Arcsecant Calculator
Enter a value of 1 or more, or −1 or less, to find its arcsecant, the angle θ = sec⁻¹(x), in degrees or radians.
Updated
Arcsecant Calculator
Our Arcsecant Calculator finds the angle whose secant is a given value, arcsec(x) or sec⁻¹(x). Type a number of 1 or more (or −1 or less), a fraction such as 5/3 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 (arcsec(2) = 60° = π/3), and the calculator also checks the result, lists the other angles with the same secant, and explains why values between −1 and 1 have no arcsecant.
Quick answer
The arcsecant of x is the angle whose secant is x, between 0° and 180°. For example, arcsec(2) = 60° because sec 60° = 1 ÷ cos 60° = 2. Enter any value of 1 or more, or −1 or less, above and the calculator shows the angle instantly.
- Formula: θ = arcsec(x) = arccos(1 ÷ x)
- Key values: arcsec(1) = 0°, arcsec(√2) = 45°, arcsec(2) = 60°, arcsec(−1) = 180°
- Domain: x ≤ −1 or x ≥ 1; values between −1 and 1 have no arcsecant
- Range: 0° to 180° (0 to π), except 90°
What Is Arcsecant?
Arcsecant is the inverse of the secant function. Secant takes an angle and gives a ratio; arcsecant takes the ratio back to the angle. In a right triangle, if you know the hypotenuse and the side next to an angle, arcsecant gives the angle:
θ = arcsec(hyp ÷ adjacent)
It is written arcsec x, sec⁻¹ x or asec x. The ⁻¹ means "inverse function", not a power: sec⁻¹ x is not 1 ÷ sec x (that is the cosine).
Arcsecant Table of Common Values
| x | Degrees | Radians |
|---|---|---|
| 1 | 0° | 0 |
| 2√3/3 ≈ 1.1547 | 30° | π/6 |
| √2 ≈ 1.4142 | 45° | π/4 |
| 2 | 60° | π/3 |
| √5 + 1 ≈ 3.2361 | 72° | 2π/5 |
| −2 | 120° | 2π/3 |
| −√2 ≈ −1.4142 | 135° | 3π/4 |
| −2√3/3 ≈ −1.1547 | 150° | 5π/6 |
| −1 | 180° | π |
2√3/3 is the same number as 2/√3. Other values have no simple exact angle, for example arcsec(1.5) ≈ 48.1897° and arcsec(3) ≈ 70.5288°. Large values give angles close to 90°, but never 90° itself.
The Arcsecant Graph
- Two branches: one for x ≥ 1 (0° up toward 90°) and one for x ≤ −1 (from 180° at x = −1 down toward 90°).
- Gap: nothing between x = −1 and x = 1, because no secant lies there.
- Horizontal asymptote: y = 90°, approached from both sides but never reached.
- End points: (1, 0°) and (−1, 180°) are on the graph.
- Symmetry: arcsec(−x) = 180° − arcsec x.
Finding Arcsec With Arccos
Calculators, Excel, Python and JavaScript have no arcsecant key or function. Because sec θ = 1 ÷ cos θ, take the reciprocal first and then use arccos:
arcsec(1.5) = arccos(0.6667)
≈ 48.19°
If x is between −1 and 1, 1 ÷ x is bigger than 1 and arccos gives an error, which is the same as saying the arcsecant does not exist.
Why Arcsecant Gives One Angle
Many angles share the same secant: sec 60° = sec 300° = sec(−60°) = 2. A function must give one answer, so arcsecant returns the principal value, the one between 0° and 180°. Every angle with sec θ = x is:
For sec θ = 2 between 0° and 360° that gives 60° and 300°. The calculator shows the second angle in its "Other angles" row.
Arcsecant Rules and Identities
- Inverse: sec(arcsec x) = x for x ≤ −1 or x ≥ 1.
- With arccos: arcsec x = arccos(1 ÷ x).
- Negative values: arcsec(−x) = 180° − arcsec x.
- With arccsc: arcsec x + arccsc x = 90° (π/2).
- Calculus: the derivative of arcsec x is 1 ÷ (|x|√(x2 − 1)).
- Other convention: some calculus books use 0° to 90° and 180° to 270°, which gives arcsec(−2) = 240° instead of 120°.
Worked Examples
Angle from a hypotenuse. A right triangle has a hypotenuse of 10 and an adjacent side of 8:
= arccos(0.8)
≈ 36.87°
A fraction. arcsec(5/3) = arccos(3/5) ≈ 53.1301°, the larger acute angle of a 3-4-5 triangle.
