91Ó°ÊÓ

Three types of symmetry can be considered for polar equations: symmetry with respect to the line \(\theta=\pi/2\), the polar axis and the pole. \n\nTo test symmetry with respect to the pole, replace \(\sin(\theta)\) by \(\sin(-\theta)\) in the equation and simplify: if we get an equivalent equation, it means the equation is symmetric with respect to the pole. \n\nIn our equation: \(r=2-\sin(-\theta)==2+\sin(\theta)\), which does not match our original equation. Hence, it's not symmetric with respect to the pole. \n\nFor symmetry with respect to the polar axis, replace \(r\) with \(-r\) and \(\sin(\theta)\) by \(\sin(-\theta)\). If we get an equivalent equation, it means the equation is symmetric with respect to the polar axis. \n\nIn our equation: \(-r=2+\sin(\theta)\), which is not equivalent to the original and hence, not symmetric about the polar axis. \n\n Lastly, for symmetry with the line \(\theta=\pi/2\), replace \(\sin(\theta)\) with \(\sin(\pi- \theta)\). Equivalent equations mean symmetry. In our equation: \(r=2-\sin(\pi - \theta) = 2 - \sin(\theta)\). This matches our original equation, so the equation is symmetric about the line \(\theta=\pi/2\).

To graph the equation, generate the set of points (r, θ) that satisfy the equation for a range of θ values (0 to 2π). Then sketch these points in the polar coordinate plane, which should indicate the whole curve. \n To do this, you could plot points for \(\theta\) ranging from 0 to 2\(\pi\). For each \(\theta\), compute the corresponding radius \(r\). For instance, when \(\theta = 0\), \(r = 2 - \sin(0) = 2\). When \(\theta = \pi/2\), \(r = 2 - \sin(\pi/2) = 1\). When \(\theta = \pi\), \(r = 2 - \sin(\pi) = 2\). And \(\theta = 3\pi/2\), \(r = 2 - \sin(3\pi/2) = 3\). When \(\theta = 2\pi\), \(r = 2 - \sin(2\pi) = 2\). \nPlot these points and draw the curve to represent them on the graph.