Polar Curves

Polar Curves

Let's start with making sure we're comfortable with coordination conversion and a quick check on complex numbers.

Multiple Choice

What is the Cartesian coordinate (1, -1) in terms of a polar coordinate?

• $\left(\sqrt{2},\ \frac{\pi}{4}\right)$

• $\left(-\sqrt{2},\frac{\pi}{4}\right)$

• $\left(\sqrt{2},\frac{7\pi}{4}\right)$

• $\left(-\sqrt{2},\frac{7\pi}{4}\right)$

Multiple Choice

 What is the value of $i^8$ written as a complex number in the form  $a+bi$  ?

•  $-1$  

•  $-i$  

•  $i$  

•  $1$  

Polar Curves

• The graph of a polar equation where $r=f\left(\theta\right)$  consists of all points P that have at least one polar representation  $\left(r,\ \theta\right)$ whose coordinates satisfy the equation.

• What curve is represented by the polar equation r=2?

• If you're thinking a circle, you are correct! In general, the equation r=a represents a circle with center  $O$ and radius  $\left|a\right|$  .

Sketching a Polar Curve

• First, sketch the curve in Cartesian coordinates.

• By doing this, we can see how the r value corresponds to increasing values of $\theta$  

• As  $\theta$  increases from  $\frac{\pi}{2}$  to  $\pi$  we can see in the Cartesian coordinate graph that r is decreasing from 2 to 1 and we see that decrease in the cardioid sketch.

• We can finish for all 360 degrees

A Completed Cardioid

It is called a cardioid because it looks like a heart.

Symmetry (Three Parts: First Rule)

• If a polar equation is unchanged when $\theta$  is replaced by  $-\theta$  , the curve is symmetric about the polar axis.

Symmetry (The Second Rule)

If the equation is unchanged when r is replaced by -r, or when θ is replaced by θ+π, the curve is symmetric about the pole.

Symmetry (The Third Rule)

 If the equation is unchanged when $\theta$ is replaced by  $\pi-\theta$  , the curve is symmetric about the vertical line  $\theta=\frac{\pi}{2}$  

Let's sketch!

• You're given the curve $r=\cos\ \left(2\theta\right)$  

• What do you think the resulting sketch on the polar axis will look like?

• To a jamboard!