Inverse Cosine Calculator — cos⁻¹ in Degrees, Radians & Gradians
Enter a cosine value or the adjacent and hypotenuse sides of a right triangle to find the angle using cos⁻¹ — output in degrees, radians, and gradians with a triangle or cosine-wave diagram.
Input method
Angle in the range 0° to 180°
- 1
Inverse cosine (radians)
arccos(0.866) = 0.52365 radPrincipal value in [0, π] rad. - 2
Convert to degrees
0.52365 × 180 ÷ π = 30.0029 °
How does this calculator work?
θ = cos⁻¹(adj/hyp) = arccos(x) ∈ [0°, 180°]. Enter a decimal cosine value or right-triangle sides. Common values: cos⁻¹(0) = 90°, cos⁻¹(0.5) = 60°, cos⁻¹(1) = 0°, cos⁻¹(−1) = 180°. Degrees × π/180 = radians; degrees × 10/9 = gradians.
Formula
How this is calculated
The inverse cosine function (cos⁻¹ or arccos) reverses the cosine: given a value x, it returns the unique angle θ in [0°, 180°] whose cosine is x. This calculator accepts two input modes. In "cosine value" mode you enter a decimal from −1 to 1 directly. In "right triangle" mode you enter the adjacent side and hypotenuse; the ratio adjacent/hypotenuse is the cosine of the angle at that corner by definition, so θ = cos⁻¹(adj/hyp).
For example, a right triangle with adjacent = 3 and hypotenuse = 5 gives cos(θ) = 0.6, so θ = cos⁻¹(0.6) ≈ 53.13°. The complementary angle (the other acute angle) is 90° − 53.13° = 36.87°. The calculator also converts the result to radians (multiply degrees by π/180) and gradians (multiply degrees by 10/9).
In "cosine value" mode the triangle diagram normalises to a unit hypotenuse: the adjacent side equals x and the opposite side equals √(1−x²), illustrating the Pythagorean identity. For obtuse angles (x < 0, meaning θ > 90°), the cosine is negative and a right triangle is not applicable — use the cosine value mode instead.
Frequently asked questions
cos⁻¹(x) (arccos) is the inverse function — it takes a cosine value and returns an angle. 1/cos(x) = sec(x) is the reciprocal of cosine for a given angle — it returns a number, not an angle. Despite the superscript −1, they are completely different operations.
Yes — arccos always returns a value in [0°, 180°], correctly handling second-quadrant angles. For example, cos⁻¹(−0.5) = 120°. In "right triangle" mode all triangle angles are acute (< 90°), so use "cosine value" mode for obtuse results.
Multiply degrees by π/180 ≈ 0.017453. So 60° = π/3 ≈ 1.0472 rad and 90° = π/2 ≈ 1.5708 rad. The calculator shows radians and gradians automatically alongside degrees.
TG we-Calculate Editorial Team. (2026). Inverse Cosine Calculator — cos⁻¹ in Degrees, Radians & Gradians [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cos-inverse-calculator
TG we-Calculate Editorial Team. "Inverse Cosine Calculator — cos⁻¹ in Degrees, Radians & Gradians." TG we-Calculate. 2026. https://we-calculate.com/calculator/cos-inverse-calculator.
TG we-Calculate Editorial Team, "Inverse Cosine Calculator — cos⁻¹ in Degrees, Radians & Gradians," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cos-inverse-calculator
@misc{wecalculate_cos_inverse_calculator, title = {Inverse Cosine Calculator — cos⁻¹ in Degrees, Radians & Gradians}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cos-inverse-calculator}}, year = {2026}, note = {TG we-Calculate} }
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