Calculate the inverse cosine (cos⁻¹) of any value from −1 to 1. Returns the angle in degrees, radians, and gradians with unit circle visual.
The **Cos Inverse Calculator** computes cos⁻¹(x) — the arccosine — and returns the angle whose cosine equals your input. Enter any value between −1 and 1 to instantly see the result in degrees, radians, and gradians, along with the quadrant, reference angle, all six trigonometric function values, and unit circle coordinates.
The inverse cosine function is one of the most fundamental operations in trigonometry. It appears constantly in physics (finding the angle between two vectors), engineering (phase angles in AC circuits), computer graphics (lighting and shading models), navigation (great-circle bearings), and statistics (angular correlation). Because the cosine function maps the interval [0°, 180°] one-to-one onto [−1, 1], the principal value of arccos always falls between 0° and 180° (0 to π radians, or 0 to 200 gradians).
This calculator goes beyond a simple angle conversion. It displays the result in three angle unit systems simultaneously, shows all six trig functions at the resulting angle, provides a visual bar chart of the unit circle position, and includes a comprehensive common-values reference table. A comparison mode lets you evaluate two inputs side by side, and collapsible reference panels cover inverse trig identities and the arccos–arcsin relationship (arccos(x) + arcsin(x) = 90° for all valid x).
Nine preset buttons cover every standard angle from cos⁻¹(1) = 0° through cos⁻¹(−1) = 180°, so you can explore the function without manual entry.
Cos Inverse Calculator (arccos / cos⁻¹) helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Angle (degrees), Angle (radians), Angle (gradians) in one pass.
θ = cos⁻¹(x), where −1 ≤ x ≤ 1 and the principal value is 0 ≤ θ ≤ π. Degrees: θ° = θ × 180/π. Gradians: θ_g = θ × 200/π. Related: sin(θ) = √(1 − x²), tan(θ) = √(1 − x²)/x.
Result: 60°
Using value=0.5, unit=degrees, the calculator returns 60°. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to cos inverse calculator (arccos / cos⁻¹) workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
Cos inverse (arccos or cos⁻¹) is the inverse of the cosine function. Given a value x between −1 and 1, it returns the unique angle θ in [0°, 180°] such that cos(θ) = x.
Cosine is not one-to-one over all angles, so arccos is restricted to the principal branch [0°, 180°] (or [0, π] radians) where cosine is strictly decreasing. This ensures a unique output for every input.
Multiply radians by 180/π to get degrees, or by 200/π to get gradians. Degrees × π/180 = radians. Degrees × 10/9 = gradians. This calculator displays all three simultaneously.
No. cos⁻¹(x) gives the angle whose cosine is x (the inverse function). 1/cos(x) = sec(x) is the reciprocal function (secant). They are completely different operations.
Cosine values only exist in the range [−1, 1], so arccos is undefined for inputs outside this interval. The calculator displays a warning and no results.
Gradians (or gons) divide a right angle into 100 equal parts, so a full circle is 400 gradians. They are used in surveying, civil engineering, and some European engineering standards.
The angle between two vectors A and B is θ = cos⁻¹((A·B) / (|A||B|)). This is the basis of cosine similarity in machine learning and the dot-product formula in physics.