How does one determine the period of a polar curve?

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SUMMARY

The period of the polar curve defined by the equation r = 3cos(θ) is determined to be π, not 2π, as it repeats itself after this interval. Horizontal tangents occur at θ = π/2, 3π/2, 5π/2, and 7π/2, while vertical tangents are found at θ = 0, π/2, π, and 3π/2. Understanding the behavior of the cosine function at these angles is crucial for identifying the repeating nature of the curve. Graphing the polar curve can provide visual confirmation of its periodicity and tangent points.

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  • Understanding of polar coordinates and polar curves
  • Knowledge of trigonometric functions, specifically cosine
  • Familiarity with tangent lines and their properties
  • Ability to graph polar equations
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  • Study the properties of polar curves and their periodicity
  • Learn how to graph polar equations using tools like Desmos or GeoGebra
  • Explore the relationship between trigonometric functions and their graphs
  • Investigate the concept of tangent lines in polar coordinates
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Students studying calculus, mathematicians interested in polar coordinates, and educators teaching polar curves and their properties.

motornoob101
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How does one know when the a polar curve repeats itself?

I usually go from 0 to 2pi but that sometimes get me into trouble.

For example:

Question says find the horizontal and vertical tangents of this curve..

r=3cos\theta

I was like well.. you have horizontal tangents when

\theta = \pi/2, 3\pi/2, 5\pi/2, 7\pi/2

vertical tangets when

\theta = 0, \pi/2, \pi, 3\pi/2

until I looked at the solution manual and they have only first half of these points.. then I realized that the curve repeats itself at pi, not 2pi. So yeah any tricks to see when it ends? Thanks!
 
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Well graphing it may help, or otherwise; at theta = pi, the line will be going in the negative direction, but cos (pi) is negative 1, so it will start going positive again, ie it starts again.
 
Ok, thanks
 

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