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Lecture 9

Behind us

Logistics

Ahead

Today: polar coordinates

Next: arc length and curvature

Read Sections 10.3, 13.3. We do not cover everything in lecture or section. In fact, we basically cover nothing in lecture.

Questions!

How does radar work?

Descartes is cool. So is radar.

The radar uses a different coordinate system . To locate a point in the plane one can:

• Find its distance $r$ from the origin.
• Find its angle $\theta$ counterclockwise from the $x$-axis.
• The pair $(r,\theta)$ uniquely determines the location of the point, so it is just as good as the classical Cartesian coordinate pair $(x,y)$.

Code name: polar coordinates

• We can convert back and forth: $x=r\cos(\theta)$, $y=r\sin(\theta)$ (like sending a code from East Berlin to West Berlin in 1974).
• Example: the point with polar coordinates $(r,\theta)=(7,\pi)$ has Cartesian coordinates $$(x,y)=(7\cos(\pi),7\sin(\pi))=(-7,0).$$
• Converting in other direction is more fun: first, $r=\sqrt{x^2+y^2}$. To get the angle, you could try to use $$\tan(\theta)=\frac{y}{x}$$ when this makes sense, or you could use one of the formulas $x=r\cos(\theta)$ or $y=r\sin(\theta)$. Use eyeballs and brain(s).
• Most fun: graphing equations .

Examples

• What is the polar equation of a circle of radius $a$ around the origin?
  • $r=a$
• What is the polar equation of the line $x=3$?
  • $r\cos(\theta)=3$
• Try one: find the polar equation of the line $y=2x+5$.
• Try another one: find the polar equation of the ellipse $\frac{1}{2}x^2+\frac{1}{3}y^2=1$

Polar graphs are beautiful

Here are some examples

You can play with the parameters and see what happens!

Examples

$r=\theta^{\textrm{power}}$, $0\leq\theta\leq \textrm{multiple}\cdot\pi$

Examples

$r=\sin(\textrm{love}\cdot\theta)$, $0\leq\theta\leq \textrm{multiple}\cdot\pi$

Examples

$r=c+\sin(\textrm{love}\cdot\theta)$, $0\leq\theta\leq \textrm{multiple}\cdot\pi$

Try some the other way!

Example: the Cartesian equation of the curve with polar equation $$r=\sin(\theta)$$ is $$x^2+\left(y-\frac{1}{2}\right)^2=\frac{1}{4}.$$

Trick: multiply both sides by $r$, yielding $r^2=r\sin\theta$, then use $r^2=x^2+y^2$ and $r\sin\theta=y$.

• What is the Cartesian equation corresponding to $r=\sin(\theta)+\cos(\theta)$?
• How about $r=\cos(\theta)$?
• Or $r^2=\sec(\theta)$?

Next time: arc length and curvature!