Inverse Cosine Calculator

What is Inverse Cosine Calculator?

An Inverse Cosine (arccos) Calculator returns the angle whose cosine equals a given ratio. If x is a real number with \(|x|\le 1\), the principal value \(\arccos(x)\) is the unique angle \(\theta\in[0,\pi]\) satisfying \(\cos\theta=x\). This value is fundamental in trigonometry, geometry, physics, and data analysis—anywhere an angle must be recovered from lengths or dot products. Key identities and definitions are shown below for clarity and quick reference.

About the Inverse Cosine Calculator

This tool accepts a numeric ratio \(x\) (or data that implies a ratio) and returns \(\arccos(x)\) in radians and degrees, with exact forms where possible. It supports three convenient modes: (1) direct input of \(x\) in \([-1,1]\), (2) vector mode using the dot‑product formula to compute the angle between vectors, and (3) triangle mode via the Law of Cosines to recover an interior angle from three side lengths. It displays step‑by‑step working, unit toggles, and domain checks, warning when inputs fall outside \([-1,1]\) due to rounding or measurement noise.

How to Use this Inverse Cosine Calculator

1. Choose a mode: Direct (enter \(x\)), Vectors (enter components of \(\mathbf a,\mathbf b\)), or Triangle (enter side lengths \(a,b,c\)).
2. Select output units (degrees or radians) and optional precision.
3. Click calculate to evaluate \(\arccos\) and view responsive formula steps and the final angle.
4. For vectors, confirm magnitudes are nonzero; for triangles, verify the triangle inequality holds.
5. Copy exact values (when available) and decimal approximations for your assignment, lab, or engineering work.

Examples

Example 1: Direct ratio

Find \(\arccos(\tfrac{1}{2})\).

Example 2: Angle between vectors

Let \(\mathbf a=(2,1)\) and \(\mathbf b=(1,3)\).

Example 3: Triangle via Law of Cosines

Given sides \(a=3\), \(b=5\), \(c=4\), find angle \(C\) opposite side \(c\).

FAQs

What range does \(\arccos\) return?

The principal value lies in \([0,\pi]\) (0° to 180°), even when the original angle might differ by multiples of \(2\pi\).

What is the valid input domain?

Real inputs must satisfy \(|x|\le1\). Values slightly outside often come from rounding; clamping to \([-1,1]\) is standard.

Why isn’t \(\arccos(\cos\theta)=\theta\) always true?

Because \(\arccos\) returns the principal value in \([0,\pi]\). Reduce \(\theta\) to that interval to compare fairly.

How do I get all solutions for a given cosine?

If \(\cos\theta=x\), then \(\theta=\pm\arccos(x)+2\pi k\) for any integer \(k\); choose the angle matching your quadrant/context.

Degrees or radians—which should I use?

Use radians for calculus and many formulas; degrees are convenient for geometry. You can switch output units at any time.

Can I enter vectors instead of a ratio?

Yes. The calculator computes \(\theta=\arccos((\mathbf a\cdot\mathbf b)/(\lVert\mathbf a\rVert\lVert\mathbf b\rVert))\).

How does this help with triangles?

Use the Law of Cosines to recover an angle from three sides; it’s robust even when no right angle is present.

What if my vectors have zero length?

The angle is undefined because the normalization requires nonzero magnitudes. Provide valid, nonzero vectors.

Will I see exact radical values?

Yes, when possible (e.g., \(\arccos(\tfrac{1}{2})=\tfrac{\pi}{3}\)). Otherwise, a decimal approximation is shown alongside.

Why do small rounding errors matter?

Cosine is flat near its extrema; tiny input changes can shift the angle noticeably. The tool warns and clamps if needed.