Is the Polar Curve r=cos(a/2) Symmetric About the Y-Axis?

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SUMMARY

The polar curve defined by r=cos(a/2) exhibits symmetry about the x-axis due to the property that cos(-a/2) equals cos(a/2). However, the discussion reveals that while initial mathematical checks suggest a lack of symmetry about the y-axis, graphical representations indicate otherwise. The periodic nature of the function, repeating over a range of 4π, further complicates the symmetry analysis, particularly when comparing the graph from θ=0 to 2π with that from θ=2π to 4π.

PREREQUISITES
  • Understanding of polar coordinates and their properties
  • Knowledge of trigonometric functions, specifically cosine
  • Familiarity with graphing techniques for polar equations
  • Basic concepts of symmetry in mathematical functions
NEXT STEPS
  • Explore the properties of polar curves and their symmetries
  • Learn about the periodicity of trigonometric functions and its implications
  • Investigate graphical representations of polar equations using tools like Desmos or GeoGebra
  • Study the mathematical proofs for symmetry in polar coordinates
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Mathematicians, educators, students studying polar coordinates, and anyone interested in the graphical analysis of trigonometric functions.

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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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?
 

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