Free Arccos Calculator (Inverse Cosine)

What Is Arccos (Inverse Cosine)?

The arccos function — also called the inverse cosine or arc cosine — reverses the operation of the standard cosine function. If $\cos(y) = x$, then $\arccos(x) = y$ (with $y$ taken from the principal value range). Because cosine is periodic, a direct inverse over all real numbers is not possible. To obtain a well-defined inverse, the domain of cosine is restricted to $[0, \pi]$ (or $0^\circ$ to $180^\circ$). This interval yields the principal values of arccos.

Domain and Range of Arccos

For real inputs, the domain of $\arccos(x)$ is:

$$-1 \leq x \leq 1$$

The corresponding principal range (output) is:

$$0 \leq y \leq \pi \quad \text{or} \quad 0^\circ \leq y \leq 180^\circ$$

This inverse trigonometry calculator automatically respects these boundaries.

Notation and Common Confusion

Arccos is written as $\arccos(x)$, $\mathrm{acos}(x)$, or $\cos^{-1}(x)$. The superscript “−1” does not denote the reciprocal ($\sec(x)$) but the inverse function. The arc cosine calculator always interprets $\cos^{-1}(x)$ as the inverse cosine.

Inverse Cosine Graph

The graph of $\arccos(x)$ is derived by reflecting the restricted cosine curve (for $x \in [0,\pi]$) across the line $y = x$. It is a decreasing function that passes through the points $(-1, \pi)$, $(0, \pi/2)$, and $(1, 0)$.

Common Arccos Values

The table below lists widely used arccos outputs in both degrees and radians:

$x$	$\arccos(x)$ (deg)	$\arccos(x)$ (rad)
$-1$	$180^\circ$	$\pi$
$-\sqrt{3}/2$	$150^\circ$	$5\pi/6$
$-\sqrt{2}/2$	$135^\circ$	$3\pi/4$
$-1/2$	$120^\circ$	$2\pi/3$
$0$	$90^\circ$	$\pi/2$
$1/2$	$60^\circ$	$\pi/3$
$\sqrt{2}/2$	$45^\circ$	$\pi/4$
$\sqrt{3}/2$	$30^\circ$	$\pi/6$
$1$	$0^\circ$	$0$

Practical Applications of Arccos

Scientific Uses

• Solving triangles – The law of cosines $c^2 = a^2 + b^2 - 2ab\cos C$ can be rearranged to find any angle: $C = \arccos\left(\dfrac{a^2 + b^2 - c^2}{2ab}\right)$.
• Catenary curves – The shape of a hanging chain or cable is described by an equation involving arccos.
• Vector angles – The dot product formula gives the angle between two vectors: $\theta = \arccos\left(\dfrac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\right)$.
• Hydraulic radius – For a partially filled pipe, the wetted perimeter calculation uses arccos.
• Molecular geometry – Bond angles in molecules like $\mathrm{H_2O}$ or $\mathrm{CH_4}$ can be estimated using inverse cosine.

Real‑Life Examples

• Construction – Determine roof pitch or staircase inclination when horizontal run and slope length are known.
• Accessibility ramps – Given ramp length and horizontal distance, arccos yields the incline angle.
• Ergonomics – Set monitor tilt or viewing angle for a comfortable workstation; this inverse cosine calculator simplifies the geometry.

Whether you need to find arccos of a cosine value, explore inverse trigonometry concepts, or solve practical problems, this arc cosine calculator serves as a fast and educational tool.