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EXAMPLE 4Graphing a Polar Equation (Rose); Finding Parametric Equations

(a) Graph the polar equation r=2 cos(2θ), 0θ2π.

(b) Find parametric equations for r=2 cos(2θ).

Graphs of polar equations of the form r=a cos(nθ) or r=a sin(nθ),a>0,n an integer, are called roses. If n is an even integer, the rose has 2n petals and passes through the pole 4n times. If n is an odd integer, the rose has n petals and passes through the pole 2n times.

Solution (a) The polar equation r=2 cos(2θ) contains cos(2θ), which has the period π. So, we construct Table 5 using common values of θ that range from 0 to 2π, noting that the values for πθ2π repeat the values for 0θπ. Then we plot the points (r,θ) and trace out the graph. Figure 46(a) on page 674 shows the graph from the point (2,0) to the point (2, π). Figure 46(b) completes the graph from the point (2, π) to the point (2, 2π).

TABLE 5
θ r=2 cos(2θ) (r,θ)
0 2(1)=2 (2,0)
π6 2(12)=1 (1,π6)
π4 2(0)=0 (0,π4)
π3 2(12)=1 (1,π3)
π2 2(1)=2 (2,π2)
2π3 2(12)=1 (1,2π3)
3π4 2(0)=0 (0,3π4)
5π6 2(12)=1 (1,5π6)
π 2(1)=2 (2, π)
7π6 2(12)=1 (1,7π6)
5π4 2(0)=0 (0,5π4)
4π3 2(12)=1 (1,4π3)
3π2 2(1)=2 (2,3π2)
5π3 2(12)=1 (1,5π3)
7π4 2(0)=0 (0,7π4)
11π6 2(12)=1 (1,11π6)
2π 2(1)=2 (2, 2π)
Figure 46 A rose with four petals.

(b) Parametric equations for r=2 cos(2θ): x=r cosθ=2 cos(2θ) cosθy=r sinθ=2 cos(2θ) sinθ

where θ is the parameter, and if 0θ2π, then the graph is traced out exactly once in the counterclockwise direction.