Trigonometry2D

Hyperbolic Trigonometry - Theory & Concepts - Trigonometry Hyperbolic Functions

Comprehensive study of Hyperbolic Sine, Cosine, Tangent, and their relationships to the hyperbola and exponential functions.

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Circular vs. Hyperbolic Functions

Compare the unit circle $x^2+y^2=1$ and the unit hyperbola $x^2-y^2=1$ . Observe how the parameters represent sector areas.

1. The Unit Circle

Equation:

x^2 + y^2 = 1

Angle / Parameter (

t

)1.00 rad (57°)

0.00π (~3.14)2π (~6.28)

COORDINATES(0.540, 0.841)

SECTOR AREAA = 0.500

x = \cos(t) = 0.540

y = \sin(t) = 0.841

2. The Unit Hyperbola

Equation:

x^2 - y^2 = 1

Hyperbolic Parameter (

u

)1.00

-2.0002.00

COORDINATES(1.543, 1.175)

SECTOR AREAA = 0.500

x = \cosh(u) = 1.543

y = \sinh(u) = 1.175

Exponential Formulations for Hyperbolic Functions

While circular functions are defined using trigonometric angles on a circle, hyperbolic functions are defined algebraically using exponents $e^u$ and $e^{-u}$ . Geometrically, both represent twice the area of their respective sectors.

Hyperbolic Cosine:

\cosh(u) = \frac{e^u + e^{-u}}{2}

Value:

\cosh(1.00) = \frac{2.718 + 0.368}{2} = 1.5431

Hyperbolic Sine:

\sinh(u) = \frac{e^u - e^{-u}}{2}

Value:

\sinh(1.00) = \frac{2.718 - 0.368}{2} = 1.1752

Hyperbolic Tangent:

\tanh(u) = \frac{\sinh(u)}{\cosh(u)} = \frac{e^u - e^{-u}}{e^u + e^{-u}}

Value:

\tanh(1.00) = \frac{1.175}{1.543} = 0.7616

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Circular vs. Hyperbolic Functions

1. The Unit Circle

2. The Unit Hyperbola

Exponential Formulations for Hyperbolic Functions

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