Arccos Calculator Degrees

The arccos calculator degrees tool helps you find the angle whose cosine is a given value, with results in degrees. This is useful in trigonometry, physics, engineering, and other technical fields where angle calculations are needed.

What is Arccos?

The arccosine function, also known as inverse cosine, is the inverse operation of the cosine function. It takes a value between -1 and 1 and returns an angle between 0 and 180 degrees (or 0 and π radians) whose cosine is that value.

In mathematical terms, if cos(θ) = x, then θ = arccos(x). The arccosine function is defined for all real numbers x in the interval [-1, 1].

Key Properties

Domain: [-1, 1]
Range: [0°, 180°]
Principal value: The angle returned is always between 0° and 180°
Symmetry: arccos(x) = 180° - arccos(-x)

How to Use This Calculator

Enter a value between -1 and 1 in the input field
Click the "Calculate" button
View the result in degrees
Use the "Reset" button to clear the calculator

The calculator will display the angle in degrees and show a visual representation of the angle on the unit circle.

Formula

The arccosine function is calculated using the following formula:

Arccos Formula

θ = arccos(x) = cos⁻¹(x)

Where:

θ is the angle in degrees
x is the cosine value (-1 ≤ x ≤ 1)

The calculator uses the JavaScript Math.acos() function which returns values in radians, then converts them to degrees by multiplying by 180/π.

Examples

Let's look at some examples of how to calculate arccosine values:

Example 1: Basic Calculation

Find the angle whose cosine is 0.5.

Solution: arccos(0.5) = 60°

Example 2: Negative Value

Find the angle whose cosine is -0.866.

Solution: arccos(-0.866) ≈ 150°

Example 3: Edge Case

Find the angle whose cosine is 1.

Solution: arccos(1) = 0°

Note

For values outside the range [-1, 1], the calculator will display an error message.

Interpreting Results

When you calculate an arccosine value, the result represents the angle whose cosine is the input value. Here's how to interpret the results:

For x = 1: The angle is 0° (cosine of 0° is 1)
For x = 0: The angle is 90° (cosine of 90° is 0)
For x = -1: The angle is 180° (cosine of 180° is -1)
For values between 0 and 1: The angle is between 0° and 90°
For values between -1 and 0: The angle is between 90° and 180°

The calculator provides a visual representation of the angle on the unit circle to help you understand the relationship between the cosine value and the resulting angle.

FAQ

What is the difference between arccos and cos?

The cosine function (cos) takes an angle and returns a value between -1 and 1. The arccosine function (arccos) takes a value between -1 and 1 and returns an angle between 0° and 180°.

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

The arccosine function is defined to return the principal value, which is the angle in the range [0°, 180°] whose cosine equals the input value. This ensures a unique solution for each valid input.

What happens if I enter a value outside the range [-1, 1]?

The calculator will display an error message because the arccosine function is only defined for values between -1 and 1. The cosine of any real angle always falls within this range.

Can I use this calculator for radians?

This calculator specifically provides results in degrees. For radians, you would need to use a different calculator or convert the result from degrees to radians.