Arccosine Calculator

What is Arccos?

As the inverse of the cosine, arccos (arc cosine) takes a value x and returns the angle θ where cos(θ) = x. Whatever the valid input, the answer always stays within [0°, 180°].

To make it concrete: arccos(1) = 0°, arccos(0.5) = 60°, arccos(0) = 90°, arccos(−1) = 180°.

Domain: −1 ≤ x ≤ 1. Any value beyond that range has no real arccos.

The arccosine function

The arccosine hands you back the angle whose cosine equals a given value: acos(x) = θ such that cos(θ) = x. Its domain runs over [−1, 1] and the conventional range is [0, π], that is, 0° up to 180°. You'll see it written as acos, cos⁻¹ or arccos. Cosine isn't one-to-one across its whole domain, so the inverse gets pinned to a single principal branch. Its derivative is (acos x)' = −1/√(1 − x²), and a useful companion is the complementary identity asen(x) + acos(x) = π/2. For a quick check, acos(0,5) = 60°.

Applications: vectors, geodesy and 3D graphics

The textbook example is finding the angle between vectors from the dot product: θ = acos((u·v)/(|u||v|)), which underpins a lot of physics, 3D animation and computer vision work. The great-circle distance between two points on Earth comes out the same way, via d = R·acos(sen φ₁ sen φ₂ + cos φ₁ cos φ₂ cos Δλ). You'll also meet arccosine in spectroscopy, where it helps compute refractive indices and angles of incidence, and in robotics, where it sets joint angles for inverse kinematics.

FAQ

Why is the domain limited to [−1, 1]? Because cosine only ever outputs values in that interval. No real angle has a cosine above 1 or below −1.

Why is the range [0°, 180°] and not [−90°, 90°]? To keep the inverse single-valued, you pick the branch where cosine is monotonically decreasing, and that branch happens to cover [0, π].

What is the difference between acos and 1/cos? They have nothing to do with each other. acos(x) is the inverse function, whereas 1/cos(x) = sec(x) is just the multiplicative reciprocal.

How does it relate to asen? Through the complementary identity asen(x) + acos(x) = π/2.