Arccosine Calculator
Enter a value from −1 to 1 to find its arccosine, the angle θ = cos⁻¹(x), in degrees or radians.
Updated
Arccosine Calculator
Our Arccosine Calculator finds the angle whose cosine is a given value, arccos(x) or cos⁻¹(x). Type a decimal, a fraction such as 2/3 or a root such as √3/2 and the angle appears straight away in degrees or radians, to 10 decimal places. Common values give exact answers (arccos(1/2) = 60° = π/3), and the calculator also checks the result, lists the other angles with the same cosine, and explains why values above 1 or below −1 have no arccosine.
Quick answer
The arccosine of x is the angle whose cosine is x, between 0° and 180°. For example, arccos(0.5) = 60° because cos 60° = 0.5. Enter any value from −1 to 1 above and the calculator shows the angle instantly.
- Formula: if cos θ = x, then θ = arccos(x) = cos⁻¹(x)
- Key values: arccos(1) = 0°, arccos(1/2) = 60°, arccos(0) = 90°, arccos(−1) = 180°
- Domain: −1 ≤ x ≤ 1; any other value has no arccosine
- Range: 0° to 180°, or 0 to π radians
What Is Arccosine?
Arccosine is the inverse of the cosine function. Cosine takes an angle and gives a ratio; arccosine takes the ratio back to the angle. In a right triangle, if you know the side next to an angle and the hypotenuse, arccosine gives the angle:
θ = arccos(adjacent ÷ hyp)
It is written arccos x, cos⁻¹ x or acos x. The ⁻¹ means "inverse function", not a power: cos⁻¹ x is not 1 ÷ cos x (that is the secant).
Arccosine Table of Common Values
| x | Degrees | Radians |
|---|---|---|
| 1 | 0° | 0 |
| √3/2 ≈ 0.8660 | 30° | π/6 |
| √2/2 ≈ 0.7071 | 45° | π/4 |
| 1/2 | 60° | π/3 |
| 0 | 90° | π/2 |
| −1/2 | 120° | 2π/3 |
| −√2/2 ≈ −0.7071 | 135° | 3π/4 |
| −√3/2 ≈ −0.8660 | 150° | 5π/6 |
| −1 | 180° | π |
Negative values give angles over 90°, because cosine is negative in quadrant II. Other values have no simple exact angle, for example arccos(2/3) ≈ 48.1897° and arccos(4/7) ≈ 55.1501°.
The Arccosine Graph
- Shape: a falling S-curve from (−1, 180°) through (0, 90°) to (1, 0°).
- Always decreasing: a bigger x always gives a smaller angle.
- Steep at the ends: near x = ±1 a tiny change in x moves the angle a lot.
- Symmetry: arccos(−x) = 180° − arccos x, symmetric about the point (0, 90°).
- Never negative: every output lies between 0° and 180°.
Why Arccosine Gives One Angle
Many angles share the same cosine: cos 60° = cos 300° = cos(−60°) = 0.5. A function must give one answer, so arccosine returns the principal value, the one between 0° and 180°. To find every angle with cos θ = x, take plus or minus that angle and add whole turns:
For cos θ = 0.5 between 0° and 360° that gives 60° and 300°. The calculator shows the second angle in its "Other angles" row.
Arccosine Rules and Identities
- Inverse: cos(arccos x) = x for −1 ≤ x ≤ 1.
- Only one way round: arccos(cos θ) = θ only for 0° ≤ θ ≤ 180°; arccos(cos 300°) = 60°.
- Negative values: arccos(−x) = 180° − arccos x.
- With arcsin: arcsin x + arccos x = 90° (π/2).
- Sine of arccosine: sin(arccos x) = √(1 − x2).
- Calculus: the derivative of arccos x is −1 ÷ √(1 − x2).
Worked Examples
Angle of a ladder. A 5 m ladder stands with its foot 1.5 m from a wall. The distance to the wall is next to the angle with the ground:
= arccos(0.3)
≈ 72.54°
Angle in a triangle. A triangle has sides 5, 6 and 7. By the law of cosines, the angle C opposite the side of 7 is:
= 0.2
C = arccos(0.2)
≈ 78.46°
Angle between two vectors. For (3, 4) and (4, 3), cos θ = (3 × 4 + 4 × 3) ÷ (5 × 5) = 0.96, so θ = arccos(0.96) ≈ 16.26°.
