Parametric & Polar Conversions Reference Cheat Sheet

Parametric and polar equations describe curves in ways that are often easier than standard Cartesian equations. This cheat sheet helps students convert between forms, graph curves, and interpret motion or direction. It is useful for precalculus, calculus, and advanced algebra topics involving circles, spirals, conics, and particle paths.

Parametric equations use a parameter such as $t$ to define $x$ and $y$ separately, while polar equations use a distance $r$ and angle $\theta$ . The most important conversions are $x = r\cos \theta$ , $y = r\sin \theta$ , $r^2 = x^2 + y^2$ , and $\tan \theta = \frac{y}{x}$ . Students should also know how to eliminate a parameter, convert equations carefully, and interpret orientation as the parameter changes.

Key Facts

A parametric curve is defined by $x = f(t)$ and $y = g(t)$ , where $t$ is the parameter that traces points on the curve.
To eliminate a parameter, solve one equation for $t$ when possible and substitute into the other equation.
The polar to Cartesian conversion formulas are $x = r\cos \theta$ and $y = r\sin \theta$ .
The Cartesian to polar conversion formulas are $r^2 = x^2 + y^2$ and $\tan \theta = \frac{y}{x}$ , with quadrant checked separately.
For a parametric curve, the slope is $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ when $\frac{dx}{dt} \ne 0$ .
A horizontal tangent occurs when $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \ne 0$ , while a vertical tangent occurs when $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \ne 0$ .
The polar equation $r = a$ represents a circle centered at the origin with radius $a$ when $a > 0$ .
The polar equation $\theta = \alpha$ represents a line through the origin making angle $\alpha$ with the positive $x$ -axis.

Vocabulary

Parametric equation: An equation system where coordinates are written separately as $x = f(t)$ and $y = g(t)$ using a parameter $t$ .
Parameter: A variable, often $t$ , that controls the position of a point on a parametric curve.
Polar coordinate: A coordinate written as $(r, \theta)$ , where $r$ is directed distance from the origin and $\theta$ is the angle from the positive $x$ -axis.
Pole: The origin in the polar coordinate system, corresponding to the point where $r = 0$ .
Eliminating the parameter: The process of removing $t$ from parametric equations to produce a relationship involving only $x$ and $y$ .
Orientation: The direction a parametric or polar curve is traced as the parameter or angle increases.

Common Mistakes to Avoid

Using $\tan \theta = \frac{y}{x}$ without checking the quadrant is wrong because tangent repeats values in opposite quadrants.
Forgetting that $r$ can be negative in polar coordinates is wrong because a point with negative $r$ is plotted in the opposite direction from angle $\theta$ .
Eliminating $t$ without keeping restrictions is wrong because the Cartesian equation may include points not reached by the original parametric equations.
Writing $\frac{dy}{dx} = \frac{dy}{dt}$ for parametric equations is wrong because slope must compare changes in $y$ to changes in $x$ , so $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ .
Replacing $r^2$ with $r$ during polar conversion is wrong because $r^2 = x^2 + y^2$ , while $r = \sqrt{x^2 + y^2}$ only gives the nonnegative distance.

Practice Questions

1 Eliminate the parameter from $x = 2t + 1$ and $y = t^2 - 3$ .
2 Convert the polar point $(4, \frac{\pi}{3})$ to Cartesian coordinates using $x = r\cos \theta$ and $y = r\sin \theta$ .
3 Convert the Cartesian equation $x^2 + y^2 = 25$ to polar form.
4 Explain why the parametric equations $x = \cos t$ and $y = \sin t$ include orientation information that the Cartesian equation $x^2 + y^2 = 1$ does not show.

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Parametric & Polar Conversions Reference Cheat Sheet

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