How do you sketch the graph of the polar equation and find the tangents at the pole of #r=2cos3theta#?

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Answer 1 · 2017-01-11T17:05:13

#theta = pi/2, 5/6pi and 7/6pi#, each twice, for the six tangents. See the graph and explanation.

graph{(x^2+y^2)^2-8x^3+6x(x^2+y^2)=0 [-5, 5, -2.5, 2.5]}

The period of #cos 3theta = (2pi)/3#.

In one rotation #theta in [0, 2pi]# that runs through three periods, 3

loops are created.

At the pole,# r = 0 to cos3theta = 0 to 3theta = pi/2, 3/2pi, 5/2pi, ...#

# to theta = pi/6, pi/2, 5/6pi, 7/6pi, 3/2pi and 11pi/6,# for the three loops.

Slope of the tangent

#= (r'sin theta+r cos theta)/(r'cos theta-r sin theta)#

#=(-6sin 3theta sin theta +2 cos 3theta cos theta)

#/(-6sin 3theta cos theta-2cos 3theta sin theta)#

#= oo, +-1#, at r = 0, against

#theta = pi/2, 5/6pi and 7/6pi#, the tangent taking the same

direction twice.