Arccosine in Excel and Code
| Tool | arccos(0.5) in degrees |
|---|---|
| Excel / Google Sheets | =DEGREES(ACOS(0.5)) |
| Python | math.degrees(math.acos(0.5)) |
| JavaScript | Math.acos(0.5) * 180 / Math.PI |
| Scientific calculator | DEG mode, then SHIFT cos (cos⁻¹) 0.5 |
ACOS and acos return radians, so convert to degrees as shown. Floating-point maths gives 60.00000000000001 here. Outside −1 to 1, Excel returns #NUM!, Python raises "math domain error" and JavaScript returns NaN.
Common Mistakes
- Reading cos⁻¹ as 1 ÷ cos. cos⁻¹ x is an angle; 1 ÷ cos θ is the secant.
- Typing an angle. The input is a cosine value (a ratio), not an angle.
- Values above 1. arccos(1.5) does not exist, because no cosine is bigger than 1.
- Expecting a negative angle. arccos(−0.5) is 120°, not −60°.
- Wrong angle mode. arccos(0.5) is 60 in degrees but 1.0472 in radians.
Key Terms
- Inverse
- A function that undoes another; arccosine undoes cosine.
- Domain
- The inputs allowed; for arccosine, −1 to 1.
- Range
- The possible outputs; for arccosine, 0° to 180°.
- Principal
- The one angle a function returns out of many that fit.
- Arcsin
- The inverse of sine; arcsin x + arccos x = 90°.
- Secant
- 1 ÷ cos θ, often confused with cos⁻¹.
How to Use the Arccosine Calculator
- Enter the value. Type the cosine value x, from −1 to 1, or press one of the buttons (1/2, √2/2, √3/2 and 1). You can type a decimal such as 0.3, a fraction such as 2/3, or a root such as √3/2 or sqrt(2)/2.
- 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 = arccos(x) appears as you type, to 10 decimal places, with the exact angle for common values (for example arccos(√2/2) = 45° = π/4 rad). The table also checks the answer and gives the other angle with the same cosine. Values above 1 or below −1 have no arccosine and the calculator says so. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter −1/2 and the result is 120°, or 2π/3 rad. The table shows that cos(120°) = −0.5 and that 240° has the same cosine.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is arccosine?
Arccosine is the inverse of cosine. arccos(x), also written cos−1(x), is the angle between 0° and 180° whose cosine is x. For example arccos(0.5) = 60°, because cos 60° = 0.5.
What is arccos 1, arccos 0.5, arccos 0 and arccos(−1)?
arccos(1) = 0°, arccos(√3/2) = 30° (π/6), arccos(√2/2) = 45° (π/4), arccos(0.5) = 60° (π/3), arccos(0) = 90° (π/2), arccos(−0.5) = 120° (2π/3) and arccos(−1) = 180° (π).
Why is arccos(2) undefined?
The cosine of an angle is always between −1 and 1, so no angle has a cosine of 2. That is why the domain of arccosine is −1 ≤ x ≤ 1. Excel returns #NUM! and Python a math domain error for such values.
What is the range of arccosine?
0° to 180°, or 0 to π radians. Arccosine never returns a negative angle: arccos(−0.5) is 120°, not −60°.
Is cos⁻¹(x) the same as 1/cos(x)?
No. cos−1(x) is the inverse function and returns an angle: cos−1(0.5) = 60°. 1 ÷ cos(x) is the secant, sec x: sec 60° = 2.
How do I find all angles with a given cosine?
Use θ = ±arccos(x) + 360° × k for any whole number k. For cos θ = 0.5 between 0° and 360° that gives 60° and 300°.
What is arccos(2/3) in degrees?
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 cosine is 311.8103°.
How do you calculate arccosine in Excel or on a calculator?
In Excel use =DEGREES(ACOS(0.5)) for degrees or =ACOS(0.5) for radians. On a scientific calculator set DEG mode and press SHIFT then cos (the cos−1 key). In code use Math.acos(x) in JavaScript or math.acos(x) in Python, both in radians.
How is arccos used to find an angle in a triangle?
With all three sides known, the law of cosines gives cos C = (a2 + b2 − c2) ÷ 2ab, and C = arccos of that. For sides 5, 6 and 7, cos C = 0.2 and C ≈ 78.46°.
Is the Arccosine Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Arccosine Calculator to find the angle for any cosine value in degrees or radians instantly.