°. In interval notation: domain (−∞, −1] ∪ [1, ∞), range [0, π/2) ∪ (π/2, π].

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

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

Some calculus books restrict the range to [0, π/2) ∪ [π, 3π/2) to simplify the derivative. This calculator uses the standard range [0, π] without π/2, the one that matches cos⁻¹(1/x).

Scientific calculators have no sec⁻¹ key, so enter cos⁻¹(1 ÷ x). On a Casio fx-991, the keys for sec⁻¹(2) are SHIFT, cos, 1 ÷ 2, ), = and the screen shows 60.

1. Set the angle mode: D for degrees, R for radians.
2. Press SHIFT cos (Casio) or 2nd COS (TI) for cos⁻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 cos, Ans, =.

Math ERROR means |x| < 1, so 1/x lies outside −1 to 1. Negative inputs need no extra step: cos⁻¹(1 ÷ −2) = 120° directly.

On the Android (Google) Calculator, open the scientific keys, tap INV, tap cos⁻¹, type 1 ÷ x and tap =. sec⁻¹(2) = cos⁻¹(1 ÷ 2) returns 60 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 cos⁻¹.
4. Type 1 ÷ 2 and tap =.

For sec⁻¹(−2), type 1 ÷ −2 inside the bracket; the result is 120.

On the iPhone Calculator, switch to Scientific, compute 1 ÷ x, then tap 2nd and cos⁻¹. For sec⁻¹(2), type 1 ÷ 2 =, tap 2nd, tap cos⁻¹: the display shows 60.

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 cos⁻¹.

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

Windows Calculator has a sec⁻¹ key: choose Scientific, type the value, open Trigonometry, select 2nd, then select sec⁻¹. The walkthrough below uses 1 ÷ x with cos⁻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 cos⁻¹.

With the sec⁻¹ key: type 2, open Trigonometry, select 2nd and select sec⁻¹. The result is 60.

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

• Excel and Google Sheets: =ACOS(1/2) returns 1.0472; =DEGREES(ACOS(1/2)) returns 60. Excel has SEC but no ASEC.
• Python: math.acos(1/2) returns 1.0472. NumPy: np.arccos(1/x).
• JavaScript: Math.acos(1 / 2) * 180 / Math.PI returns 60.00000000000001.
• MATLAB and Octave: asec(2) for radians and asecd(2) for degrees.
• Wolfram Alpha: ArcSec[2] returns π/3.

Inputs between −1 and 1 make 1/x larger than 1 in size, so ACOS returns #NUM! in Excel and NaN in JavaScript. MATLAB returns a complex number for those inputs.

Without a calculator, use exact values for special angles, or the arcsine series of 1/x subtracted from 90°.

1. Special values: sec 30° = 2/√3, sec 45° = √2 and sec 60° = 2, so sec⁻¹(2/√3) = 30°, sec⁻¹(√2) = 45° and sec⁻¹(2) = 60°.
2. Series: with u = 1/x, sec⁻¹ x = π/2 − (u + u³/6 + 3u⁵/40 + 5u⁷/112 + …).
3. Negative inputs: sec⁻¹(−x) = 180° − sec⁻¹(x).

Example: sec⁻¹(3), with u = 1/3: 1.570796 − (0.333333 + 0.006173 + 0.000309 + 0.000019) = 1.230962 rad = 70.529°. The true value is 1.230959 rad.

Large inputs converge fastest: for x = 10, u = 0.1 and 2 terms land within 0.000001 rad.

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

• Right branch: starts at (1, 0°) and approaches 90° from below as x grows.
• Left branch: approaches 90° from above as x → −∞ and ends at (−1, 180°).
• Horizontal asymptote y = π/2; no part of the graph lies between x = −1 and x = 1.
• Vertical tangents at x = ±1, where the slope 1/(|x|√(x² − 1)) grows without bound.
• Point symmetry about (0, π/2): sec⁻¹(−x) = π − sec⁻¹(x).

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

Exact values to know: sec⁻¹(1) = 0, sec⁻¹(2/√3) = π/6, sec⁻¹(√2) = π/4, sec⁻¹(2) = π/3 and sec⁻¹(−1) = π.

No row lies between −1 and 1: those inputs fall outside the domain.

The inverse secant of a negative number is an obtuse angle: sec⁻¹(−x) = 180° − sec⁻¹(x). sec⁻¹(−2) = 120°, sec⁻¹(−√2) = 135° and sec⁻¹(−1) = 180°.

Negative secants come from negative cosines, which live in quadrant II between 90° and 180°.

What is sec⁻¹(−2)?

sec⁻¹(−2) = 120° = 2π/3 rad. cos(120°) = −0.5, so sec(120°) = 1/(−0.5) = −2. Through arccosine: cos⁻¹(1/−2) = cos⁻¹(−0.5) = 120°.

arcsec x equals arccos(1/x), and arcsec x + arccsc x = 90° for every |x| ≥ 1. Arcsecant takes hypotenuse/adjacent, the reciprocal of the cosine ratio.

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

Inverse secant finds an angle from a slanted length and its horizontal span, the setup of cables, ramps and rafters.

• Guy wires: a 13 m wire anchored 5 m from a mast makes sec⁻¹(13/5) = 67.38° with the ground.
• Rafters: a 5 m rafter spanning 4 m of floor rises at sec⁻¹(5/4) = 36.87°.
• Light through glass: the path length through a plate of thickness t at angle θ is t·sec θ, so θ = sec⁻¹(path / t).
• Calculus: integrals of the form ∫ dx / (x√(x² − a²)) give (1/a)·sec⁻¹(x/a) + C.
• Air mass in astronomy: light from a star at zenith angle z crosses about sec z atmospheres, so an air mass of 2 means z = sec⁻¹(2) = 60°.