Chapter 7: Additional Applications of Integration
Section 7.1: Polar Coordinates
Example 7.1.10
Obtain the polar coordinates of the points of intersection of the curves defined by r=cosθ and r=1+sinθ.
Solution
Mathematical Solution
In Figure 7.1.10(a), use the slider to draw the cardioid r=1+sinθ with θ∈−π,π. The circle r=cosθ, with θ∈−π/2,π/2, is added to each frame of the "animation."
From the figure, there are clearly two points of intersection. The points
r,θ=0,−π/2 and r,θ=1,0
obtained by solving cosθ=1+sinθ for θ, satisfy both equations simultaneously. Indeed,
cosθ
=1+sinθ
cos2θ
=1+sinθ2
1−sin2θ
=1+2 sinθ+sin2θ
0
=2 sinθsinθ+1
from which it follows that θ=0 and θ=arcsin−1=−π/2.
θ = = °
Figure 7.1.10(a) Circle and cardioid
Maple Solution
To graph the circle and cardioid, use the Plot Builder to draw separate graphs of the circle and cardioid. Then copy/paste one graph onto the other.
Alternatively, use following command. (Select Evaluate in the Context Panel.)
plotcosθ,θ,θ=−π..π,1+sinθ,θ,θ=−π..π,coords=polar
Obtain intersections
Write the relevant equation.
Context Panel: Solve≻Solve
cosθ=1+sinθ→solveθ=−π2,θ=0
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