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

Question
Alternate coordinate systems
asked 2021-01-28
Find the Equivalent Polar Equation for a given Equation with Rectangular Coordinates:
\(\displaystyle{r}{\cos{\theta}}=\ -{1}\)

Answers (1)

2021-01-29
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 \(\displaystyle{x}=\ -{1}\)
0

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Given:
\(\displaystyle\ {x}=\ {r}{\cos{\theta}}\)
\(\displaystyle\ {y}=\ {r}{\sin{\theta}}\)
b. From rectangular to polar:
\(\displaystyle{r}=\pm\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}\)
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Given: equation in rectangular-coordinate is \(\displaystyle{y}={x}\).
Converting into equivalent polar equation -
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Thus, desired equivalent polar equation would be \(\displaystyle\theta={1}\)
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