Arctan (Inverse Tangent) Calculator

1. What is the Arctan (Inverse Tangent) Calculator?

Definition: This calculator computes the inverse tangent (\( y = \arctan(x) \)) of a given value \( x \), where \( x \) can be any real number. The result is the angle \( y \) whose tangent is \( x \).

Purpose: It is used in mathematics, physics, and engineering to find angles based on tangent values, often in problems involving trigonometry, geometry, or slope analysis.

2. How Does the Calculator Work?

The calculator uses the inverse tangent function:

\( y = \arctan(x) \), where \( x \in \mathbb{R} \)

Where:

\( x \): The input value (tangent of the angle)
\( y \): The angle in various units

Unit Conversions:

Angle (\( y \)):
- Radians (rad): The default output of the inverse tangent function
- Degrees (deg): \( \text{deg} = \text{rad} \times \frac{180}{\pi} \)
- Gradians (gon): \( \text{gon} = \text{rad} \times \frac{200}{\pi} \)
- Turns (tr): \( \text{tr} = \text{rad} \times \frac{1}{2\pi} \)
- Minutes of Arc (arcmin): \( \text{arcmin} = \text{rad} \times \frac{180}{\pi} \times 60 \)
- Seconds of Arc (arcsec): \( \text{arcsec} = \text{rad} \times \frac{180}{\pi} \times 3600 \)
- Milliradians (mrad): \( \text{mrad} = \text{rad} \times 1000 \)
- Microradians (urad): \( \text{urad} = \text{rad} \times 1000000 \)
- π Radians (x π rad): \( \text{x π rad} = \text{rad} \times \frac{1}{\pi} \)

Steps:

Enter the value \( x \) (any real number).
Click "Calculate" to compute \( y = \arctan(x) \).
Select the desired unit (e.g., deg, rad, gon) for the result.
The result is displayed with 4 decimal places.

3. Importance of Arctan Calculation

Calculating the inverse tangent is crucial for:

Trigonometry: Finding angles in right triangles or other geometric shapes.
Physics: Determining angles in mechanics, optics, or electromagnetism.
Engineering: Calculating slopes, angles in navigation, or control systems.

4. Using the Calculator

Example: Calculate the inverse tangent of \( x = 1 \).

Enter \( x = 1 \) in the input field.
Click "Calculate" to compute:
- In radians: \( y = \arctan(1) = \frac{\pi}{4} \approx 0.7854 \, \text{rad} \)
- In degrees: \( y = 0.7854 \times \frac{180}{\pi} = 45.0000 \, \text{deg} \)
- In gradians: \( y = 0.7854 \times \frac{200}{\pi} \approx 50.0000 \, \text{gon} \)
- In turns: \( y = 0.7854 \times \frac{1}{2\pi} \approx 0.1250 \, \text{tr} \)
- In minutes of arc: \( y = 45 \times 60 = 2700.0000 \, \text{arcmin} \)
- In seconds of arc: \( y = 45 \times 3600 = 162000.0000 \, \text{arcsec} \)
- In milliradians: \( y = 0.7854 \times 1000 = 785.3982 \, \text{mrad} \)
- In microradians: \( y = 0.7854 \times 1000000 = 785398.1634 \, \text{urad} \)
- In π radians: \( y = 0.7854 \times \frac{1}{\pi} \approx 0.2500 \, \text{x π rad} \)
Select the unit to view the result in the desired format.

5. Frequently Asked Questions (FAQ)

Q: What is the inverse tangent (arctan)?
A: The inverse tangent (\( \arctan(x) \)) is the angle \( y \) such that \( \tan(y) = x \), where \( x \) can be any real number. The result is typically between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians (or -90° to 90°).

Q: What is the range of the arctan function?
A: The arctan function returns angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians (or -90° to 90°), regardless of the input value.

Q: What are the different angle units?
A: Angles can be measured in various units:

Degrees (deg): 360° in a full circle.
Radians (rad): \( 2\pi \) in a full circle.
Gradians (gon): 400 gon in a full circle.
Turns (tr): 1 turn is a full circle.
Minutes of Arc (arcmin): 60 arcmin per degree.
Seconds of Arc (arcsec): 3600 arcsec per degree.
Milliradians (mrad): 1000 mrad per radian.
Microradians (urad): 1000000 urad per radian.
π Radians (x π rad): Expressed as a multiple of π.