Tanh vs Tan Inverse: Which is Better?

Both Tanh (Hyperbolic Tangent) and Tan⁻¹ (Inverse Tangent / Arctan) are mathematical functions used in different contexts. They have distinct properties and applications.

1️⃣ Tanh (Hyperbolic Tangent)

Formula: tanh⁡(x)=ex−e−xex+e−x\tanh(x) = \frac{e^x – e^{-x}}{e^x + e^{-x}}tanh(x)=ex+e−xex−e−x
Range: (-1, 1)
Behavior:
- Maps large negative values close to -1 and large positive values close to 1.
- Smooth and differentiable everywhere.
- Used as an activation function in neural networks.
Derivative: ddxtanh⁡(x)=1−tanh⁡2(x)\frac{d}{dx} \tanh(x) = 1 – \tanh^2(x)dxdtanh(x)=1−tanh2(x)
Applications:
✅ Commonly used in machine learning as an activation function.
✅ Used in signal processing and hyperbolic geometry.

Example in Python (Tanh)

import numpy as np
x = np.array([-2, -1, 0, 1, 2])
tanh_output = np.tanh(x)
print(tanh_output)  # [-0.9640 -0.7616  0.0000  0.7616  0.9640]

2️⃣ Tan⁻¹ (Inverse Tangent / Arctan)

Formula: tan⁡−1(x)=arctan⁡(x)\tan^{-1}(x) = \arctan(x)tan−1(x)=arctan(x)
Range: (-π/2, π/2) ≈ (-1.57, 1.57)
Behavior:
- Maps large negative values close to -π/2 and large positive values close to π/2.
- Grows more slowly than Tanh, without a strict upper or lower bound of -1 and 1.
- Used to determine angles in trigonometry.
Derivative: ddxtan⁡−1(x)=11+x2\frac{d}{dx} \tan^{-1}(x) = \frac{1}{1 + x^2}dxdtan−1(x)=1+x21
Applications:
✅ Used in trigonometry and geometry.
✅ Appears in control systems, physics, and signal processing.

Example in Python (Arctan)

arctan_output = np.arctan(x)
print(arctan_output)  # [-1.107 -0.785  0.000  0.785  1.107]

🔑 Key Differences

Feature	Tanh	Tan⁻¹ (Arctan)
Formula	ex−e−xex+e−x\frac{e^x – e^{-x}}{e^x + e^{-x}}ex+e−xex−e−x	tan⁡−1(x)\tan^{-1}(x)tan−1(x)
Range	(-1, 1)	(-π/2, π/2) ≈ (-1.57, 1.57)
Growth Rate	Steep	Slower
Derivative	1−tanh⁡2(x)1 – \tanh^2(x)1−tanh2(x)	11+x2\frac{1}{1 + x^2}1+x21
Best for	Neural networks (activation function)	Trigonometry, angle calculations
Asymptotes	Yes, at -1 and 1	Yes, at -π/2 and π/2

🛠️ When to Use Each?

Use Tanh when working with machine learning models or when you need a bounded, smooth function between -1 and 1.
Use Arctan when dealing with trigonometry, angles, and geometric computations.

🚀 Which is Better?

For deep learning: Tanh is better.
For mathematical and trigonometric calculations: Arctan is better.

Let me know if you need further details! 🚀

ApexDelight