WOLFRAM|DEMONSTRATIONS PROJECT

Representations of Trigonometric and Hyperbolic Functions in Terms of Sector Areas

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show functions
trigonometric
hyperbolic
sin θ, sinh t
tan θ, tanh t
θ (degrees)
45
hyperbolic t
1
sin 45.0° = 0.707
cos 45.0° = 0.707
sinh 1.00 = 1.175
cosh 1.00 = 1.543
A sector of angle
θ
of a unit circle
2
x
+
2
y
=1
has an area equal to
2θ
radians. So half the area can serve as the argument for the trigonometric functions via parametric equations for
x(θ)
and
y(θ)
. The two constructions shown are consistent with the trigonometric identities
2
sin
θ+
2
cos
θ=1
and
2
sec
θ-
2
tan
θ=1
. (As a consequence, circular functions are alternatively called trigonometric functions.)
An analogous set of relations exists for the hyperbolic functions, based on the unit hyperbola
2
x
-
2
y
=1
. The asymptote
y=x
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
x
axis at
(1,0)
. Half this area, designated by
t
, can then serve as the argument in parametric representations of the hyperbolic functions. The integral over the area can be evaluated to give
t=ln(x+y)
, consistent with
x(t)=cosht=
t
e
+
-t
e
2
and
y(t)=sinht=
t
e
-
-t
e
2
. The two hyperbolic constructions are consistent with the identities
2
cosh
t-
2
sinh
t=1
and
2
tanh
t+
2
sech
t=1
. The construction for
secht
is not as neat as its analog for
secθ
.
The ranges of
θ
and
t
are constrained to fit within the scale of the graphics, but the behavior at extrapolated values should be evident.