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Determine whether the tests for symmetry detect symmetry with respect to a. The polar axis. Replace (r,theta) by (r,-theta). b. The line theta=(pi)/(2). Replace (r,theta) by (r,pi-theta). c. The pole. Replace (r,theta) by (-r,theta). r^(2)=9sin 2theta a. Yes b. Inconclusive c. Yes a. Inconclusive b. Yes c. Yes a. Inconclusive b. Inconclusive c. Yes a. Yes b. Yes c. Inconclusive

Determine whether the tests for symmetry detect symmetry with respect to\newlinea. The polar axis. Replace (r,θ) (r, \theta) by (r,−θ) (r,-\theta) .\newlineb. The line θ=π2 \theta=\frac{\pi}{2} . Replace (r,θ) (r, \theta) by (r,π−θ) (r, \pi-\theta) .\newlinec. The pole. Replace (r,θ) (r, \theta) by (−r,θ) (-r, \theta) .\newliner2=9sin⁡2θ r^{2}=9 \sin 2 \theta \newlinea. Yes b. Inconclusive c. Yes\newlinea. Inconclusive b. Yes c. Yes\newlinea. Inconclusive b. Inconclusive c. Yes\newlinea. Yes b. Yes c. Inconclusive

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Q. Determine whether the tests for symmetry detect symmetry with respect to\newlinea. The polar axis. Replace (r,θ) (r, \theta) by (r,−θ) (r,-\theta) .\newlineb. The line θ=π2 \theta=\frac{\pi}{2} . Replace (r,θ) (r, \theta) by (r,π−θ) (r, \pi-\theta) .\newlinec. The pole. Replace (r,θ) (r, \theta) by (−r,θ) (-r, \theta) .\newliner2=9sin⁡2θ r^{2}=9 \sin 2 \theta \newlinea. Yes b. Inconclusive c. Yes\newlinea. Inconclusive b. Yes c. Yes\newlinea. Inconclusive b. Inconclusive c. Yes\newlinea. Yes b. Yes c. Inconclusive
  1. Test Symmetry Polar Axis: Test for symmetry with respect to the polar axis by replacing (r,θ)(r, \theta) with (r,−θ)(r, -\theta). Original equation: r2=9sin⁡(2θ)r^{2} = 9\sin(2\theta) Replace θ\theta with −θ-\theta: r2=9sin⁡(2(−θ))=9sin⁡(−2θ)r^{2} = 9\sin(2(-\theta)) = 9\sin(-2\theta) Since sin⁡(−2θ)=−sin⁡(2θ)\sin(-2\theta) = -\sin(2\theta), the equation becomes r2=−9sin⁡(2θ)r^{2} = -9\sin(2\theta), which is not the same as the original equation.
  2. Test Symmetry Line Theta: Test for symmetry with respect to the line θ=π2\theta=\frac{\pi}{2} by replacing (r,θ)(r, \theta) with (r,π−θ)(r, \pi-\theta). Original equation: r2=9sin⁡(2θ)r^{2} = 9\sin(2\theta) Replace θ\theta with π−θ\pi-\theta: r2=9sin⁡(2(π−θ))=9sin⁡(2π−2θ)r^{2} = 9\sin(2(\pi-\theta)) = 9\sin(2\pi - 2\theta) Since sin⁡(2π−2θ)=sin⁡(−2θ)\sin(2\pi - 2\theta) = \sin(-2\theta), and we know sin⁡(−2θ)=−sin⁡(2θ)\sin(-2\theta) = -\sin(2\theta), the equation does not remain the same.
  3. Test Symmetry Pole: Test for symmetry with respect to the pole by replacing (r,θ)(r, \theta) with (−r,θ)(-r, \theta). Original equation: r2=9sin⁡(2θ)r^{2} = 9\sin(2\theta) Replace rr with −r-r: (−r)2=9sin⁡(2θ)(-r)^{2} = 9\sin(2\theta) Since (−r)2=r2(-r)^{2} = r^{2}, the equation remains the same.

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