Part 3/6:
The term "hyperbolic" stems from the analogy between hyperbolic functions and the geometric properties of hyperbolas, similar to how trigonometric functions are associated with circles.
Trigonometric Functions and Circles
To understand this relationship, consider the equation of a unit circle:
[ x^2 + y^2 = 1 ]
In the context of this circle, if a point ( P ) exists on its circumference, it can be expressed in terms of an angle ( T ):
( x = \cos(T) )
( y = \sin(T) )
From these relationships, one can derive the identity:
[ \cos^2(T) + \sin^2(T) = 1 ]
This relationship makes trigonometric functions circular functions because they describe ratios derived from a circle's geometry.