Inverse Cosine
Calculate inverse cosine (arccos/cos⁻¹) of any value x in [-1, 1]. Free online arccos calculator returns the angle in degrees and radians with instant domain validation.
About This Calculator
Our Inverse Cosine Calculator (Arccos Calculator) computes the inverse cosine function, denoted as arccos(x), cos&supmin;¹(x), or acos(x), for any input value within the valid domain of [-1, 1]. The inverse cosine function returns the angle whose cosine equals the given value, with results displayed in both degrees (0° to 180°) and radians (0 to π).
This calculator is essential for students learning trigonometry, engineers solving geometric problems, and professionals working with angles in fields such as physics, computer graphics, and construction. The inverse cosine is widely used in the law of cosines to find angles when all three sides of a triangle are known, and in the dot product formula to compute the angle between two vectors.
How Inverse Cosine Works
Mathematically, arccos is defined as the inverse of the cosine function restricted to the interval [0, π] (or [0°, 180°]). Since cosine is periodic, restricting its domain ensures a one-to-one mapping. For a given value x, arccos(x) = y where cos(y) = x and y ∈ [0, π].
The calculator uses JavaScript's built-in Math.acos() function for high-precision computation. Results are displayed to 4 decimal places in both degree and radian units. Input is validated to ensure it falls within the [-1, 1] domain before computation.
Key Inverse Cosine Values Reference
| x | arccos(x) in degrees | arccos(x) in radians |
|---|---|---|
| -1 | 180° | π |
| -√3/2 | 150° | 5π/6 |
| -√2/2 | 135° | 3π/4 |
| -1/2 | 120° | 2π/3 |
| 0 | 90° | π/2 |
| 1/2 | 60° | π/3 |
| √2/2 | 45° | π/4 |
| √3/2 | 30° | π/6 |
| 1 | 0° | 0 |
Frequently Asked Questions
What is inverse cosine (arccos)?
Inverse cosine, also written as arccos, cos&supmin;¹, or acos, is the inverse function of the cosine. For a given value x in the domain [-1, 1], arccos(x) returns the angle whose cosine is x. The result is always in the range [0°, 180°] or [0, π] radians.
What is the domain of inverse cosine?
The domain of arccos(x) is [-1, 1] for real number results. Any value outside this range will produce a complex number. The calculator validates your input and shows an error if you enter a value outside the valid domain.
What is the range of inverse cosine?
The range (principal values) of arccos(x) is [0°, 180°] in degrees or [0, π] in radians. This means arccos always returns an angle between 0 and 180 degrees, unlike arcsin which returns values between -90° and 90°.
Is arccos the same as 1/cos?
No. Arccos is the inverse cosine function (cos&supmin;¹), not the reciprocal (secant). arccos(x) returns the angle whose cosine is x, while 1/cos(x) is the secant function. This is a common misconception in trigonometry.
What is arccos(0)?
Arccos(0) = 90° (π/2 radians). Since cos(90°) = 0, the inverse cosine of 0 is 90 degrees. This is one of the standard trigonometric values worth memorizing.
What are common arccos values?
Common arccos values include: arccos(1) = 0°, arccos(√3/2) = 30°, arccos(√2/2) = 45°, arccos(1/2) = 60°, arccos(0) = 90°, arccos(-1/2) = 120°, arccos(-√2/2) = 135°, arccos(-√3/2) = 150°, arccos(-1) = 180°.
How is inverse cosine used in real life?
Inverse cosine is used in physics to find angles between vectors via the dot product formula, in engineering to calculate roof pitch and ramp inclinations, in chemistry to estimate bond angles in molecules, and in computer graphics for rotation calculations.
What is the difference between arccos and arcsec?
Arccos is the inverse of cosine (domain [-1, 1], range [0°, 180°]), while arcsec is the inverse of secant (domain |x| ≥ 1, range [0°, 180°] excluding 90°). They are related by arcsec(x) = arccos(1/x) for |x| ≥ 1.