Please explain to me the "period" of a polar equation

[paranoidandroid]

Let's say r = (2cos(2theta))/costheta

Between which angles would the curve be traced out ONCE ? How do I know this?

I'm struggling to visualize this concept and would really appreciate a practical explanation. Thanks!

 

[tkhunny]

Let's say r = (2cos(2theta))/costheta

Between which angles would the curve be traced out ONCE ? How do I know this?

I'm struggling to visualize this concept and would really appreciate a practical explanation. Thanks!

What is the period of \(\displaystyle \cos(2\theta)\)?

What is the period of \(\displaystyle \cos(\theta)\)?

Does this lead you to a guess?

 

[paranoidandroid]

---period of cos(2theta) on a graph is pi

---period of cos(theta) on a graph is 2pi

Yet, a circle represented by r = cos(theta) would trace once around between 0 and pi. This is where I'm confused.

 

[Steven G]

Let's say r = (2cos(2theta))/costheta

Between which angles would the curve be traced out ONCE ? How do I know this?

I'm struggling to visualize this concept and would really appreciate a practical explanation. Thanks!

The answer would probable be something like, pi/4, 2pi/4, 3pi/4,..... Try these and see when it starts repeating. Then try to figure out why it starts to repeat where it does.

We can help you, but you need to get your hands dirty by trying aswell. Eventually it will start repeating!!

 

[Dr.Peterson]

Let's say r = (2cos(2theta))/costheta

Between which angles would the curve be traced out ONCE ? How do I know this?

I'm struggling to visualize this concept and would really appreciate a practical explanation. Thanks!

The only way to learn how to visualize something is to experience it! You do have to try graphing this (and a few more) in order to really know what to look for. Make a table of values and plot it. What I just did was to sketch a graph of y = (2 cos(2x))/cos(x), noting important points (intercepts, asymptotes, maxima, ...), and use that to sketch the polar graph. You may need more than that your first time through.

What you will find is that r = f(theta) retraces the same point when either you get to exactly the same point (same theta, same r), or you get to the same point from the opposite side (theta differing by pi, opposite sign for r). I'm not sure there is a word to describe this properly; but you'll know what I mean when you do it.

 

[Steven G]

---period of cos(2theta) on a graph is pi

---period of cos(theta) on a graph is 2pi

Yet, a circle represented by r = cos(theta) would trace once around between 0 and pi. This is where I'm confused.

Cos(x) is an even function.

 

[Dr.Peterson]

---period of cos(2theta) on a graph is pi

---period of cos(theta) on a graph is 2pi

Yet, a circle represented by r = cos(theta) would trace once around between 0 and pi. This is where I'm confused.

Have you tried drawing this graph by hand, to see how it happens?

What point do you plot for theta = 0? How about theta = pi? Try some other angles as well. You should see what I was talking about in my previous answer.

This phenomenon is really unrelated to the period of the trig functions involved.