A curve is defined in polar coordinates by for

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

A curve is defined in polar coordinates by r=2cos⁡θr=2\cos\theta for 0≤θ≤π0\le \theta\le \pi. Three specific points on the curve are given in polar form: • P1 (r,θ)=(1,π3)P_1\,(r,\theta)=\bigl(1,\tfrac{\pi}{3}\bigr) • P2 (r,θ)=(2,0)P_2\,(r,\theta)=\bigl(2,0\bigr) • P3 (r,θ)=(12,2π3)P_3\,(r,\theta)=\bigl(\tfrac12,\tfrac{2\pi}{3}\bigr) Part (a) through Part (d) guide an investigation of the curve’s symmetry and Cartesian description. (a) Convert P1P_1, P2P_2, and P3P_3 to Cartesian coordinates. Show all steps in your calculations. (b) Use the conversions x=rcos⁡θx=r\cos\theta and y=rsin⁡θy=r\sin\theta to eliminate θ\theta and derive a Cartesian equation that represents the entire curve r=2cos⁡θr=2\cos\theta. Simplify your result completely. (c) Based on your Cartesian equation from Part (b) and the converted points from Part (a), determine whether the curve is symmetric about the xx-axis, the yy-axis, the origin, or any combination of these. Provide a clear justification. (d) Explain how the algebraic form of the Cartesian equation supports the symmetry conclusion you reached in Part (c).

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