Polar & Parametric Equations Cheat Sheet

Key Facts

• Polar coordinates convert to rectangular coordinates using $x=r\cos\theta$ and $y=r\sin\theta$.
• Rectangular coordinates convert to polar coordinates using $r^2=x^2+y^2$ and $\tan\theta=\frac{y}{x}$, with quadrant checked carefully.
• For a parametric curve $x=x(t)$ and $y=y(t)$, the slope is $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ when $\frac{dx}{dt}\ne 0$.
• The second derivative for a parametric curve is $\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$ when $\frac{dx}{dt}\ne 0$.
• The arc length of a parametric curve from $t=a$ to $t=b$ is $L=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$.
• The area enclosed by a polar curve from $\theta=\alpha$ to $\theta=\beta$ is $A=\frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta$.
• The slope of a polar curve $r=f(\theta)$ is $\frac{dy}{dx}=\frac{\frac{dr}{d\theta}\sin\theta+r\cos\theta}{\frac{dr}{d\theta}\cos\theta-r\sin\theta}$ when the denominator is not zero.
• The arc length of a polar curve from $\theta=\alpha$ to $\theta=\beta$ is $L=\int_{\alpha}^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$.

Vocabulary

Polar coordinates

A coordinate system that locates a point by its distance $r$ from the origin and its angle $\theta$ from the positive $x$-axis.

Parametric equations

Equations that describe $x$ and $y$ separately as functions of a parameter, usually written $x=x(t)$ and $y=y(t)$.

Parameter

An independent variable such as $t$ that controls the position of a point on a parametric curve.

Initial line

The polar axis from which the angle $\theta$ is measured, usually the positive $x$-axis.

Arc length

The distance along a curve, found by integrating a square root expression based on rates of change.

Tangent slope

The slope of the tangent line to a curve at a point, often found using derivative ratios in polar or parametric form.