Tanh Calculator
Calculate hyperbolic tangent and inverse hyperbolic tangent with detailed explanations
Calculate Hyperbolic Tangent
Any real number
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Hyperbolic Tangent Results
Exponential Form
Formula: tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x))
Calculation: tanh(0) = (1.000000 - 1.000000) / (1.000000 + 1.000000) = 0.000000
Alternative Form
Using sinh and cosh: tanh(x) = sinh(x) / cosh(x)
Calculation: tanh(0) = 0.000000 / 1.000000 = 0.000000
Properties
Domain: All real numbers
Range: (-1, 1)
Function type: Odd function
Monotonicity: Strictly increasing
Other Hyperbolic Functions
Example Calculations
Common Values
tanh(0) = 0 (exact value)
tanh(1) ≈ 0.7616
tanh(0.5) ≈ 0.4621
tanh(∞) → 1 (approaches but never reaches)
tanh(-∞) → -1 (approaches but never reaches)
Step-by-step: tanh(1)
Step 1: Calculate e^1 = 2.7183
Step 2: Calculate e^(-1) = 0.3679
Step 3: Apply formula: tanh(1) = (2.7183 - 0.3679) / (2.7183 + 0.3679)
Result: tanh(1) = 2.3504 / 3.0862 ≈ 0.7616
Tanh Function Properties
Domain
All real numbers (-∞, ∞)
Range
(-1, 1) - open interval
Symmetry
Odd function: tanh(-x) = -tanh(x)
Monotonicity
Strictly increasing everywhere
Key Formulas
Definition
tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x))
Ratio Form
tanh(x) = sinh(x) / cosh(x)
Derivative
d/dx tanh(x) = sech²(x) = 1 - tanh²(x)
Inverse
arctanh(x) = ½ ln((1+x)/(1-x))
Quick Tips
tanh approaches ±1 but never reaches them
arctanh is only defined for |x| < 1
tanh(0) = 0 exactly
Used in neural networks as activation function
Related to hyperbolic geometry and relativity
Understanding Hyperbolic Tangent (tanh)
What is tanh?
The hyperbolic tangent function (tanh) is one of the hyperbolic functions, analogous to the circular trigonometric functions but based on hyperbolas instead of circles. It's defined as:
tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x))
Key Characteristics
- •Bounded: Output always between -1 and 1
- •Smooth: Infinitely differentiable everywhere
- •Odd function: tanh(-x) = -tanh(x)
- •Monotonic: Always increasing
Applications
- •Neural Networks: Popular activation function in deep learning
- •Physics: Special relativity and hyperbolic geometry
- •Engineering: Signal processing and control systems
- •Mathematics: Solving differential equations
Inverse Function
The inverse hyperbolic tangent (arctanh) is defined only for values between -1 and 1:
arctanh(x) = ½ ln((1+x)/(1-x))
Domain: (-1, 1), Range: (-∞, ∞)