Arccos Calculator
Calculate arccos(x) = cos⁻¹(x) with exact values, domain validation, results in degrees and radians.
cos⁻¹(x)
Updates as you type| x | Degrees | Radians |
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Formula & definition
- x
- A real number between −1 and 1 (inclusive). Represents a cosine ratio — the x-coordinate of a point on the unit circle.
- θ
- The principal-value angle whose cosine equals x. Arccos returns only one solution even though infinitely many angles share the same cosine.
- Notation
- arccos(x), cos⁻¹(x), and acos(x) all mean the same function. The superscript −1 denotes the inverse, not a reciprocal.
- Quadrants
- The principal value lies in Q1 when x > 0 (angle 0°–90°) and Q2 when x < 0 (angle 90°–180°). Unlike arcsin, arccos never returns a negative angle.
- Identity
- arccos(−x) = π − arccos(x). Other angles with the same cosine are ±θ + 360°k for any integer k.
Step-by-step
Frequently Asked Questions
What is arccos and what does it do?
Arccos (also written cos⁻¹ or acos) is the inverse cosine function. Given a cosine value x between −1 and 1, it returns the unique angle θ in [0°, 180°] whose cosine equals x. It is the function you use to recover an angle when you only know the cosine ratio.
Why does arccos only accept inputs between −1 and 1?
The cosine of any real angle is bounded by [−1, 1] — that is the range of cos(θ). So an input outside that range cannot correspond to the cosine of any real angle, and arccos is undefined there. (It is defined for complex numbers, but this calculator works in the reals.)
Why does arccos return only one angle when many angles have the same cosine?
Cosine is periodic and not one-to-one — for example cos(60°) = cos(300°) = cos(420°) = ½. To make the inverse a function, mathematicians restrict the output to the principal range [0°, 180°]. All other valid angles can be obtained as ±θ + 360°·k for any integer k.
What is the difference between arccos and 1/cos?
They are unrelated. arccos(x) is the inverse function — it returns an angle. 1/cos(x) = sec(x) is the secant — it returns a ratio. The notation cos⁻¹(x) means the inverse, not the reciprocal.
When would I use arccos in real problems?
Arccos is essential whenever you know a cosine ratio and need the angle. Common uses: finding the angle between two vectors using the dot product (θ = arccos(u·v / (|u||v|))), recovering an angle in a right triangle from the adjacent/hypotenuse ratio, and law-of-cosines triangle problems where you solve for an interior angle.