4 Polar Coordinates Free Online FlashCards

What do $r$ and $\theta$ represent in polar coordinates?

In polar coordinates, $r$ is the directed distance from the pole, and $\theta$ is the directed counterclockwise angle from the positive $x$-axis.

How do you convert polar coordinates to rectangular coordinates?

Use $x=r\cos\theta$ and $y=r\sin\theta$.

What does a negative radial coordinate mean?

A negative $r$ plots the point in the direction opposite the terminal side of $\theta$.

What curve does $r=a$ represent?

The equation $r=a$ represents a circle centered at the pole with radius $|a|$.

Convert $r=4\sin\theta$ to rectangular form.

Multiplying by $r$ gives $r^2=4r\sin\theta$. Substitution yields $x^2+y^2=4y$, or $x^2+(y-2)^2=4$.

How is symmetry about the polar axis tested?

For $r=f(\theta)$, test polar-axis symmetry by replacing $\theta$ with $-\theta$.

How many petals does a standard rose curve have?

For $r=a\cos(n\theta)$ or $r=a\sin(n\theta)$, there are $n$ petals if $n$ is odd and $2n$ if $n$ is even.

What is the slope formula for $r=f(\theta)$?

For a polar curve, $\frac{dy}{dx}=\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$.

What is the area formula for a polar curve?

The area swept from $\theta=\alpha$ to $\theta=\beta$ is $A=\frac{1}{2}\int_{\alpha}^{\beta}[f(\theta)]^2\,d\theta$.

How do you find the area between two polar curves?

The area is $A=\frac{1}{2}\int_{\alpha}^{\beta}(r_{\rm out}^2-r_{\rm in}^2)\,d\theta$, using outer minus inner radius squared.

What is the arc-length formula for a polar curve?

The polar arc length is $L=\int_{\alpha}^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$.

What must be checked when finding polar-curve intersections?

To find intersections, set the radial equations equal, check the pole separately, and account for equivalent representations such as $(r,\theta)$ and $(-r,\theta+\pi)$.