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Q 42.

Expert-verifiedFound in: Page 731

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

${r}^{2}=\mathrm{sin}\theta $

The required equation is ${\left({x}^{2}+{y}^{2}\right)}^{\frac{3}{2}}=y$.

The given equation in polar coordinates is:

${r}^{2}=\mathrm{sin}\theta $

${r}^{2}=\mathrm{sin}\theta $

First, multiply both sides by* r,*

$r\xb7{r}^{2}=r\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}{r}^{3}=y\left[r\mathrm{sin}\theta =y\right]\phantom{\rule{0ex}{0ex}}{\left(\sqrt{{x}^{2}+{y}^{2}}\right)}^{3}=y\left[{r}^{2}={x}^{2}+{y}^{2}\mathrm{and}r=\sqrt{{x}^{2}+{y}^{2}}\right]\phantom{\rule{0ex}{0ex}}{\left({x}^{2}+{y}^{2}\right)}^{\frac{3}{2}}=y$

Therefore, the equation in rectangular coordinates is ${\left({x}^{2}+{y}^{2}\right)}^{\frac{3}{2}}=y$.

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