# Lecture 001

## Parametric Curves

### Eliminating Variables

Normally: don't forget the domain t
Cos-and-Sin: use \cos{t}^2 + \sin(t)^2 = 1

\begin{cases}
x(t) = r\cos(t) \\
y(t) = r\sin(t)
\end{cases}

is a parameterization of the circle x^2 + y^2 = r^2

### Solving the Tangent Line

Note that the solution might be one-sided (ie. limit does not exist)
You need to be careful of the domain of output rectangular equation.

### Hyperbolic sine / cosine

\begin{cases}
x = \cosh{t} = \frac{e^t + e^{-t}}{2} \\
y = \sinh{t} = \frac{e^t - e^{-t}}{2}
\end{cases}

\cosh^2{t} - sinh^2{t} = 1