# To fill: The blanck spaces in the statement " The origin in the rectangular coordinate system concedes with the ? in polar coordinates. The positive x-axis in rectangular coordinates coincides with the ? in polar coordinates".

Question
Alternate coordinate systems
To fill: The blanck spaces in the statement " The origin in the rectangular coordinate system concedes with the ? in polar coordinates. The positive x-axis in rectangular coordinates coincides with the ? in polar coordinates".

2021-02-10
The two different ways of locating a point in a plane are the rectangular coordinates and polar coordinates. In polar coordinate system, each point on a plane is determined by a distance from a reference point and an angle from the reference direction. It is represented by $$\displaystyle{\left({r},\theta\right)}$$, where r denotes the distance from the reference point and $$\displaystyle\theta$$ denotes the angle that it makes with the reference direction.
But in the rectangular coordinate system, each point in a plane is determined by a pair of coordinates and represented in the form of (x, y).
So, in rectangular coordinate system the origin coincides with origin in polar and axis in the rectangular coordinate system is the polar axis in the polar coordinate system because an extended line from the reference point is the axis and it denotes the direction of point from the pole in both the coordinate systems.
So, both representations are similar but their nature will vary.
The pole and the polar coordinate arerepresented as the figure given below.

### Relevant Questions

To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
$$\displaystyle{x}^{2}+{y}^{2}+{8}{x}={0}$$

Solve the given Alternate Coordinate Systems and give a correct answer 10) Convert the equation from Cartesian to polar coordinates solving for $$r^2$$:
$$\frac{x^2}{9} - \frac{y^2}{16} = 25$$

The equivalent polar coordinates for the given rectangular coordinates.
A rectangular coordinate is given as (0, -3).
What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?
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 $$\displaystyle\theta={1}$$
Convert the following polar equation into a cartesian equation. Specifically describe the graph of the equation in rectangular coordinates: $$\displaystyle{r}={5}{\sin{\theta}}$$
The coordinates of the point in the $$\displaystyle{x}^{{{p}{r}{i}{m}{e}}}\ {y}^{{{p}{r}{i}{m}{e}}}$$ - coordinate system with the given angle of rotation and the xy-coordinates.
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.
The systems used to find the location of the point in a plane and find the rectangular and polar coordinates for the point P.