1—Basic Stuff 2This is the reason that the radian was invented. The radian is the unit designed so that the propor-
tionality constant is one.
C= 1radian−^1 then s=
(
1 radian−^1)
Rθ
In practice, no one ever writes it this way. It’s the custom simply to omit theC and to say that
s=Rθwithθrestricted to radians — it saves a lot of writing. How big is a radian? A full circle has
circumference 2 πR, and this equalsRθwhen you’ve takenCto be one. It says that the angle for a
full circle has 2 πradians. One radian is then 360 / 2 πdegrees, a bit under 60 ◦. Why do you always use
radians in calculus? Only in this unit do you get simple relations for derivatives and integrals of the
trigonometric functions.
Hyperbolic Functions
The circular trigonometric functions, the sines, cosines, tangents, and their reciprocals are familiar, but
their hyperbolic counterparts are probably less so. They are related to the exponential function as
coshx=
ex+e−x
2
, sinhx=
ex−e−x
2
, tanhx=
sinhx
coshx
=
ex−e−x
ex+e−x
(1.1)
The other three functions are
sechx=
1
coshx
, cschx=
1
sinhx
, cothx=
1
tanhx
Drawing these is left to problem1.4, with a stopover in section1.8of this chapter.
Just as with the circular functions there are a bunch of identities relating these functions. For
the analog ofcos^2 θ+ sin^2 θ= 1you have
cosh^2 θ−sinh^2 θ= 1 (1.2)
For a proof, simply substitute the definitions ofcoshandsinhin terms of exponentials and watch
the terms cancel. (See problem4.23for a different approach to these functions.) Similarly the other
common trig identities have their counterpart here.
1 + tan^2 θ= sec^2 θ has the analog 1 −tanh^2 θ= sech^2 θ (1.3)
The reason for this close parallel lies in the complex plane, becausecos(ix) = coshxandsin(ix) =
isinhx. See chapter three.
The inverse hyperbolic functions are easier to evaluate than are the corresponding circular func-
tions. I’ll solve for the inverse hyperbolic sine as an example