Mathematical Tools for Physics

1—Basic Stuff 2

It results from one figure and the relation between the radius of the circle, the angle drawn, and the length
of the arc shown. If you remember the equations=Rθ, does that mean that for a full circleθ = 360◦so
s= 360R? No. For some reason this equation is valid only in radians. The reasoning comes down to a couple of
observations. You can see from the drawing thatsis proportional toθ— doubleθand you doubles. The same
observation holds about the relation betweensandR, a direct proportionality. Put these together in a single
equation and you can conclude that
s=CRθ

whereCis some constant of proportionality. Now what isC?
You know that the whole circumference of the circle is 2 πR, so ifθ= 360◦, then

2 πR=CR 360 ◦, and C=

π

180

degree−^1

It has to have these units so that the left side,s, comes out as a length when the degree units cancel. This is an
awkward equation to work with, and it becomesveryawkward when you try to do calculus.

d

dθ

sinθ=

π

180

cosθ

This is the reason that the radian was invented. The radian is the unit designed so that the proportionality
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 theCand to say thats=Rθwithθ
restricted to radians — it saves a lot of writing. How big is a radian? A full circle has circumference 2 πR, and
this isRθ. 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)