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Stitz-Zeager_College_Algebra_e-book

# 13 focusing on z and w from 12 13 assuming w 1

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Unformatted text preview: -plane r = 4 − 2 sin(θ) in the xy -plane. 2. The ﬁrst thing to note when graphing r = 2 + 4 cos(θ) on the θr-plane over the interval [0, 2π ] is that the graph crosses through the θ-axis. This corresponds to the graph of the curve passing through the origin in the xy -plane, and our ﬁrst task is to determine when this 1 happens. Setting r = 0 we get 2 + 4 cos(θ) = 0, or cos(θ) = − 2 . Solving for θ in [0, 2π ] 2π 4π gives θ = 3 and θ = 3 . Since these values of θ are important geometrically, we break the π π π ππ π interval [0, 2π ] into six subintervals: 0, π , π , 23 , 23 , π , π , 43 , 43 , 32 and 32 , 2π . As 2 2 θ ranges from 0 to π , r decreases from 6 to 2. Plotting this on the xy -plane, we start 6 units 2 out from the origin on the positive x-axis and slowly pull in towards the positive y -axis. r y 6 θ runs from 0 to π 2 4 2 x 4π 3 2π 3 π 2 π 3π 2 2π θ −2 π On the interval π , 23 , r decreases from 2 to 0, which means the graph is heading into (and 2 π will eventually cross through) th...
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