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\centerline{\bigrm Polynomials and Trigonometry}
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\noindent
{\bf Purpose:} In this project you will investigate the feasibility of
approximating the
$\sin$ and $\cos$ functions using polynomials.
There is a general method using
calculus to determine how closely a function can be approximated using
polynomials. You will investigate how well polynomials you
derive approximate the $\sin$ and $\cos$ functions.
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\noindent
{\bf Procedure:}  You are to follow the steps below.  You do not need to
do the steps in exactly the order indicated as long as you cover
everything you are asked to do.
After completing the outline below you should write your
results and calculations {\bf neatly}.  Be sure to include relevant
graphs
in what you turn in. Your grade will be determined
in part by your presentation.
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\noindent
{\bf Project Outline:}
Elmer is writing an educational computer program for kids.  The program
contains bright graphics and cute sounds.  In order to make sectors of
circles appear on the screen the program needs to be able to compute
$\sin x$ and $\cos x$ for random values of $x$.
Elmer is using C++ to write the
program.  In his version of C++, if he includes the standard library of
math functions it will increase his code size beyond what would be
feasible for the market.  He therefore wishes to write his own functions
in C++ which would give him a good approximation for $\sin x$ and
$\cos x$, so he
does not need to include the whole math library.  He wishes to
find polynomials $p(x)$ and $q(x)$ so that $|p(x)-\sin(x)|<.0005$ and
$|q(x)-\cos x| < .0005$ for any value of
$x$ between $-\pi\over 2$ and $\pi\over 2$.  Elmer thinks he remembers
from his calculus course several years ago that for any desired degree
of accuracy it is possible to find
two polynomials to approximate the $\sin$ and $\cos$ functions
for every real value $x$.
His daughter Anna, who
is taking precalculus says that would be impossible.
Who is right? Is it possible for Elmer to find polynomials $p$ and $q$
with the properties given above?  If so what are the lowest degree
polynomials he can find that meet the condition?
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\item{1.}Plot $\sin x$ and $mx+b$ on the same axes for various values of
         $m$ and $b$.  Experiment until you find the values for $m$ and
         $b$ which give a line that seems to be the best approximation
         for $\sin x$ when $x$ is {\bf very close to 0}.  Do the same for
         $\cos x$.
\item{2.}The next step is to prove the approximation for $\sin \theta$
         found in part 1 when $\theta$ is close to 0 is a good
         approximation. Follow the steps
         below.
 \itemitem{a.}Draw a sector of a circle of radius 1
              having angle $\theta$ with one
              ray on the $x$-axis and the other in the first quadrant.
              Draw two right triangles using the angle of the sector as
              one angle.
              For the first triangle put the right angle at the point
              (1,0). For the second, drop a perpendicular from the point
              on the circle meeting the other side of the sector to the
              $x$-axis.
 \itemitem{b.}Compare the area of the sector with the areas
              of the triangles to get inequalities.
 \itemitem{c.}Simplify the inequalities found in c to get an
              approximation for $\sin\theta$. State why this proves your
              approximation in part 1 is a good approximation.

\item{3.}Next you will use what you learned in part 1 to find
         polynomial approximations for $\sin x$ and $\cos x$.
         The philosophy you
         should follow is that you have a linear approximation for
         $\sin x$ and $\cos x$ when $x$ is close to zero. Use the double
         angle formula to rewrite $\sin x$ in terms of trig functions of
         angles smaller than $x$. At this point you can approximate all
         the occurrences of $\sin$ and $\cos$ with the linear
         approximations you found above. Try using the double angle
         formula several times before making the approximations to see
         if you get a better polynomial approximation. Do the same for
         $\cos$. (Mathematica will simplify your calculations!)

\item{4.}Plot some graphs to see how close the approximations you found
         in part 2 are to $\sin x$ and $\cos x$.

\item{5.}Try removing some of the highest order terms to see if you lose
         much accuracy in the approximation.  What are the lowest degree
         polynomials you can find that meet Elmer's conditions for
         approximating the $\sin$ and $\cos $ functions?  Plot graphs to
         show that the error is acceptable to Elmer.

\item{6.}Try to approximate $\pi$ using what you found above. (Be
         crafty!)

\item{7.}Who is right, Elmer or Anna? Discuss why.

\item{8.}Discuss why you would expect the process you did in step 3 to
         work.

\bye