Find an equation of the followingcurve in polar coordinates and describe the curve. x = (1 + cos t) \cos t, y = (1 + \cos t) \sin t. 0 \leq t \leq 2\pi

facas9 2021-09-14 Answered
Find an equation of the following curve in polar coordinates and describe the curve.
\(\displaystyle{x}={\left({1}+{\cos{{t}}}\right)}{\cos{{t}}},{y}={\left({1}+{\cos{{t}}}\right)}{\sin{{t}}}.{0}\leq{t}\leq{2}\pi\)

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Expert Answer

Ezra Herbert
Answered 2021-09-15 Author has 17619 answers

Given that the cylindrical coordinates,
To convert cylindrical coordinates to Cartesian coordinates,
\(\displaystyle{r}={t}^{{{2}}},\theta={\frac{{\pi}}{{{2}}}}\)
Apply the rule of conversion
\(\displaystyle{x}={r}{\cos{\theta}},{z}={t}\)
Substituting \(\displaystyle{r}={t}^{{{2}}},\theta={\frac{{\pi}}{{{2}}}}\) in Cartesian coordinates, \(\displaystyle{x}={\cos{\theta}}\)
\(\displaystyle{x}={t}^{{{2}}}{\cos{{\frac{{\pi}}{{{2}}}}}}\)
\(\displaystyle={t}^{{{2}}}\)
x=0
and
\(\displaystyle{y}={r}{\sin{\theta}}\)
\(\displaystyle={t}^{{{2}}}{\sin{{\frac{{\pi}}{{{2}}}}}}\)
\(=t^{2}*(1) \because sin\frac{\pi}{2}=1\)
\(y=t^{2}\)
Substitute z=t in equation (1)
\(\displaystyle{y}={t}^{{{2}}}\)
\(\displaystyle{y}={\left({z}\right)}^{{{2}}}\)
\(\displaystyle{y}={z}^{{{2}}}\)
Hence the parametric equation of Cartesian coordinates is,
\(\displaystyle{y}={z}^{{{2}}}\) And the curve is a parabola in plane

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