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Chapter 6: Problem 26

Test for symmetry and then graph each polar equation. $$r=2-3 \sin \theta$$

Short Answer

Expert verified
The given polar equation, \( r = 2 -3 \sin \theta \), does not exhibit symmetry about the line \( \theta = \pi/2 \) or about the pole. Its corresponding Cartesian equation is \( x^2 + y^2 = 2 \sqrt{x^2 + y^2} - 3y \) which can be plotted to reveal the nature of the polar equation.

Step by step solution

01

Test for Symmetry

Verify the symmetry of the polar equation about the line \( \theta = \pi/2 \) or the pole- Symmetry about the line \( \theta = \pi/2 \): Substitute \( -\theta \) for \( \theta \) in the equation. If the equation remains unchanged, then it's symmetric about the line \( \theta = \pi/2 \).r = 2-3sin(-胃) = 2+3sin(胃)The equation is not the same as the given equation so it's not symmetric about the line \( \theta = \pi/2 \).- Symmetry about the pole: Replace r with -r and 胃 with \( 胃+ \pi \) in the equation. If the equation remains the same, then it is symmetric about the pole.- r = 2-3sin(胃+蟺) = -rThis equation doesn't meet the criteria, meaning the given polar equation is not symmetric about the pole.
02

Convert to Cartesian coordinates

The polar equation \( r=2-3 \sin \theta \) can be converted to Cartesian coordinates using the relationships \( r^2 = x^2+y^2 \) and \( y = r \sin \theta \). Replacing r in terms of x and y and sin胃 in terms of y: \( r = 2 - 3 \frac{y}{r} \)To eliminate fractions multiply both sides by r \( r^2 = 2r - 3y \)Then, replace \( r^2 \) by \( x^2 + y^2 \)\( x^2+y^2 = 2 \sqrt{x^2 + y^2} -3y \)
03

Plot the graph in Cartesian coordinates

At this point, you can use a graphing tool to plot the Cartesian equation \( x^2+y^2 = 2 \sqrt{x^2 + y^2} -3y \), which corresponds to the polar equation \( r=2-3 \sin \theta \). The graph will reveal the shape and important features of the given polar equation.

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