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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

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

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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
  1. Test Symmetry Polar Axis: a. Test for symmetry with respect to the polar axis by replacing (r,θ)(r, \theta) with (r,−θ)(r, -\theta).\newlineOriginal equation: r2=9sin⁡(2θ)r^2 = 9\sin(2\theta)\newlineReplace θ\theta with −θ-\theta: r2=9sin⁡(2(−θ))=9sin⁡(−2θ)r^2 = 9\sin(2(-\theta)) = 9\sin(-2\theta)\newlineSince sin⁡(−x)=−sin⁡(x)\sin(-x) = -\sin(x), we have: r2=9(−sin⁡(2θ))=−9sin⁡(2θ)r^2 = 9(-\sin(2\theta)) = -9\sin(2\theta)\newlineThis does not match the original equation, so there is no symmetry with respect to the polar axis.
  2. Test Symmetry Line: b. 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).\newlineOriginal equation: r2=9sin⁡(2θ)r^2 = 9\sin(2\theta)\newlineReplace 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)\newlineSince sin⁡(2π−x)=sin⁡(x)\sin(2\pi - x) = \sin(x), we have: r2=9sin⁡(2θ)r^2 = 9\sin(2\theta)\newlineThis matches the original equation, so there is symmetry with respect to the line θ=π2\theta = \frac{\pi}{2}.
  3. Test Symmetry Pole: c. Test for symmetry with respect to the pole by replacing (r,θ)(r, \theta) with (−r,θ)(-r, \theta).\newlineOriginal equation: r2=9sin⁡(2θ)r^2 = 9\sin(2\theta)\newlineReplace rr with −r-r: (−r)2=9sin⁡(2θ)(-r)^2 = 9\sin(2\theta)\newlineSince (−r)2=r2(-r)^2 = r^2, we have: r2=9sin⁡(2θ)r^2 = 9\sin(2\theta)\newlineThis matches the original equation, so there is symmetry with respect to the pole.

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