Question

To determine: a) Whether the statement, " The point with Cartesian coordinates \[( -2,\ 2 )\] has polar coordinates \[\bf \left( 2 \sqrt{2},\ \frac{3

Alternate coordinate systems
ANSWERED
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.

Answers (1)

2021-02-10

a)
The polar coordinate \(\displaystyle{\left[\begin{array}{cc} {r}&\theta\end{array}\right]}\) corresponding to Cartesian coordinate \(\displaystyle{\left[\begin{array}{cc} {x}&\ {y}\end{array}\right]}{i}{s}\ {g}{i}{v}{e}{n}\ {b}{y}{\left[{r}=\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}\ {\quad\text{and}\quad}\theta={{\tan}^{{-{1}}}{\left({\frac{{{y}}}{{{x}}}}\right)}}\right]}.\)
From the given data, \(\displaystyle{\left[{\left({x},\ {y}\right)}={\left(-{2},\ {2}\right)}\right]}.\)
Substitute \(\displaystyle{\left[{\left({x},\ {y}\right)}={\left(-{2},\ {2}\right)}\right]}\in{\left[{r}=\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}\right]}\) and obtain the value of r.
\(\displaystyle{r}=\sqrt{{{2}^{{{2}}}\ +\ {\left(-{2}\right)}^{{{2}}}}}\)
\(\displaystyle=\pm\sqrt{{{8}}}\)
\(\displaystyle=\pm\ {2}\sqrt{{{2}}}\)
Similary, calculate the value of \(\displaystyle{\left[\theta\right]}\) as follows.
\(\displaystyle\theta={{\tan}^{{-{1}}}{\left(-{\frac{{{2}}}{{{2}}}}\right)}}\)
\(\displaystyle={{\tan}^{{-{1}}}{\left(-{1}\right)}}\)
\(\displaystyle={\frac{{{3}\pi}}{{{4}}}}\ +\pi{n},\ {n}\ \in{\mathbb{{Z}}}\)
For \(\displaystyle{\left[{r}={2}\sqrt{{{2}}}\ {\quad\text{and}\quad}\ {n}={0}\right]}{t}{h}{e}\ {p}{o}{l}{a}{r}\ {c}\ in\ coo{r}{d}in{a}{t}{e}\ {i}{s}{\left[\begin{array}{cc} {2}\sqrt{{{2}}}&{\frac{{{3}\pi}}{{{4}}}}\end{array}\right]}.\)
For \(\displaystyle{\left[{r}={2}\sqrt{{{2}}}\ {\quad\text{and}\quad}\ {n}={2}\backslash\right]}{t}{h}{e}\ {p}{o}{l}{a}{r}\ {c}\ in\ coo{r}{d}in{a}{t}{e}\ {i}{s}{\left[{\left({2}\sqrt{{{2}}},{\frac{{{11}\pi}}{{{4}}}}\right)}\backslash\right]}.\)
For \(\displaystyle{\left[{r}={2}\sqrt{{{2}}}\ {\quad\text{and}\quad}\ {n}=\ -{2}\backslash\right]}{t}{h}{e}\ {p}{o}{l}{a}{r}\ {c}\ in\ coo{r}{d}in{a}{t}{e}\ {i}{s}{\left[{\left({2}\sqrt{{{2}}},\ -{\frac{{{5}\pi}}{{{4}}}}\right)}\backslash\right]}\).
For \(\displaystyle{\left[{r}={2}\sqrt{{{2}}}\ {\quad\text{and}\quad}\ {n}=\ -{1}\backslash\right]}{t}{h}{e}\ {p}{o}{l}{a}{r}\ {c}\ in\ coo{r}{d}in{a}{t}{e}\ {i}{s}{\left[{\left(-{2}\sqrt{{{2}}},\ -{\frac{{\pi}}{{{4}}}}\right)}\backslash\right]}\)
Hence the given stament is true.
b) Note that the Cartesian coordinates corresponding to polar coordinates \(\displaystyle{\left[\begin{array}{cc} {r}&\theta\end{array}\right]}\) are given by \(\displaystyle{\left[{x}={r}{\cos{\theta}}\ {\quad\text{and}\quad}\ {y}={r}{\sin{\theta}}\right]}.\)
From the given data it can be concluded that,
\(\displaystyle{x}={r}{\cos{\theta}}\)
\(\displaystyle={4}\)
\(\displaystyle{y}={r}{\sin{\theta}}\)
\(\displaystyle=\ -{2}\)
