cos⁻¹ Calculator — Inverse Cosine (cos-1)
Enter a cosine value (from −1 to 1) to find the angle whose cosine equals that value — output in degrees, radians, and as a fraction of π, with sin and tan of the result.
Principal value: always in the range 0° to 180°
- 1
Apply inverse cosine
arccos(0.5) = 1.047198 radThe principal value is always in [0, π] rad. - 2
Convert to degrees
1.047198 × 180 ÷ π = 60 ° - 3
sin(θ) = √(1 − x²)
√(1 − 0.5²) = √0.75 = 0.866025
How does this calculator work?
cos⁻¹(x) returns the angle θ ∈ [0°, 180°] whose cosine is x. Common values: cos⁻¹(1) = 0°, cos⁻¹(0.5) = 60°, cos⁻¹(0) = 90°, cos⁻¹(−0.5) = 120°, cos⁻¹(−1) = 180°. sin(θ) = √(1−x²). Input must be in [−1, 1].
Formula
How this is calculated
The inverse cosine function cos⁻¹(x), also written arccos(x), answers "which angle has this cosine value?" Because cosine is periodic it would normally have infinitely many answers; by convention cos⁻¹ returns only the principal value — the unique angle in [0°, 180°] (equivalently [0, π] radians) whose cosine is x.
The result is derived from the Pythagorean identity: cos²(θ) + sin²(θ) = 1, so sin(θ) = √(1 − x²) (positive because θ is in the first or second quadrant where sine is non-negative). Then tan(θ) = sin(θ)/cos(θ) = √(1−x²)/x, which is undefined when x = 0 (the angle is 90°). The cosine wave visualisation shows the full cycle with your angle marked by the phase offset, so you can see where the value sits on the curve.
Common exact values: cos⁻¹(1) = 0°, cos⁻¹(√3/2) ≈ 30°, cos⁻¹(√2/2) ≈ 45°, cos⁻¹(0.5) = 60°, cos⁻¹(0) = 90°, cos⁻¹(−0.5) = 120°, cos⁻¹(−1) = 180°. Inputs outside [−1, 1] are outside the real domain of arccos.
Frequently asked questions
cos⁻¹(x) (also written arccos(x)) is the inverse cosine function: it returns the angle θ in [0°, 180°] whose cosine equals x. It undoes the cosine function: cos(cos⁻¹(x)) = x for x in [−1, 1].
Cosine is not one-to-one — many angles share the same cosine value (e.g., cos(60°) = cos(300°) = 0.5). To define a unique inverse, cos⁻¹ is restricted to the principal value range [0°, 180°]. For angles outside this range, use your quadrant information to find the correct solution.
In a right triangle, cos(θ) = adjacent / hypotenuse. Divide the adjacent side by the hypotenuse and enter that ratio here. For example, adjacent = 3, hypotenuse = 5 → cos⁻¹(3/5) = cos⁻¹(0.6) ≈ 53.13°.
TG we-Calculate Editorial Team. (2026). cos⁻¹ Calculator — Inverse Cosine (cos-1) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cos-1-calculator
TG we-Calculate Editorial Team. "cos⁻¹ Calculator — Inverse Cosine (cos-1)." TG we-Calculate. 2026. https://we-calculate.com/calculator/cos-1-calculator.
TG we-Calculate Editorial Team, "cos⁻¹ Calculator — Inverse Cosine (cos-1)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cos-1-calculator
@misc{wecalculate_cos_1_calculator, title = {cos⁻¹ Calculator — Inverse Cosine (cos-1)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cos-1-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?