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\centerline{MATH 392 -- Geometry Through History}
\centerline{Problem Set 3}
\centerline{Due: Monday, February 29}
\vskip 10pt
\noindent
I.  Show under the assumption of HAA that given any angle $\angle ABC < 180^\circ$, there
is a line $\ell$ that is simultaneously parallel to $\overrightarrow{BA}$ in the 
direction of $A$ and to $\overrightarrow{BC}$ in the direction of $C$.  Hint: use point (3)
on page 47 of McCleary. 
\vskip 10pt
\noindent 
II.  Note:  All of the following facts about 3-dimensional geometry
are proved in McCleary.  Suggestion:  try to 
work out proofs for yourselves using just properties following
from Euclid's Postulates I-IV (not V), then use the text to check your work.  
\begin{enumerate}
\item[(A)]  Show that the 
following are equivalent for a line $\ell$ intersecting a plane $T$ at a point $P$.
\begin{enumerate}
\item[(1)] $\ell \perp T$  (i.e. $\ell$ meets every line through $P$ in $T$
at a right angle -- see McCleary and the handout for Wednesday and Friday).
\item[(2)] There exist lines $m_1\ne m_2$ through $P$ in $T$ such that $\ell$ meets
$m_1$ and $m_2$ at right angles. 
\end{enumerate}
\item[(B)]
If $\ell \perp T$ with $\ell \cap T = P$, let $m$ be any line in $T$ and $A \in m$
such that $\overline{AP}\perp m$.  If $B$ is any point on $\ell$, then $\overline{AB}\perp m$.
\item[(C)]
If $T$ and $T'$ are perpendicular planes, and $\ell$ is a line on $T$, then 
$\ell \perp T'$ is equivalent to $\ell \perp m$ for $m = T\cap T'$.
\item[(D)]  If $\ell_1 \perp T$ and $\ell_2 \perp T$, then $\ell_1,\ell_2$
are coplanar and non-intersecting (but not necessarily parallel -- this is a
difference in the hyperbolic case!)
\item[(E)]
Given a line $\ell$ and a plane $T$ not containing $\ell$, there exists a unique plane
$T'$ containing $\ell$ that is perpendicular to $T$.  
\end{enumerate}
\vskip 10pt
\noindent  
III.  Let $\cosh(t) = \frac{e^t + e^{-t}}{2}$ and $\sinh(t) = \frac{e^t - e^{-t}}{2}$
be the hyperbolic functions.
\begin{enumerate}
\item[(A)]  Sketch the graphs $y = \cosh(t)$ and $y = \sinh(t)$.  (Suggestion: plot
$y = e^t$ and $y = e^{-t}$ first, then combine them in appropriate ways.)
\item[(B)]  Show that $\lim_{t\to\pm\infty} \tanh(t) = 1$ and sketch the graph
$y = \tanh(t)$.  
\item[(C)]  Determine the Taylor series of the functions $\cosh(t)$ and $\sinh(t)$
at $t = 0$.  
\item[(D)]  Show that $\cosh^2(t) - \sinh^2(t) = 1$ for all real $t$.
Explain why this shows that the parametric curve $x = \cosh(t), y = \sinh(t)$ lies
on the hyperbola with equation $x^2 - y^2 = 1$.  What part of that hyperbola
is obtained from the parametrization?
\end{enumerate}
\vskip 10pt
\noindent
IV.  Why is the formula
$$\cosh(c/k) = \cosh(a/k)\cosh(b/k)$$
called the hyperbolic form of the Pythagorean theorem?  Hint: Use III, part (C) and
think about what happens as $k \to \infty$.  
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