Inverse Cosine Degrees Calculator

The inverse cosine function, also known as arccos, calculates the angle whose cosine is a given value. This calculator computes the inverse cosine in degrees, providing both the principal value and all possible solutions within the specified range.

What is Inverse Cosine?

The inverse cosine function, written as arccos(x), is the inverse operation of the cosine function. While cosine takes an angle and returns a ratio, arccos takes a ratio and returns an angle. The function is defined for inputs between -1 and 1, with outputs ranging from 0° to 180°.

Inverse cosine is used in various fields including physics, engineering, and computer graphics to determine angles from known trigonometric values. It's particularly useful when working with right triangles or when analyzing wave patterns.

How to Use This Calculator

Enter a value between -1 and 1 in the input field.
Select whether you want the principal value (0°-180°) or all possible solutions (0°-360°).
Click "Calculate" to see the result.
Review the explanation and chart visualization if available.

Note: The calculator will display an error if you enter a value outside the valid range of -1 to 1.

Formula

The inverse cosine function in degrees is calculated using the arccos function from the Math library, then converted to degrees:

θ = arccos(x) × (180° / π)

For all possible solutions, the calculator also considers the negative angle:

θ = ±arccos(x) × (180° / π)

Examples

Example 1: Principal Value

If you enter 0.5 in the calculator and select "Principal value only", the result will be 60° because cos(60°) = 0.5.

Example 2: All Solutions

For the same input of 0.5 but selecting "All possible solutions", the calculator will show both 60° and 300° because cos(300°) = 0.5 as well.

Example 3: Edge Case

Entering 1 will always return 0° in the principal value mode, as cos(0°) = 1.

Interpreting Results

The result from the inverse cosine calculator represents the angle whose cosine equals your input value. The principal value (0°-180°) is the most commonly used solution, while the full range (0°-360°) provides all possible angles that satisfy the equation.

In practical applications, you might need to consider which solution makes sense in your specific context. For example, in navigation problems, you might need the smallest positive angle, while in wave analysis, you might need all possible solutions.

FAQ

What is the domain of the inverse cosine function?

The domain of the inverse cosine function is all real numbers x such that -1 ≤ x ≤ 1. Any input outside this range will result in an error.

What is the range of the inverse cosine function?

The range of the inverse cosine function is 0° to 180° for the principal value, and 0° to 360° when considering all possible solutions.

Why does the calculator show two solutions for some inputs?

The calculator shows two solutions when you select "All possible solutions" because cosine is periodic and symmetric. For example, cos(60°) = cos(300°).