Calculate Arc Tangent in Degrees

The arc tangent function, also known as the inverse tangent, calculates the angle whose tangent is a given number. This calculator computes the result in degrees, which is useful for many practical applications in geometry, engineering, and physics.

What is Arc Tangent?

The arc tangent function, often written as arctan(x) or tan⁻¹(x), is the inverse of the tangent function. While the tangent function takes an angle and returns a ratio, the arc tangent function takes a ratio and returns the angle.

In trigonometry, the tangent of an angle θ in a right-angled triangle is the ratio of the opposite side to the adjacent side. The arc tangent function reverses this process: given a ratio, it finds the angle θ.

The range of the arc tangent function is typically -90° to 90° (or -π/2 to π/2 radians), as the tangent function is periodic with a period of 180° (or π radians).

How to Calculate Arc Tangent in Degrees

To calculate the arc tangent of a number in degrees:

Identify the ratio (opposite side/adjacent side) for which you want to find the angle.
Use the arc tangent function to find the angle in radians.
Convert the result from radians to degrees by multiplying by 180/π.

This calculator automates these steps for you, providing a quick and accurate result.

Formula

The formula for calculating the arc tangent in degrees is:

arctan(x) in degrees = arctan(x) in radians × (180/π)

Where:

x is the ratio (opposite side/adjacent side)
arctan(x) in radians is the inverse tangent function in radians
180/π is the conversion factor from radians to degrees

The result will be in degrees, providing a more intuitive angle measurement for many applications.

Example Calculation

Let's calculate the arc tangent of 0.5 in degrees:

First, calculate arctan(0.5) in radians: approximately 0.4636 radians.
Convert to degrees: 0.4636 × (180/π) ≈ 26.565°.

So, arctan(0.5) ≈ 26.565°.

This means the angle whose tangent is 0.5 is approximately 26.565 degrees.

Applications

The arc tangent function has several practical applications:

Geometry: Calculating angles in right-angled triangles.
Engineering: Determining angles for structural calculations.
Physics: Analyzing motion and forces.
Computer Graphics: Rotating objects in 3D space.
Navigation: Calculating bearings and directions.

By providing results in degrees, this calculator makes it easier to apply the arc tangent function in real-world scenarios.

FAQ

What is the range of the arc tangent function?

The range of the arc tangent function is typically -90° to 90° (or -π/2 to π/2 radians). This is because the tangent function is periodic with a period of 180° (or π radians).

How do I convert radians to degrees?

To convert radians to degrees, multiply the radian value by 180/π. For example, 1 radian × (180/π) ≈ 57.2958 degrees.

What is the difference between tangent and arc tangent?

The tangent function takes an angle and returns a ratio, while the arc tangent function takes a ratio and returns the angle. They are inverse functions of each other.

Can the arc tangent function be used for angles outside -90° to 90°?

No, the standard arc tangent function returns values only in the range -90° to 90°. For angles outside this range, additional information about the quadrant is needed.