Arccosine Calculator
Calculate the exact angle whose cosine equals a given numerical value. Supports radians, exact $\pi$ fraction representations, degrees, and dynamic unit circle angle plots.
Unit Circle Principal Range [0, π]
(±x, √(1−x²))Step-by-Step Arccosine Evaluation & Range Analysis Range: 0 ≤ θ ≤ π (0° ≤ θ ≤ 180°)
How to Calculate the Arccosine of a Number
The arccosine of a number x (written as arccos(x) or cos⁻¹(x)) returns the angle θ such that cos(θ) = x. Because the cosine of any real angle is restricted to [-1, 1], the input x must satisfy -1 ≤ x ≤ 1. The output angle θ is returned within the principal branch [0, π] radians (or [0°, 180°]). Positive values of x yield acute angles in Quadrant I, while negative values of x yield obtuse angles in Quadrant II according to arccos(-x) = π - arccos(x).
Anatomy of the Arccosine Function & Mathematical Definition
The arccosine function, denoted as $\arccos(x)$, $\text{acos}(x)$, or $\cos^{-1}(x)$, is the inverse operation of the trigonometric cosine function.
Because the standard cosine function $\cos(\theta)$ is periodic and non-monotonic across the entire real number line, it fails the horizontal line test. To construct a well-defined single-valued inverse function, mathematicians restrict the domain of $\cos(\theta)$ to $[0, \pi]$.
All possible cosine ratio values.
Quadrants I and II of the unit circle.
Strictly valid for all $x \in [-1, 1]$.
Domain [-1, 1] & Principal Range [0, π] / [0°, 180°]
The behavior of $\arccos(x)$ is cleanly partitioned across its domain:
- $x > 0$ (Positive Cosine): Produces an acute angle $\theta \in (0, \pi/2)$ located in Quadrant I.
- $x = 0$ (Zero Cosine): Produces the quadrantal right angle $\theta = \pi/2 = 90^\circ$.
- $x < 0$ (Negative Cosine): Produces an obtuse angle $\theta \in (\pi/2, \pi)$ located in Quadrant II.
Exact Values of Arccosine on the Unit Circle
| Cosine Value (x) | Exact Radians | Degrees | Quadrant |
|---|---|---|---|
| 1 | 0 | 0° | Positive x-axis |
| √3 / 2 ≈ 0.8660 | π / 6 | 30° | Quadrant I |
| √2 / 2 ≈ 0.7071 | π / 4 | 45° | Quadrant I |
| 1 / 2 = 0.5 | π / 3 | 60° | Quadrant I |
| 0 | π / 2 | 90° | Positive y-axis |
| −1 / 2 = −0.5 | 2π / 3 | 120° | Quadrant II |
| −√2 / 2 ≈ −0.7071 | 3π / 4 | 135° | Quadrant II |
| −√3 / 2 ≈ −0.8660 | 5π / 6 | 150° | Quadrant II |
| −1 | π | 180° | Negative x-axis |
Negative Argument & Reflection Identities: arccos(-x) = π − arccos(x)
Unlike arcsine and arctangent (which are odd functions where $\arcsin(-x) = -\arcsin(x)$), the arccosine function has point symmetry about $(0, \pi/2)$:
arccos(−x) = π − arccos(x)
In degree mode: $\arccos(-x) = 180^\circ - \arccos(x)$.
Co-function Identity: $\arcsin(x) + \arccos(x) = \frac{\pi}{2} = 90^\circ$ for all $x \in [-1, 1]$.
Calculus Properties: Derivative & Integral of arccos(x)
Strictly negative on $(-1, 1)$, indicating $\arccos(x)$ is monotonically decreasing.
Derived via integration by parts ($u = \arccos x, dv = dx$).
Step-by-Step Worked Examples (Exact & Decimal Solutions)
Find the exact and decimal value of arccos(−0.5).
1. Evaluate positive base angle: arccos(0.5) = π/3 rad (60°).
2. Apply reflection identity: arccos(−0.5) = π − π/3 = 2π/3 radians.
3. Convert to degrees: 2π/3 × (180/π) = 2 × 60° = 120.000°.
4. Decimal approximation: 2(3.14159)/3 ≈ 2.09440 radians.
Common Pitfalls & Secant Notation Confusion
$\cos^{-1}(x)$ denotes the functional inverse $\arccos(x)$, NOT the multiplicative reciprocal. The reciprocal of cosine is $\sec(x) = \frac{1}{\cos(x)} = (\cos(x))^{-1}$.
Entering numbers like $x = 1.5$ or $x = -2$ into $\arccos(x)$ produces no real solution because the range of $\cos(\theta)$ is strictly $[-1, 1]$.
Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.
Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.