Rose Curve Proof

[Mathsgla]

So I'm doing a project on Rose Curves and have a formula I can't quite prove. I have that for r=acos(kx)+b if k is divisible by 4 then the curve intercepts the X and y axes. However if b not equal to 0, then the graph of r=acos(kx)+b does not cross the y axis if k is not divisible by 4. I'm having trouble proving this second part

 

[Steven G]

So I'm doing a project on Rose Curves and have a formula I can't quite prove. I have that for r=acos(kx)+b if k is divisible by 4 then the curve intercepts the X and y axes. However if b not equal to 0, then the graph of r=acos(kx)+b does not cross the y axis if k is not divisible by 4. I'm having trouble proving this second part

Are you sure that you have this correct??

 

[Mathsgla]

Are you sure that you have this correct??

Ive managed to come up with a proposition:

Let r=acos(kx)+b be a rose curve.

If k is even, divisible by 4, then the curve crosses the x and y axes at +and - (a+b).
If k is even, not divisible by 4, then the curve crosses the x axis at a+b and the y axis at -a+b
If k is odd then the curve crosses the x axis at + and - (a+b) and crosses the y axis at + and - b.

Ive managed to prove this eventually and added in some diagrams for visual aid.

 

[Mathsgla]

Are you sure that you have this correct??

In addition to the proposition I gave, I've now come across another problem.

I have that if k is odd the curve crosses the x-axis at pi at (-a+b) however the curve does not cross the X axis at pi if b is in the interval (-a,a) and the curve crosses the x-axis twice at x=2pi. Now having problems proving this.

 

[Steven G]

Ive managed to prove this eventually and added in some diagrams for visual aid.

Great! Can we see your work?

 

[Mathsgla]

Great! Can we see your work?

So for 3.

Let k=2m+1 m in Z st k is odd.

then
r=acos(kx)+b
r=acos((2m+1)x)+b
r=acos(2m+1)pi) +b
r=-a+b evaluated at x=pi

similarly r=a+b at x=2pi.

however this doesn't hold for some values of a,b,k.

i should be getting it crosses the x axis at (a+b) and (a-b).

dont know what I'm doing wrong!

 

[Steven G]

So I'm doing a project on Rose Curves and have a formula I can't quite prove. I have that for r=acos(kx)+b if k is divisible by 4 then the curve intercepts the X and y axes. However if b not equal to 0, then the graph of r=acos(kx)+b does not cross the y axis if k is not divisible by 4. I'm having trouble proving this second part

Usually you have y being a function of x, not r is a function of x

Usually you have r being a function of theta, not r being a function of x.

So I'm confused when you talk about crossing the x-axis.

 

[Mathsgla]

Usually you have y being a function of x, not r is a function of x

Usually you have r being a function of theta, not r being a function of x.

So I'm confused when you talk about crossing the x-axis.

Yeah didn't know how to put theta in this. Working in polars

 

[Steven G]

Yeah didn't know how to put theta in this. Working in polars

OK, cool. It doesn't take much to confuse me. You can use Q for theta or at least state that x = theta.

Unless it is obvious, you must always always define your variables.