Domain of a Polar Equation?

**EnterReality** · January 6th 2008, 11:41 AM

I have 11 problems to do--all relating to finding the domains of polar equations--a concept that I don't understand.

Here are the graphs of the equations whose domains I need to find. The dashed lines indicate where the graph is undefined. If you can do them all, awesome, but at least 3 or 4 examples should suffice. I've bolded the ones I think are the most important in case no one has time or doesn't want to do them all:

1. http://i265.photobucket.com/albums/i...olargraph3.jpg (General equation: r = 2cos2θ)

2.http://i265.photobucket.com/albums/i...olargraph4.jpg (General equation: r = 3sin2θ)

3. http://i265.photobucket.com/albums/i...olargraph5.jpg (General equation: r = 2cosθ)

4. http://i265.photobucket.com/albums/i...olargraph6.jpg (General equation: r = -2sinθ)

5. http://i265.photobucket.com/albums/i...olargraph7.jpg (General equation: r = -sin3θ)

6. http://i265.photobucket.com/albums/i...olargraph8.jpg (General equation: r = -sin3θ)

7. http://i265.photobucket.com/albums/i...olargraph9.jpg (General equation: r = -sin3θ)

8. http://i265.photobucket.com/albums/i...largraph10.jpg (General equation: r = -sin3θ yet again)

9. http://i265.photobucket.com/albums/i...largraph11.jpg (General equation: r = 3cos4θ--this one was deemed "super difficult" by my teacher)

**mr fantastic** · January 6th 2008, 03:50 PM

Originally Posted by EnterReality

I have 11 problems to do--all relating to finding the domains of polar equations--a concept that I don't understand.

Here are the graphs of the equations whose domains I need to find. The dashed lines indicate where the graph is undefined. If you can do them all, awesome, but at least 3 or 4 examples should suffice. I've bolded the ones I think are the most important in case no one has time or doesn't want to do them all:

[snip]

3. http://i265.photobucket.com/albums/i...olargraph5.jpg (General equation: r = 2cosθ)

[snip]

By definition of polar coordinates: $x = r \cos \theta$ and $y = r \sin \theta$ .

Substitute $r = 2 \cos \theta$ : $x = 2 \cos^2 \theta$ and $y = 2 \cos \theta \sin \theta = \sin(2 \theta)$ .

Since there are no restrictions on $\theta$ :

$0 \leq 2 \cos^2 \theta \leq 2$ therefore $0 \leq x \leq 2$ .

$-1 \leq \sin(2 \theta) \leq 1$ therefore $-1 \leq y \leq 1$ .

After thought: Do your dotted lines mean that part of the curve is not included? If so, re-consider the above in light of the restiction on $\theta$ .

**mr fantastic** · January 6th 2008, 04:31 PM

Originally Posted by EnterReality

I have 11 problems to do--all relating to finding the domains of polar equations--a concept that I don't understand.

Here are the graphs of the equations whose domains I need to find. The dashed lines indicate where the graph is undefined. If you can do them all, awesome, but at least 3 or 4 examples should suffice. I've bolded the ones I think are the most important in case no one has time or doesn't want to do them all:

[snip]
5. http://i265.photobucket.com/albums/i...olargraph7.jpg (General equation: r = -sin3θ)

6. http://i265.photobucket.com/albums/i...olargraph8.jpg (General equation: r = -sin3θ)

7. http://i265.photobucket.com/albums/i...olargraph9.jpg (General equation: r = -sin3θ)

8. http://i265.photobucket.com/albums/i...largraph10.jpg (General equation: r = -sin3θ yet again)

[snip]

Using the same approach as my earlier post (and a couple of standard trig identities):

$x = -3 \sin(3 \theta) \cos \theta = -\frac{1}{2} \left[ \sin(2 \theta) + \cos(4 \theta) \right] = \sin^2 (2 \theta) - \frac{1}{2} \sin (2 \theta) - \frac{1}{2}$ .

$y = -3 \sin(3 \theta) \sin \theta = -\frac{1}{2} \left[ \cos(2 \theta) - \cos(4 \theta) \right] = \cos^2 (2 \theta) - \frac{1}{2} \cos (2 \theta) - \frac{1}{2}$ .

Now you need to consider the ranges of $\sin^2 (2 \theta) - \frac{1}{2} \sin (2 \theta) - \frac{1}{2}$ and $\cos^2 (2 \theta) - \frac{1}{2} \cos (2 \theta) - \frac{1}{2}$ under the given rstrictions on $\theta$ .

I'll outline one way of doing this for $x = \sin^2 (2 \theta) - \frac{1}{2} \sin (2 \theta) - \frac{1}{2}$ (the method can be used for the other one):

Let $a = \sin (2 \theta)$ .

Then $x = a^2 - \frac{a}{2} - \frac{1}{2}$ which is a parabola.

By considering the restriction on $\theta$ you can get the restriction on $\sin (2 \theta)$ and hence the restriction on a.

So sketch a graph of $x = a^2 - \frac{a}{2} - \frac{1}{2}$ over the restricted values of a and use it to read off the range of values of x.

The spade work is left for you.

**mr fantastic** · January 6th 2008, 04:33 PM

Originally Posted by EnterReality

I have 11 problems to do--all relating to finding the domains of polar equations--a concept that I don't understand.

Here are the graphs of the equations whose domains I need to find. The dashed lines indicate where the graph is undefined. If you can do them all, awesome, but at least 3 or 4 examples should suffice. I've bolded the ones I think are the most important in case no one has time or doesn't want to do them all:

[snip]
9. http://i265.photobucket.com/albums/i...largraph11.jpg (General equation: r = 3cos4θ--this one was deemed "super difficult" by my teacher)

After reading my earlier posts you should see how to tackle this one. You might need to apply some more trig identies.

**EnterReality** · January 6th 2008, 07:01 PM

Your examples utterly confused me.

Trig identities are not needed to do any of these questions, or so my teacher says. Also, don't worry about restricting any variables other than θ.

**mr fantastic** · January 7th 2008, 12:56 AM

Originally Posted by EnterReality

Your examples utterly confused me.

[snip]

Well ain't that a poke in the eye. For you and for me. Even the $r = 2 cos\theta$ one I did?

Originally Posted by EnterReality

[snip]
Trig identities are not needed to do any of these questions, or so my teacher says.
[snip]

Really? To use a line from Get Smart, I find that very hard to believe.

Your teacher obviously has a more elementary approach in mind. Not being able to see your class notes or textbook, I really can't tell what that might be. For all I know, your teacher wants you to spreadsheet the values of x and y. Or read them off from an accurately drawn curve. Or use a graphics calculator to draw the polar curves. Or convert to cartesian coordinates and plot points. Or graph $x = x(\theta)$ and $y = y(\theta)$ with a graphics calculator.

Does your teacher expect you to use technology? Because I don't see any way of avoiding trig identities if algebraic 'by-hand' solutions are required.

Perhaps someone else can make a suggestion.

Here's a thought - maybe you can provide some detailed information on what your background is and what you've actually done in this topic. Has your teacher done any similar examples in class? - if so, what method did s/he use.

There are usually several ways of solving a problem - but understanding them is totally dependent on ones mathematical background. Of course, I have no idea what your background is - have you studied trig identities (compound angle formulae, double angle formulae etc.)?

Originally Posted by EnterReality

[snip]
Also, don't worry about restricting any variables other than θ.

I didn't and I haven't. Study my second post more carefully.

Good luck.

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