Inverse Cosine Calculator Degrees

The inverse cosine calculator (arccos) computes the angle whose cosine is the given value, expressed in degrees. This is useful in trigonometry, physics, and engineering for solving problems involving right triangles and circular motion.

What is Inverse Cosine?

The inverse cosine function, also known as arccos, 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 result is always between 0° and 180°.

Inverse cosine is used in various fields including:

Trigonometry to find angles in right triangles
Physics for calculating angles of projection
Engineering for analyzing forces and vectors
Computer graphics for 3D transformations

Key Properties

Domain: -1 to 1 (input must be between -1 and 1)
Range: 0° to 180° (output is always in this range)
Special cases: arccos(1) = 0°, arccos(-1) = 180°

How to Use This Calculator

Enter a cosine value between -1 and 1 in the input field
Click "Calculate" to compute the angle in degrees
View the result and visualization
Use the "Reset" button to clear the form

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

Formula

Inverse Cosine Formula

θ = arccos(x) × (180° / π)

Where:

θ = angle in degrees
x = cosine value (-1 ≤ x ≤ 1)
π ≈ 3.141592653589793

The formula converts the radian result from the arccos function to degrees by multiplying by 180° and dividing by π.

Worked Example

Let's calculate the angle whose cosine is 0.5:

Identify x = 0.5
Compute θ = arccos(0.5) ≈ 60° (since arccos(0.5) ≈ 1.0472 radians)
Convert to degrees: 1.0472 × (180° / π) ≈ 60°

The result is 60°, which matches the known value for arccos(0.5).

FAQ

What is the domain of the inverse cosine function?: The domain is all real numbers between -1 and 1, inclusive. Values outside this range will produce an error.
What is the range of the inverse cosine function?: The range is 0° to 180° for all valid inputs. This means the function can only return angles in the first and second quadrants.
How is the inverse cosine different from the cosine function?: The cosine function takes an angle and returns a ratio, while the inverse cosine function takes a ratio and returns an angle. They are mathematical inverses of each other.
Can I use this calculator for radians?: No, this calculator specifically provides results in degrees. For radian results, use our inverse cosine calculator radians.
What happens if I enter a value outside the domain?: The calculator will display an error message indicating that the input must be between -1 and 1.