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\centerline{\bigrm Trigonometry without Geometry}
\smallskip
\centerline{or}
\smallskip
\centerline{\medrm Non--Circular Trigonometry}
\vfill

While reviewing trigonometry Ted decides that the definitions for
trigonometric functions are circular.  It bothers Ted that the
definitions are in terms of the arc length of a circle. He
prefers definitions of functions which do not depend on geometry.
After all, when you define $f(x)=x^2$ you do not rely on
geometry, you only multiply $x$ by itself.
Ted approaches you for assistance
in developing definitions for trigonometric functions which do not
depend on the circle.  You point out that there are several ways to
define the trigonometric functions and suggest the following procedure.


You are to complete the steps below.  When you write your report,
explain carefully what each of the functions you define should be in
terms of the trigonometric functions.  You should also point out that
formulas you derive are consistent with the usual trigonometric
identities.  Keep in the spirit that this is an alternate approach to
defining the trigonometric functions. You are not to use
identities unless you prove them first.  Be sure to show your
calculations.

\vfill

\item{1.}Define $A(x)=\int_0^x {1\over 1+t^2} dt$ for all $x$.
         Note that we know
         (or will know) from class that $A(x)=\arctan (x)$.  You are not
         to use this fact since you are trying to give alternate
         definitions for the trigonometric functions.  Using this
         definition compute $A(0)$ and $A'(x)$ for every $x$.  What is
         $A'(0)$?

\item{2.}You may assume that $p=\lim_{x\to \infty}A(x)$ exists.  With this
         assumption compute $\lim_{x\to -\infty}A(x)$ in terms of $p$.
         Then show that $p$ is between 1 and 2. State
         the exact value of $p$ without proof.

\item{3.}Show that $A(x)$ has an inverse function.  Let $T(x)$ be the
         inverse function for $A(x)$.  State what the domain and range
         are for both $A(x)$ and $T(x)$.

\item{4.}Compute the derivative $T'(x)$. (Your answer should be in terms
         of $T(x)$.)  What is $T'(0)$?

\item{5.}Now, define $C(x)={1\over \sqrt{T'(x)}}.$  Compute $C(0)$,
         $C'(x)$ for every $x$ (in terms of $T(x)$), and $C'(0)$.

\item{6.}Next define $S(x)=-C'(x)$.  Compute $S(0)$, $S'(x)$ in terms of
         $T(x)$, and $S'(0)$.

\item{7.}Find a relation between $S^2(x)$ and $C^2(x)$ and prove it.

\item{8.}Find $S'(x)$ in terms of $C(x)$ and prove your answer.

\item{9.}How would you extend the domain of the functions $S(x)$ and
         $C(x)$ to be consistent with the trigonometric functions?

\item{10.}For extra credit you may estimate $\pi$ numerically by
         approximating $\lim_{x\to\infty} A(x)$.  You may either use a
         computer or calculator to do the estimate.  You should also
         give some indication of how close your estimate is to the
         correct value and how you know it is as close as you say it is.

\bye