Arcus Tangent Calculator

1. What is an Arcus Tangent Calculator?

Definition: This calculator computes the arcus tangent (inverse tangent, or arctan) of a number \( x \), which is the angle \( y \) such that \( \tan(y) = x \). It also supports reverse calculation, computing \( \tan(y) \) given an angle \( y \) in degrees. The angle output can be displayed in degrees (default) or radians.

Purpose: It aids in trigonometry education and applications by finding angles from tangent values or vice versa, useful in fields like geometry, physics, and engineering for solving problems involving right triangles or angular relationships.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Arcus Tangent (\( x \to y \)):

In radians: \( y = \arctan(x) \)
In degrees: \( y_{\text{deg}} = y \cdot \frac{180}{\pi} \)

Tangent (\( y \to x \)):

Convert to radians: \( y_{\text{rad}} = y \cdot \frac{\pi}{180} \)
Tangent: \( x = \tan(y_{\text{rad}}) \)

Steps:

Select the calculation mode: Arcus Tangent or Tangent.
Input the number \( x \) (for Arcus Tangent) or angle \( y \) in degrees (for Tangent).
Validate inputs: For Tangent, ensure \( y \neq 90^\circ + k \cdot 180^\circ \).
Compute the result using the formulas above.
For Arcus Tangent, select the output unit (degrees or radians, defaulting to degrees).
Display the result to 4 decimal places.

3. Importance of Arcus Tangent Calculations

Arcus tangent calculations are essential for:

Geometry: Finding angles in right triangles when the ratio of the opposite to adjacent sides is known.
Physics: Determining angles in vector analysis or projectile motion.
Machine Learning: Using arctan as an activation function due to its asymptotic behavior.

4. Using the Calculator

Examples:

Arcus Tangent: Number \( x = 1 \), Output in degrees
\( y = \arctan(1) = \frac{\pi}{4} \) radians
Convert to degrees: \( y = \frac{\pi}{4} \cdot \frac{180}{\pi} = 45.0000^\circ \).
Arcus Tangent: Number \( x = 0 \), Output in degrees
\( y = \arctan(0) = 0 \) radians
Convert to degrees: \( y = 0.0000^\circ \).
Tangent: Angle \( y = 30^\circ \)
Convert to radians: \( y_{\text{rad}} = 30 \cdot \frac{\pi}{180} = \frac{\pi}{6} \)
Tangent: \( \tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3} \approx 0.5774 \).

5. Frequently Asked Questions (FAQ)

Q: What is the arcus tangent?
A: The arcus tangent (arctan) is the inverse of the tangent function, finding the angle \( y \) such that \( \tan(y) = x \). Its domain is all real numbers, and its range is \((-90^\circ, 90^\circ)\).

Q: Why is the tangent undefined at certain angles?
A: The tangent function is undefined at angles where the cosine is zero, i.e., \( 90^\circ + k \cdot 180^\circ \), because \( \tan(y) = \frac{\sin(y)}{\cos(y)} \), and division by zero is undefined.

Q: Why does the angle output default to degrees?
A: Degrees are the default unit for angles as they are commonly used in educational and practical applications, but you can switch to radians using the dropdown.