# Hyperbolic trig functions

Homework Helper
Gold Member
I just got a clue as to why 0.5(e^x + e^-x) was called "hyperbolic cosine" and 0.5(e^x - e^-x) is called "hyperbolic sine". It is because the "complex version" reads

$$cos(x)=\frac{e^{ix}+e^{-ix}}{2}$$

$$sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$

That explains the "cos" and "sin" part in "cosh" and "sinh", but what does the "h" (hyperbolic) part comes from?

lurflurf
Homework Helper
sin(x) and cos(x)
are called circular functions because
x^2+y^2=1
is the the equation of a (unit) circle
and if x(t) and y(t) points on the circle under the natural parametritization where t is the distance along the curve from (1,0) to (x(t),y(t)) then
x(t)=cos(t)
y(t)=sin(t)
likewise
x^2-y^2=1
is the the equation of a (unit) hyperbola
and if x(t) and y(t) points on the hyperbola under the natural parametritization where t is the distance along the curve from (1,0) to (x(t),y(t)) then
x(t)=cosh(t)
y(t)=sinh(t)
if we take t>=0 we get one forth the hyperbola, we can get the whole thing by using different signs
x(t)={+,-}cosh(t)
y(t)={+,-}sinh(t)
the interpatation of t as distance changes slightly though the sign of cosh determines if the starting point is (1,0) of (-1,0) the sign of sinh determines which direction is considered positive (or if t is kept nonnegitive wether we travel up or down from the starting point).

Tide
Homework Helper
Notice that $x^2 + y^2 = constant$ represents a circle while $x^2-y^2 = constant$ represents a hyperbola. Compare these with the identities $\cos^2 z + \sin^2 z = 1$ for the circular functions and $\cosh^2 z - \sinh^2z=1$ for the hyperbolic functions. :)

Homework Helper
Gold Member
Haha, very nice.

SGT
lurflurf said:
sin(x) and cos(x)
are called circular functions because
x^2+y^2=1
is the the equation of a (unit) circle
and if x(t) and y(t) points on the circle under the natural parametritization where t is the distance along the curve from (1,0) to (x(t),y(t)) then
x(t)=cos(t)
y(t)=sin(t)
likewise
x^2-y^2=1
is the the equation of a (unit) hyperbola
and if x(t) and y(t) points on the hyperbola under the natural parametritization where t is the distance along the curve from (1,0) to (x(t),y(t)) then
x(t)=cosh(t)
y(t)=sinh(t)
if we take t>=0 we get one forth the hyperbola, we can get the whole thing by using different signs
x(t)={+,-}cosh(t)
y(t)={+,-}sinh(t)
the interpatation of t as distance changes slightly though the sign of cosh determines if the starting point is (1,0) of (-1,0) the sign of sinh determines which direction is considered positive (or if t is kept nonnegitive wether we travel up or down from the starting point).

Actually t is the area between the radius(the segment between (0,0) and (x,y)), the curve and the x axis. In the case of the unit circle the area is numerically equal to the arc, but not in the hyperbola.

lurflurf
Homework Helper
SGT said:
Actually t is the area between the radius(the segment between (0,0) and (x,y)), the curve and the x axis. In the case of the unit circle the area is numerically equal to the arc, but not in the hyperbola.
Oops. I took the analogy too far. Area is what generalizes not arc length.
That would also hold with parabolic trig functions
cosp(t)=t
sinp(t)=t^2/2

lurflurf said:
Oops. I took the analogy too far. Area is what generalizes not arc length.
That would also hold with parabolic trig functions
cosp(t)=t
sinp(t)=t^2/2

Parabolic trig. functions?

re

Trig and hyperbolic trig functions are exacltly mirror images of one another, mathematically of course. (duality)