Very hard question about polar coordinates

In summary, the conversation discusses the requirements for a polar-curve to be symmetric around the x-axis and the y-axis. The graph of r=cos(a/2) is shown to be symmetric around the x-axis but not around the y-axis, however, when looking at the graph over a range of 4*pi, it is clear that the function is symmetric about the y-axis.
  • #1
hyper
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Hello, this question is about symmetry of polar coordinates.

For a polar-curve to be symmetric around the x-axis we require that if (r,a) lies on the graph then (r,-a) or (-r,Pi-a) lies on the graph.

To be symmetric about the y-axis we require that (-r,-a) or (r,Pi-a) lies on the graph.

Now let's look at the graph r=cos(a/2)

Since cos(-a/2)=cos(a/2) then the curve is symmetric about the x-axis. Of the requirements of symmetri I wrote earlier it doesn't seem to be symmetric about the y-axis. But when I draw it, symmetry about the y-axis occours. How can this be shown mathematically?


hyper
 
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  • #2
The function in question obviously repeats over a range of 4*pi. What does the graph of the function over the range theta=0 to 2*pi versus theta=2*pi to 4*pi tell you about symmetry with respect to the y axis?
 

1. What are polar coordinates?

Polar coordinates are a coordinate system in which a point in a plane is determined by its distance from a fixed point (called the pole) and the angle it makes with a fixed line (called the polar axis).

2. How are polar coordinates different from Cartesian coordinates?

Polar coordinates use distance and angle to locate a point, while Cartesian coordinates use x and y coordinates. Polar coordinates are useful for representing circular and rotational motion, while Cartesian coordinates are better for representing linear motion.

3. How do you convert polar coordinates to Cartesian coordinates?

To convert polar coordinates to Cartesian coordinates, you can use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
where r is the distance from the pole and theta is the angle in radians.

4. What is the relationship between polar and rectangular equations?

Polar and rectangular equations are two different ways of representing the same curve. Polar equations use r and theta to describe the distance and angle of a point, while rectangular equations use x and y coordinates. Both types of equations can be used to graph the same curve.

5. How are polar coordinates used in real life?

Polar coordinates are commonly used in physics and engineering to describe rotational motion and in navigation systems to track the position and direction of objects. They are also used in GPS technology to determine the location of a point on the Earth's surface.

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