# Representations of Trigonometric and Hyperbolic Functions in Terms of Sector Areas

Representations of Trigonometric and Hyperbolic Functions in Terms of Sector Areas

A sector of angle of a unit circle +=1 has an area equal to radians. So half the area can serve as the argument for the trigonometric functions via parametric equations for and . The two constructions shown are consistent with the trigonometric identities θ+θ=1 and θ-θ=1. (As a consequence, circular functions are alternatively called trigonometric functions.)

θ

2

x

2

y

2θ

x(θ)

y(θ)

2

sin

2

cos

2

sec

2

tan

An analogous set of relations exists for the hyperbolic functions, based on the unit hyperbola -=1. The asymptote is shown as a dashed line. The corresponding area is the sector swept out by a path from the origin following the hyperbola beginning on the axis at . Half this area, designated by , can then serve as the argument in parametric representations of the hyperbolic functions. The integral over the area can be evaluated to give , consistent with and . The two hyperbolic constructions are consistent with the identities t-t=1 and t+t=1. The construction for is not as neat as its analog for .

2

x

2

y

y=x

x

(1,0)

t

t=ln(x+y)

x(t)=cosht=+

t

e

-t

e

2

y(t)=sinht=-

t

e

-t

e

2

2

cosh

2

sinh

2

tanh

2

sech

secht

secθ

The ranges of and are constrained to fit within the scale of the graphics, but the behavior at extrapolated values should be evident.

θ

t