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

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.

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.