Negative values. arcsec(−3) ≈ 109.4712° and arcsec(−5/2) ≈ 113.5782°. A negative secant always gives an angle between 90° and 180°.
Arcsecant in Excel and Code
| Tool | arcsec(2) in degrees |
|---|---|
| Excel / Google Sheets | =DEGREES(ACOS(1/2)) |
| Python | math.degrees(math.acos(1 / 2)) |
| JavaScript | Math.acos(1 / 2) * 180 / Math.PI |
| Scientific calculator | DEG mode, cos⁻¹(1 ÷ 2) |
Floating-point maths gives 60.00000000000001 here. For x between −1 and 1, Excel returns #NUM!, Python raises "math domain error" and JavaScript returns NaN.
Common Mistakes
- Reading sec⁻¹ as 1 ÷ sec. sec⁻¹ x is an angle; 1 ÷ sec θ is just cos θ.
- Skipping the reciprocal. arcsec(2) is arccos(1/2) = 60°, not arccos(2), which is an error.
- Values between −1 and 1. arcsec(0.5) does not exist.
- Expecting 90°. No finite number has an arcsecant of 90°.
- Wrong angle mode. arcsec(2) is 60 in degrees but 1.0472 in radians.
Key Terms
- Inverse
- A function that undoes another; arcsecant undoes secant.
- Secant
- 1 ÷ cos θ, or hypotenuse ÷ adjacent.
- Domain
- The inputs allowed; for arcsecant, x ≤ −1 or x ≥ 1.
- Range
- The possible outputs; here 0° to 180°, except 90°.
- Arccos
- The inverse of cosine; arcsec x = arccos(1 ÷ x).
- Arccsc
- The inverse of cosecant; arcsec x + arccsc x = 90°.
How to Use the Arcsecant Calculator
- Enter the value. Type the secant value x, which must be 1 or more, or −1 or less, or press one of the buttons (2√3/3, √2, 2 and −2). You can type a decimal such as 1.5, a fraction such as 5/3, 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 = arcsec(x) appears as you type, between 0° and 180°, to 10 decimal places, with the exact angle for common values (for example arcsec(√2) = 45° = π/4 rad). The table also checks the answer and gives the other angle with the same secant. Values between −1 and 1 have no arcsecant 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 120°, or 2π/3 rad. The table shows that sec(120°) = −2 and that 240° has the same secant.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is arcsecant?
Arcsecant is the inverse of secant. arcsec(x), also written sec−1(x), is the angle between 0° and 180° whose secant is x. For example arcsec(2) = 60°, because sec 60° = 1 ÷ cos 60° = 2.
What is arcsec 1, arcsec √2, arcsec 2 and arcsec(−1)?
arcsec(1) = 0°, arcsec(2√3/3) = 30° (π/6), arcsec(√2) = 45° (π/4), arcsec(2) = 60° (π/3), arcsec(−2) = 120° (2π/3) and arcsec(−1) = 180° (π).
Why is arcsec(0.5) undefined?
Secant is 1 ÷ cos θ, and cosine is never bigger than 1 or smaller than −1, so a secant is always 1 or more, or −1 or less. No angle has a secant of 0.5, so arcsec(0.5) does not exist.
What are the domain and range of arcsecant?
The domain is x ≤ −1 or x ≥ 1. The range is 0° to 180° (0 to π) without 90°, because sec 90° is undefined. Large positive or negative x gives angles close to 90°.
Is sec⁻¹(x) the same as 1/sec(x)?
No. sec−1(x) is the inverse function and returns an angle: sec−1(2) = 60°. 1 ÷ sec(x) is simply cos(x).
How do you calculate arcsec on a calculator?
Calculators have no arcsec key. Use arcsec(x) = arccos(1 ÷ x): in DEG mode work out 1 ÷ x, then press SHIFT cos (cos−1). For x = 2 that is cos−1(0.5) = 60°. In Excel use =DEGREES(ACOS(1/2)).
How do I find all angles with a given secant?
Use θ = ±arcsec(x) + 360° × k for any whole number k. For sec θ = 2 between 0° and 360° that gives 60° and 300°.
What is arcsec(1.5) in degrees?
arcsec(1.5) = arccos(2/3) ≈ 48.1897°, or about 0.8411 radians. It has no simple exact form. The other angle from 0° to 360° with the same secant is 311.8103°.
Can arcsec ever equal 90°?
No. sec 90° is undefined because cos 90° = 0. As x grows very large, arcsec(x) gets closer and closer to 90° (arcsec(1,000,000) ≈ 89.99994°) but never reaches it.
Is the Arcsecant Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Arcsecant Calculator to find the angle for any secant value in degrees or radians instantly.