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**Unformatted text preview: **of r = 5 sin(2θ) in the θr-plane, which in
this case, occurs as θ ranges from 0 to π . We partition our interval into subintervals to help
π
π
us with the graphing, namely 0, π , π , π , π , 34 and 34 , π . As θ ranges from 0 to π , r
4
42
2
4
increases from 0 to 5. This means that the graph of r = 5 sin(2θ) in the xy -plane starts at
the origin and gradually sweeps out so it is 5 units away from the origin on the line θ = π .
4 r
y 5 π
4 π
2 3π
4 π x
θ −5 Next, we see that r decreases from 5 to 0 as θ runs through π , π , and furthermore, r is
42
heading negative as θ crosses π . Hence, we draw the curve hugging the line θ = π (the y -axis)
2
2
as the curve heads to the origin. 806 Applications of Trigonometry
r
y 5 π
4 π
2 3π
4 π x
θ −5 π
As θ runs from π to 34 , r becomes negative and ranges from 0 to −5. Since r ≤ 0, the curve
2
pulls away from the negative y -axis into Quadrant IV.
r
y 5 π
4 π
2 3π
4 π x
θ −5 For 3π
4 ≤ θ ≤ π , r increases from −5 to 0, so...

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