Arccos Calculator - Inverse Cosine Calculator

The Arccos Calculator is a powerful mathematical tool that calculates the inverse cosine (arccos) of any value. The arccos function, also known as the inverse cosine function, returns the angle whose cosine equals the given value. This calculator provides accurate results in both radians and degrees, making it essential for trigonometry, engineering, physics, and various mathematical applications.

What is Arccos (Inverse Cosine)?

The arccos function is the inverse of the cosine function. If cos(θ) = x, then arccos(x) = θ. The arccos function is defined for values between -1 and 1 (inclusive) and returns angles in the range [0, π] radians or [0°, 180°].

Key Properties of Arccos Function

Domain: [-1, 1] - The input values must be between -1 and 1
Range: [0, π] radians or [0°, 180°] - The output angles
Monotonic: Decreasing function (as input increases, output decreases)
Continuous: The function is continuous on its domain

Important Arccos Values

cos(θ)	arccos(θ) in Radians	arccos(θ) in Degrees
1	0	0°
√3/2 ≈ 0.866	π/6	30°
√2/2 ≈ 0.707	π/4	45°
1/2 = 0.5	π/3	60°
0	π/2	90°
-1/2 = -0.5	2π/3	120°
-1	π	180°

Mathematical Formula

The arccos function can be expressed using the following mathematical relationship:

If cos(θ) = x, then θ = arccos(x)

Where:

θ is the angle in radians or degrees
x is the cosine value (must be between -1 and 1)
arccos(x) returns the angle whose cosine is x

Applications of Arccos Function

1. Trigonometry and Geometry

Used to find angles in triangles when the cosine value is known. Essential for solving trigonometric equations and geometric problems.

2. Physics and Engineering

Applied in wave analysis, signal processing, and mechanical engineering calculations involving angular measurements and periodic functions.

3. Computer Graphics and Animation

Used in 3D graphics programming for calculating angles between vectors, camera positioning, and object rotation calculations.

4. Navigation and Astronomy

Essential for celestial navigation, calculating angles between celestial objects, and determining positions based on angular measurements.

How to Use the Arccos Calculator

Enter the cosine value: Input a number between -1 and 1 in the cosine value field
Select output unit: Choose between radians or degrees for the result
Get instant results: The calculator automatically computes and displays the arccos value
Copy results: Use the copy button to copy the result to your clipboard

Common Use Cases

Example 1: Finding an Angle in a Right Triangle

If you know the adjacent side is 4 and the hypotenuse is 5, then cos(θ) = 4/5 = 0.8. Using arccos(0.8) ≈ 36.87°, you can find the angle θ.

Example 2: Solving Trigonometric Equations

To solve cos(x) = 0.5, you would use arccos(0.5) = 60° or π/3 radians.

Example 3: Vector Calculations

When calculating the angle between two vectors using the dot product formula, arccos is used to find the angle from the cosine value.

Tips for Using Arccos Calculator

Input validation: Always ensure your input is between -1 and 1
Unit selection: Choose the appropriate unit (radians or degrees) for your application
Precision: The calculator provides high precision results up to 10 decimal places
Error handling: The calculator will display an error message for invalid inputs

Relationship with Other Trigonometric Functions

The arccos function is related to other inverse trigonometric functions:

arccos(x) + arcsin(x) = π/2 (for x in [-1, 1])
arccos(-x) = π - arccos(x)
arccos(x) = arctan(√(1-x²)/x) (for x > 0)

Frequently Asked Questions

What is the domain and range of the arccos function?

The arccos function has a domain of [-1, 1], meaning you can only input values between -1 and 1 (inclusive). The range is [0, π] radians or [0°, 180°], meaning the output angles are always between 0 and 180 degrees.

Can I use arccos for values outside the range [-1, 1]?

No, the arccos function is only defined for values between -1 and 1. If you input a value outside this range, the calculator will display an error message. This is because cosine values are always between -1 and 1, so there's no real angle whose cosine would be outside this range.

What's the difference between arccos and cos⁻¹?

arccos and cos⁻¹ are different notations for the same function - the inverse cosine function. Both represent the angle whose cosine equals the given value. The notation cos⁻¹ is more common in mathematical contexts, while arccos is often used in programming and calculators.

How accurate are the results from this calculator?

This calculator provides results with high precision up to 10 decimal places. The accuracy is sufficient for most mathematical, engineering, and scientific applications. The calculations use JavaScript's built-in Math.acos() function, which provides excellent precision for practical purposes.

When should I use radians vs degrees?

Use radians when working with calculus, physics, or advanced mathematics, as most mathematical formulas use radians. Use degrees for geometry, trigonometry, and everyday applications. The calculator allows you to switch between both units, and you can see both values in the calculation details section.

Can I use this calculator for complex numbers?

This calculator is designed for real numbers only. It calculates the principal value of arccos for real inputs between -1 and 1. For complex number calculations, you would need specialized mathematical software that handles complex analysis.