Is r=5sinθ,r=5sinθ, the value θ=π2θ=π2 willĪdditional information by calculating values When θ=π2,θ=π2, and if our polar equation Similarly, the maximum value of the sine function is 1 Is 5cosθ,5cosθ, and the value θ=0θ=0 will yield The maximum value of the cosineįunction is 1 when θ=0,θ=0, so our polar equation Of θθ into the equation that result in the maximum valueĬonsider r=5cosθ r=5cosθ the maximum distance between Polar equation is found by substituting those values Set r=0,r=0, and solveįor many of the forms we will encounter, the maximum value of a We use the same processįor polar equations. Recall that, toįind the zeros of polynomial functions, we set the equation equal To find the zeros of a polar equation, we solve for the values Polar equations, but its application is not perfect. Testing for symmetry is a technique that simplifies the graphing of Reflecting points across the apparent axis of symmetry or the pole.
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Instances, we can confirm that symmetry exists by plotting Line θ=π2,θ=π2, the polar axis, or the pole. Necessarily indicate that a graph will not be symmetric about the
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However, failing the symmetry tests does not One or more of the symmetry tests verifies that symmetry will beĮxhibited in a graph. Not mean that it is not symmetric with respect to the pole. The equation has failed the symmetry test, but that does In the third test, we consider symmetry with respect to the poleĭetermine if the tested equation is equivalent to the original The graph of this equation exhibits symmetry with respect to the Replace (r,θ)(r,θ) with (r,−θ)(r,−θ) or (−r,π−θ)(−r,π−θ) toĭetermine equivalency between the tested equation and the original. In the second test, we consider symmetry with respect to the This equation exhibits symmetry with respect to the Replace (r,θ)(r,θ) with (−r,−θ)(−r,−θ) toĭetermine if the new equation is equivalent to the originalĮquation. In the first test, we consider symmetry with respect to the Key points, zeros, and maximums of r)r) to determine the Further, we will use symmetry (in addition to plotting Tests, we will see how to apply the properties of symmetry to polarĮquations. Half over that axis, the portion of the graph on one side wouldĬoincide with the portion on the other side. With respect to an axis, it means that if we folded the graph in If an equation has a graph that is symmetric Symmetry is a property that helps us recognize and plot the
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Points that satisfy the polar equation are on the graph. (origin) rr units in the direction of θ.θ. We move counterclockwise from the polar axis Recall that the coordinate pair (r,θ)(r,θ) indicates that Relationship between rr and θθ on a polar grid.