Find the Equivalent Polar Equation for a given Equation with Rectangular Coordinates: r cos theta= -1

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asked 2020-11-22

To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
\(\displaystyle\ {x}=\ {r}{\cos{\theta}}\)
\(\displaystyle\ {y}=\ {r}{\sin{\theta}}\)
b. From rectangular to polar:
\(\displaystyle{r}=\pm\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}\)
\(\displaystyle{\cos{\theta}}={\frac{{{x}}}{{{r}}}},{\sin{\theta}}={\frac{{{y}}}{{{r}}}},{\tan{\theta}}={\frac{{{x}}}{{{y}}}}\)
Calculation:
Given: equation in rectangular-coordinate is \(\displaystyle{y}={x}\).
Converting into equivalent polar equation -
\(\displaystyle{y}={x}\)
Put \(\displaystyle{x}={r}{\cos{\theta}},\ {y}={r}{\sin{\theta}},\)
\(\displaystyle\Rightarrow\ {r}{\sin{\theta}}={r}{\cos{\theta}}\)
\(\displaystyle\Rightarrow\ {\frac{{{\sin{\theta}}}}{{{\cos{\theta}}}}}={1}\)
\(\displaystyle\Rightarrow\ {\tan{\theta}}={1}\)
Thus, desired equivalent polar equation would be \(\displaystyle\theta={1}\)

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asked 2020-11-03

Interraption: To show that the system \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le\dot{{{\left\lbrace{r}\right\rbrace}}}={\left\lbrace{r}\right\rbrace}{\left\lbrace{\left({\left\lbrace{1}\right\rbrace}-{\left\lbrace{r}\right\rbrace}^{{{2}}}\right)}\right\rbrace},\dot{{\theta}}={\left\lbrace{1}\right\rbrace}\) is equivalent to \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le\dot{{{\left\lbrace{x}\right\rbrace}}}={\left\lbrace{x}\right\rbrace}-{\left\lbrace{y}\right\rbrace}-{\left\lbrace{x}\right\rbrace}{\left\lbrace{\left({\left\lbrace{x}\right\rbrace}^{{{2}}}+{\left\lbrace{y}\right\rbrace}^{{{2}}}\right)}\right\rbrace},\dot{{{\left\lbrace{y}\right\rbrace}}}={\left\lbrace{x}\right\rbrace}+{\left\lbrace{y}\right\rbrace}-{\left\lbrace{y}\right\rbrace}{\left\lbrace{\left({\left\lbrace{x}\right\rbrace}^{{{2}}}+{\left\lbrace{y}\right\rbrace}^{{{2}}}\right)}\right\rbrace}\) for polar to Cartesian coordinates.
A limit cycle is a closed trajectory. Isolated means that neighboring trajectories are not closed.
A limit cycle is said to be unstable or half stable, if all neighboring trajectories approach the lemin cycle.
These systems oscillate even in the absence of external periodic force.

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asked 2021-02-09

To determine:
a) Whether the statement, " The point with Cartesian coordinates \(\displaystyle{\left[\begin{array}{cc} -{2}&\ {2}\end{array}\right]}\) has polar coordinates \(\displaystyle{\left[{b}{f}{\left({2}\sqrt{{{2}}},\ {\frac{{{3}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},{\frac{{{11}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},\ -{\frac{{{5}\pi}}{{{4}}}}\right)}\ {\quad\text{and}\quad}\ {\left(-{2}\sqrt{{2}},\ -{\frac{{\pi}}{{{4}}}}\right)}\right]}\) " is true or false.
b) Whether the statement, " the graphs of \(\displaystyle{\left[{r}{\cos{\theta}}={4}\ {\quad\text{and}\quad}\ {r}{\sin{\theta}}=\ -{2}\right]}\) intersect exactly once " is true or false.
c) Whether the statement, " the graphs of \(\displaystyle{\left[{r}={4}\ {\quad\text{and}\quad}\ \theta={\frac{{\pi}}{{{4}}}}\right]}\) intersect exactly once ", is true or false.
d) Whether the statement, " the point \(\displaystyle{\left[\begin{array}{cc} {3}&{\frac{{\pi}}{{{2}}}}\end{array}\right]}{l}{i}{e}{s}{o}{n}{t}{h}{e}{g}{r}{a}{p}{h}{o}{f}{\left[{r}={3}{\cos{\ }}{2}\ \theta\right]}\) " is true or false.
e) Whether the statement, " the graphs of \(\displaystyle{\left[{r}={2}{\sec{\theta}}\ {\quad\text{and}\quad}\ {r}={3}{\csc{\theta}}\right]}\) are lines " is true or false.

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asked 2020-11-01

The equivalent polar coordinates for the given rectangular coordinates.
A rectangular coordinate is given as (0, -3).

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asked 2020-12-29

The equivalent polar coordinates for the given rectangular coordinates:
\(\displaystyle{\left[\begin{array}{cc} {4}&\ {0}\end{array}\right]}\)

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asked 2020-12-02

The equivalent rectangular coordinates for the given polar coordinates:
\(\displaystyle{\left[\begin{array}{cc} {6}&\ -{135}^{{\circ}}\end{array}\right]}\)

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asked 2020-12-09

The equivalent polar coordinates for the given rectangular coordinates: \(\displaystyle{\left[\begin{array}{cc} {5}&\ {180}^{{\circ}}\end{array}\right]}\)

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asked 2021-02-08

To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{x}\right\rbrace}^{{{2}}}+{\left\lbrace{y}\right\rbrace}^{{{2}}}+{\left\lbrace{8}\right\rbrace}{\left\lbrace{x}\right\rbrace}={\left\lbrace{0}\right\rbrace}\)

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asked 2021-02-26

The equivalent polar equation for the given rectangular - coordinate equation.
\(\displaystyle{y}=\ -{3}\)

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asked 2020-12-24

The equivalent polar equation for the given rectangular-coordinate equation:
\(\displaystyle{x}^{{{2}}}\ +\ {y}^{{{2}}}\ +\ {8}{x}={0}\)

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Answer 1

The transformation formula for coordinate systems is:
a) \(\displaystyle{x}={r}{\cos{\theta}}\)
\(\displaystyle{y}={r}{\sin{\theta}}\)
b) \(\displaystyle{r}^{{{2}}}={x}^{{{2}}}\ +\ {y}^{{{2}}}\ \Rightarrow\ {r}=\pm\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}\)
\(\displaystyle{\cos{\theta}}={\frac{{{x}}}{{{r}}}},\ {\sin{\theta}}={\frac{{{y}}}{{{r}}}},\ {\tan{\theta}}={\frac{{{x}}}{{{y}}}}\)
Calculation:
Convert formula to equivalent polar coordinates:
\(\displaystyle{r}{\cos{\theta}}=\ -{1}\)
Put \(\displaystyle{x}={r}{\cos{\theta}},\ {y}={r}{\sin{\theta}},\)
\(\displaystyle\Rightarrow\ {x}=\ -{1}\)
Hence the desired equivalent polar coordinates would be \(\displaystyle{x}=\ -{1}\)

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Find the Equivalent Polar Equation for a given Equation with Rectangular Coordinates: r cos theta= -1

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