What Is the Inverse of Cosine (Arccos)?
Arccos (written arccos(x), acos(x), or cos⁻¹(x)) is the inverse of the cosine function — specifically, it answers the question "which angle has this cosine?" Given a known cosine value x, it returns the angle θ whose cosine equals x:
arccos(x) = θ iff cos(θ) = x
Because cosine is periodic, it cannot be inverted in the strict sense — infinitely many angles share the same cosine value. We solve this by restricting the domain to an interval where cosine is monotonic. For cosine, the common choice is [0, π]. This range is called the set of principal values, and arccos always returns an angle inside it:
- Inverse cosine domain of x for a real result: −1 ≤ x ≤ 1
- Inverse cosine range of usual principal value: 0 ≤ θ ≤ π (i.e. 0° ≤ θ ≤ 180°)
arccos(x) is the most commonly used notation. Be careful with cos⁻¹(x) — the inverse cosine is not the reciprocal of the function:
cos⁻¹(x) ≠ 1 / cos(x)
Inverse Cosine Graph
A function f has an inverse if and only if it is one-to-one. The full cosine function is not one-to-one, since cos(x) = cos(x + 2πn) for every integer n. To fix this we restrict the domain of the basic periodic cosine to [0, π]. Because cosine always lies in [−1, 1] and we chose the domain [0, π], the inverse cosine simply swaps these two intervals:
- Inverse cosine domain of x for a real result: [−1, 1]
- Inverse cosine range of usual principal value: [0, π]
The graph of y = arccos(x) is a decreasing curve running from the point (−1, π) down to (1, 0), passing through (0, π/2). Key points include:
- arccos(−1) = 180° (= π rad)
- arccos(−√3/2) = 150° (= 5π/6 rad)
- arccos(−√2/2) = 135° (= 3π/4 rad)
- arccos(−1/2) = 120° (= 2π/3 rad)
- arccos(0) = 90° (= π/2 rad)
- arccos(1/2) = 60° (= π/3 rad)
- arccos(√2/2) = 45° (= π/4 rad)
- arccos(√3/2) = 30° (= π/6 rad)
- arccos(1) = 0° (= 0 rad)
Inverse Cosine — Why Should I Care? Some Arccos Applications
Arccos may look abstract, but it turns up whenever a known ratio needs to be turned back into an angle. A few examples:
- Physics & vectors: the angle between two vectors comes straight from the dot product, θ = arccos( (a·b) / (|a|·|b|) ) — used in mechanics, computer graphics, and machine learning (cosine similarity).
- Chemistry: molecular bond angles (for example the ≈109.47° tetrahedral angle, arccos(−1/3)) are derived with inverse cosine.
- Construction & ergonomics: finding the tilt of a ramp, roof, or workstation from a rise-over-length ratio, or checking a comfortable viewing angle, both reduce to an arccos.
- Navigation & astronomy: the great-circle distance formula and many spherical-trig problems use arccos to recover an angle from a cosine.
Arccos is also tied to the other inverse trig functions through several identities:
- arccos(x) + arcsin(x) = π/2 (complementary relationship)
- arccos(−x) = π − arccos(x)
- cos(arccos(x)) = x for −1 ≤ x ≤ 1
- d/dx arccos(x) = −1 / √(1 − x²) (derivative)
- ∫ arccos(x) dx = x·arccos(x) − √(1 − x²) + C (integral)
Note: The notation cos⁻¹(x) means the inverse function, not 1/cos(x).
The abbreviation acos(x) is used in most programming languages.
Example: How to Use the Arccos Calculator
- Enter the cosine value in the input field (any number from −1 to 1). For example, type 0.5.
- Click Calculate.
- The calculator instantly shows the angle in degrees, radians, radians as a multiple of π, and gradians.
- For x = 0.5: arccos(0.5) = 60° = π/3 rad ≈ 1.0472 rad = 66.667 grad.
- The Verification line confirms that cos(60°) = 0.5 ✓
Use the Common Values quick-buttons to jump straight to standard angles like 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°.
Common Arccos Values Table
| x (cosine value) | arccos(x) — Degrees | arccos(x) — Radians | arccos(x) — Gradians |
|---|---|---|---|
| −1 | 180° | π | 200 grad |
| −√3/2 ≈ −0.8660 | 150° | 5π/6 | 166.667 grad |
| −√2/2 ≈ −0.7071 | 135° | 3π/4 | 150 grad |
| −1/2 = −0.5 | 120° | 2π/3 | 133.333 grad |
| 0 | 90° | π/2 | 100 grad |
| 1/2 = 0.5 | 60° | π/3 | 66.667 grad |
| √2/2 ≈ 0.7071 | 45° | π/4 | 50 grad |
| √3/2 ≈ 0.8660 | 30° | π/6 | 33.333 grad |
| 1 | 0° | 0 | 0 grad |
FAQs
What is the difference between arccos and cos⁻¹?
They are the same thing. arccos(x) and cos⁻¹(x) both denote the inverse cosine function. The cos⁻¹ notation can be confusing because it may look like a reciprocal (1/cos x), so mathematicians often prefer "arccos".
Why is the domain of arccos limited to [−1, 1]?
The cosine of any real angle always lies between −1 and 1. There is no real angle whose cosine is, say, 2 — so arccos(2) is undefined in the real numbers.
What range of angles can arccos return?
The principal value of arccos is always in [0°, 180°] (that is, [0, π] radians). Other angles with the same cosine exist (e.g. cos(300°) = cos(60°) = 0.5), but they are not the principal value returned by this calculator.
How are degrees, radians, and gradians related?
360° = 2π rad = 400 grad. Degrees split a full circle into 360 parts; radians measure arc length on a unit circle; gradians divide a right angle into 100 equal parts (used in surveying). This calculator shows all three for easy conversion.
What is the derivative of arccos(x)?
d/dx arccos(x) = −1 / √(1 − x²), defined for |x| < 1. Notice it is exactly the negative of the derivative of arcsin(x), which follows from the identity arccos(x) + arcsin(x) = π/2.