How Do You Determine the Key Angles for a Three-Leaf Rose in Polar Coordinates?

In summary, the conversation is discussing finding the area of curves in polar coordinates, specifically using the equation r=2cos3\theta. The author is struggling to understand how to find the intervals at which \theta=pi/6, 3pi/6, 5pi/6, 7pi/6, 9pi/6, and 11pi/6, which are important for graphing the curve. Suggestions are given to graph the equation and use knowledge of the unit circle to determine these values.
  • #1
mateomy
307
0
I don't remember ANYTHING from this section when I took Trig but we're finding the area of curves in the polar coords. Looking at the book they give us this equation

[tex]
r=2cos3\theta
[/tex]

I can see, and I know how to figure out its a 3 leaf "rose" symmetrical about the theta= zero axis, but I can't figure out the next part which is the author stating "Finding the intervals we see that [itex]\theta[/itex]=pi/6, 3pi/6, 5pi/6, 7pi/6, 9pi/6, and 11pi/6." Maybe (probably) its simple trig stuff that I am overlooking but how the H do they find those values?

Pointers, suggestions, and/or degrading comments will be greatly appreciated. Thanks.
 
Last edited:
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  • #2
mateomy said:
I don't remember ANYTHING from this section when I took Trig but we're finding the area of curves in the polar coords. Looking at the book they give us this equation

[tex]
r=2cos\theta
[/tex]

I can see, and I know how to figure out its a 3 leaf "rose" symmetrical about the theta= zero axis, but I can't figure out the next part which is the author stating "Finding the intervals we see that [itex]\theta[/itex]=pi/6, 3pi/6, 5pi/6, 7pi/6, 9pi/6, and 11pi/6." Maybe (probably) its simple trig stuff that I am overlooking but how the H do they find those values?

Pointers, suggestions, and/or degrading comments will be greatly appreciated. Thanks.

You may want to double check what the graph looks like. This isn't a 3-petal 'rose'.

Also, think back to your unit circle and how it relates to theta - all of the thetas they give should come out relatively nice. (Maybe think about them in their degree equivalents?)
 
  • #3
Did you see my edit on the LaTex? its actually 3theta not just theta.
 
  • #4
mateomy said:
Did you see my edit on the LaTex? its actually 3theta not just theta.

Trickery!

I suggest graphing the 2sin3[itex]\theta[/itex] as an x-y plot. Using the graph you're more used to seeing, you should be able to identify how r varies a little better. For example: whenever 2sin3x = 0, 'drawing' your petal should be at the origin.

I think they're looking for 'intervals' where you have a continuous petal (since when [itex]\theta = 0[/itex], you're r = 2 and doesn't really 'close' a loop over the next [itex]\frac{\pi}{3}[/itex]).


(Still learning latex myself, sorry :p)
 
Last edited by a moderator:
  • #5
For which values of t (t = theta) is cos(3*t) = 0? These would be the t-values at which the figure passes through the origin (if that is what you are after).

RGV
 
  • #6
Ray Vickson said:
For which values of t (t = theta) is cos(3*t) = 0? These would be the t-values at which the figure passes through the origin (if that is what you are after).

RGV

I know cos(theta) will equal zero and pi/2 and all of its multiples, but I can't figure out how theyre getting the pi/6. cos(3theta) will be zero at 3pi/2 right? I always hated trig graphing.
 
  • #7
mateomy said:
I know cos(theta) will equal zero and pi/2 and all of its multiples, but I can't figure out how theyre getting the pi/6. cos(3theta) will be zero at 3pi/2 right? I always hated trig graphing.

What's 3(pi/6) ?
 

FAQ: How Do You Determine the Key Angles for a Three-Leaf Rose in Polar Coordinates?

1. What are polar coordinates?

Polar coordinates are a system used to locate points in a two-dimensional space. They are represented by a distance (r) from the origin and an angle (θ) measured counterclockwise from a reference direction.

2. How are polar coordinates related to Cartesian coordinates?

Polar coordinates and Cartesian coordinates are two different systems used to locate points in a two-dimensional space. In polar coordinates, a point is represented by (r, θ) while in Cartesian coordinates, a point is represented by (x, y). The relationship between the two can be expressed as x = r cosθ and y = r sinθ.

3. What is the conversion formula between polar and Cartesian coordinates?

The conversion formula between polar and Cartesian coordinates is x = r cosθ and y = r sinθ. To convert from polar to Cartesian coordinates, we can substitute the values of r and θ into these equations. To convert from Cartesian to polar coordinates, we can use the equations r = √(x² + y²) and θ = tan⁻¹(y/x).

4. What are some common applications of polar coordinates?

Polar coordinates are commonly used in mathematics, physics, and engineering applications. Some examples include describing the movement of objects in circular motion, analyzing electrical currents and fields, and solving problems in vector calculus and differential equations.

5. How is integration different in polar coordinates compared to Cartesian coordinates?

In polar coordinates, integration is done over a circular or annular region rather than a rectangular region as in Cartesian coordinates. This requires a different set of integration formulas and techniques. In polar coordinates, we also have to consider the area element (rdrdθ) in the integration process, which is not present in Cartesian coordinates.

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