# 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

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