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Arccos Calculator (Inverse Cosine)

Calculate the inverse cosine (arccos) of a value. Returns the angle whose cosine equals the input.

By Christopher FloiedPublished Updated

The cosine value (must be between -1 and 1)

Results

Angle (degrees)

60°

Angle (radians)

1.047198 rad

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Arccos Calculator (Inverse Cosine). Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Arccos Calculator (Inverse Cosine) computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Arccos Calculator (Inverse Cosine) when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Arccos Calculator (Inverse Cosine)

The arccosine function (also written cos⁻¹ or arccos) is the inverse of the cosine function. Given a value x between −1 and 1, arccos(x) returns the angle θ whose cosine equals x. The principal value range is [0°, 180°] or [0, π] radians. The arccosine is commonly used when you know the adjacent side and hypotenuse of a right triangle and need the angle. It is also the function used to find angles in the dot product formula: cos(θ) = (A·B)/(|A||B|), so θ = arccos((A·B)/(|A||B|)). This appears frequently in computer graphics for lighting calculations, in physics for work done by a force at an angle, and in machine learning for cosine similarity. This calculator provides the result in both degrees and radians.

The Math Behind It

To define arccos, we restrict the cosine function to the domain [0, π], where it is strictly decreasing and covers the full range [−1, 1]. The inverse arccos: [−1, 1] → [0, π] is continuous and differentiable on (−1, 1). Its derivative is d/dx arccos(x) = −1/√(1 − x²), which is the negative of the arcsine derivative. This means arcsin(x) + arccos(x) = π/2, a fundamental identity. The integral ∫ dx/√(1 − x²) = arcsin(x) + C, and equivalently −arccos(x) + C. The Taylor series of arccos near x = 0 is arccos(x) = π/2 − x − x³/6 − 3x⁵/40 − .... In the dot product formula, the angle between vectors u and v is θ = arccos(u·v / (|u||v|)). In the Law of Cosines, the angle C = arccos((a² + b² − c²)/(2ab)). The arccosine function is monotonically decreasing, unlike arcsine which is increasing.

Formula Reference

Inverse Cosine

θ = arccos(x) = cos⁻¹(x)

Variables: x ∈ [−1, 1], θ ∈ [0°, 180°]

Worked Examples

Example 1: Angle between vectors

Vectors u = (1, 0) and v = (1, 1). Find the angle between them.

Step 1:u·v = 1×1 + 0×1 = 1
Step 2:|u| = 1, |v| = √2
Step 3:cos(θ) = 1/√2 ≈ 0.7071
Step 4:θ = arccos(0.7071)

θ = 45°

Common Mistakes & Tips

  • !Entering a value outside [−1, 1].
  • !Confusing arccos with arcsec (1/cos).
  • !Forgetting that arccos returns values in [0°, 180°], never negative.

Related Concepts

Arcsin Calculator

Inverse sine function.

Cosine Calculator

The forward cosine function.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Arcsin Calculator (Inverse Sine)

Cosine Calculator

Frequently Asked Questions

What is the relationship between arcsin and arccos?

arcsin(x) + arccos(x) = 90° (or π/2 radians) for all x in [−1, 1].

Why does arccos return angles between 0° and 180°?

This is the principal value range chosen so that arccos is a proper function (single-valued). Cosine is one-to-one on [0°, 180°].

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