About

The Inverse Cosine function, denoted as arccos(x) or cos^-1(x), is the reverse operation of the cosine function. While cosine takes an angle and returns a ratio (adjacent side / hypotenuse), arccos takes that ratio and returns the angle. This is fundamental in physics for calculating dot products of vectors, finding angles in triangles using the Law of Cosines, and resolving forces in engineering.

Because the cosine function is periodic, the inverse cosine is restricted to a specific range (principal values) to ensure a unique result. The output is always defined within the range [0, π] radians or [0°, 180°].

Formulas

The relationship between the angle and the value is defined as:

θ = arccos(x)

Where:

−1 ≤ x ≤ 1

If the input x is outside this domain, the result is undefined (complex).

Reference Data

Input Value (x)	Angle (Radians)	Angle (Degrees)	Note
1	0	0°	Max Value
√3 / 2 (0.866)	π / 6	30°	Standard
√2 / 2 (0.707)	π / 4	45°	Standard
0.5	π / 3	60°	Standard
0	π / 2	90°	Orthogonal
-0.5	2π / 3	120°	Obtuse
-√2 / 2 (-0.707)	3π / 4	135°	Obtuse
-1	π	180°	Min Value

Frequently Asked Questions

The cosine of an angle represents the ratio of the adjacent side to the hypotenuse in a right triangle. Since the hypotenuse is always the longest side, this ratio cannot exceed 1 or be less than -1. Therefore, arccos(2) is geometrically impossible in real numbers.

They are completely different. Arccos is the *inverse function* (finding the angle), while Secant is the *reciprocal* of cosine (1/cos). Do not confuse cos⁻¹(x) with (cos(x))⁻¹.

Multiply the radian value by 180 and divide by Pi (π). For example: (π/2) * (180/π) = 90 degrees.

The principal value range for arccos is [0, π] in radians or [0°, 180°]. This means the calculator will never return a negative angle, unlike arcsin or arctan.