1. ## Polar Functions

Could someone teach me a few things about polar functions?

How do they differ from the standard f(x) functions?

I tried drawing the same formula, using different functions.

The red one is a polar function, and the navy one is a standard function.

Why do these two functions differ so very much?

Differences
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In the rectangular coordinate system, the one you love and use all the time a point $\displaystyle (x,y)$ is determined by its vertical distance and horizontal distance. So $\displaystyle (-1,2)$ means 1 unit to the left and 2 up.

In the polar coordinate system, which because more popular in Calculus, a point $\displaystyle (r,\theta)$ is determined by its angle it makes with the x-axis (the counterclockwise angle, like in trigonometry) and its length (basically its radius). For example, $\displaystyle (1,\pi/2)$ means a point which makes and angle of $\displaystyle \pi/2 = 90^o$ and whose lengh is one. Can you find an equivalent in rectangular coordinates? Yes. It is $\displaystyle (0,1)$. What about $\displaystyle (1,\pi/4)$ a little bit more though suggests that $\displaystyle \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ is that point. That is the idea of this coordinate system.

Now we cannot conversion formula to convert from one coordinate system to another.
The rules are as follows,
$\displaystyle 1) \ \ \ r^2 = x^2 + y^2$
$\displaystyle 2) \ \ \ x = r\cos \theta$
$\displaystyle 3) \ \ \ y= r\sin \theta$
See if you can show why these formulas are right.

Some graphs look simpler in polar coordinates and some look simpler in rectangular coordinates.

First what does it means "a graph in polar coordinates"? It means the same thing as in rectangular. Just pick several $\displaystyle \theta$'s and find their cooresponding $\displaystyle r$'s and connect the dots.

For example, $\displaystyle x^2+y^2 = 1$ is a circle in rectangular coordinates. But using the conversion formula we get, $\displaystyle r^2 = 1$ thus, $\displaystyle r=1$. So the graph $\displaystyle r=1$ in polar form is a circle, look how much easier it looks!

Now let us show when polar form is worse. For example, $\displaystyle y=1$ is a basic simple honest horizontal line. But if we use the conversion formulas we get $\displaystyle r\sin \theta = 1$ thus $\displaystyle r = \frac{1}{\sin \theta} = \csc \theta$. Look how it overcomplicates it.

3. Thanks a lot TPH.

Let me see if i can figure those formulas out.

$\displaystyle r^2 = x^2 + y^2$ That's just Pyhtagoras right?

$\displaystyle cos \theta = \frac{x}{r}$

$\displaystyle x = r cos \theta$

$\displaystyle sin \theta = \frac{y}{r}$

$\displaystyle y = r sin \theta$

Am i right?

4. Originally Posted by janvdl

Am i right?
Exactly

5. Originally Posted by ThePerfectHacker
Exactly
Thanks for the help and explanation.

6. Hello, janvdl!

In rectangular coordinates, we are giving directions "in a city".

Let the origin be some agreed-upon landmark, say, City Hall.

To get to a particular corner, the directions might be: $\displaystyle (2,\,3)$
. . which translates to: "Go two blocks east and three blocks north".

In polar coordinates, we are giving directions "in the woods".

Suppose we place the origin at your camp.
. . And you want direction to my camp.

We could establish our own coding system for these directions.

For example: $\displaystyle \left(2,\,\frac{\pi}{3}\right)$
This would mean: "Face east, turn $\displaystyle \frac{\pi}{3}$ radians $\displaystyle (60^o)$ counterclockwise,
. . walk forward 2 miles."

Now imagine getting a pair of coordinates, say, $\displaystyle (4.2,\,1.7)$
. . and you don't know which system is being used . . .
Choose the right system and you arrive at your favorite restaurant;
. . choose the other and you end up in a local river.

No wonder the graphs look drastically different.

7. Awesome explanation, Soroban! Thanks

8. Thanks Soroban. But polar coordinates arent really that hard. There are so many things they could teach us in school, but they prefer to stick to the boring stuff.