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