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Q29E

Expert-verified
Found in: Page 523

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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# Sketch the curve with the given polar equation by first sketching the graph as a function of$${\rm{c}}$$ Cartesian coordinates.$${\rm{r = 4sin3\theta }}$$.

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## Step 1: Cartesian coordinate.

In the Cartesian diagram,$${\rm{r = 4sin(3\theta )}}$$ the value of the function increases from$${\rm{0}}$$ to$${\rm{4}}$$ in the interval$${\rm{0}} \le {\rm{\theta }} \le \frac{{\rm{\pi }}}{{\rm{6}}}$$, and then decreases$${\rm{0}}$$ as the angle travels$$\frac{{\rm{\pi }}}{{\rm{3}}}$$. The first leaf of the rose on the right side represents the same curve in polar coordinates. Similarly, when the angle moves from $$\frac{{\rm{\pi }}}{{\rm{3}}}$$to$$\frac{{\rm{\pi }}}{{\rm{2}}}$$, the function in the Cartesian coordinate decreases $${\rm{ - 4}}$$to$${\rm{0}}$$ and subsequently raises to$${\rm{0}}$$ when the angle $${\rm{\theta }}$$moves from $$\frac{{\rm{\pi }}}{{\rm{2}}}$$tos$$\frac{{{\rm{2\pi }}}}{{\rm{3}}}$$. On the right, the polar curve's corresponding curve is marked ($${\rm{2}}$$).

## Step 2: Cartesian diagram.

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