Which is noting but a single point \(\displaystyle{\left[\begin{array}{cc} {4}&\ -{2}\end{array}\right]}\).
Therefore, the given statement is true.
c)
The given polar equations are \(\displaystyle{\left[{r}={2}\ {\quad\text{and}\quad}\ \theta={\frac{{\pi}}{{{4}}}}\right]}\).
Note that \(\displaystyle{\left[{r}={2}\right]}\) is an equation of a circle centered at the origin and having radius 2 while \(\displaystyle{\left[\theta={\frac{{\pi}}{{{4}}}}\right]}\) is an equation of a straight line passing through the origin.
Thus the line \(\displaystyle{\left[\theta={\frac{{\pi}}{{{4}}}}\right]}\) intersects the circle \(\displaystyle{r}={2}\) twise.
Therefore, the statement is false.
d)
The given point is \(\displaystyle{\left[{\left({3},{\frac{{\pi}}{{{2}}}}\right)}={\left({r},\theta\right)}\right]}\).
Change the radial coordinate to -3 and subtract \(\displaystyle{\left[\pi\right]}\) from the angle to obtain the alternate representation of the point as follows.
\(\displaystyle{\left[{\left(-{3},{\frac{{\pi}}{{{2}}}}\ -\pi\right)}={\left(-{3},-{\frac{{\pi}}{{{2}}}}\right)}\right]}\)
Thus, point \(\displaystyle{\left[\begin{array}{cc} -{3}&-{\frac{{\pi}}{{{2}}}}\end{array}\right]}{\quad\text{and}\quad}{\left[\begin{array}{cc} {3}&{\frac{{\pi}}{{{2}}}}\end{array}\right]}\) represent the same point.
Substitute \(\displaystyle{\left[{r}=\ -{3}\ {\quad\text{and}\quad}\ \theta=\ -{\frac{{\pi}}{{{2}}}}\ \in\ {r}={3}{\cos{{2}}}\theta\right]}\)
\(\displaystyle{\left[-{3}={3}{\cos{\ }}{2}{\left(-{\frac{{\pi}}{{{2}}}}\right)}\right.}\)
\(\displaystyle={3}{\cos{\ }}{\left(-\pi\right)}\)
\(\displaystyle={3}{\cos{\pi}}\)
\(\displaystyle=-{3}\)
Therefore, the point \(\displaystyle{\left[\begin{array}{cc} -{3}&-{\frac{{\pi}}{{{2}}}}\end{array}\right]}\) lies on the graph of \(\displaystyle{\left[{r}={3}{\cos{{2}}}\theta\right]}\) that is the point \(\displaystyle{\left[\begin{array}{cc} {3}&{\frac{{\pi}}{{{2}}}}\end{array}\right]}\) also lies on the graph of \(\displaystyle{\left[{r}={3}{\cos{{2}}}\theta\right]}\).
Therefore, the statement is true.
e)
The given equations are \(\displaystyle{\left[{r}={2}{\sec{\theta}}\right]}{\quad\text{and}\quad}{\left[{r}={3}{\csc{\theta}}\right]}.\)
Rewrite \(\displaystyle{\left[{r}={2}{\sec{\theta}}\right]}{\quad\text{and}\quad}{\left[{r}={3}{\csc{\theta}}\right]}\) as follows.
\(\displaystyle{r}={2}{\sec{\theta}}\)
\(\displaystyle{r}{\cos{\theta}}={2}\)
\(\displaystyle{r}={3}{\csc{\theta}}\)
\(\displaystyle{r}{\sin{\theta}}={3}\)
Note that \(\displaystyle{r}{\cos{\theta}}={x}{]}{\quad\text{and}\quad}{\left[{r}{\sin{\theta}}={y}\right]}\).
Thus, the given equations \(\displaystyle{\left[{x}={2}\right]}{\quad\text{and}\quad}{\left[{y}={3}\right]}\) represent the lines in the xy - plane.
Therefore, the given statement is true.

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