Question

How do you write the polar equation $r = - 3$ in rectangular form?

Answer 1

We will use the Cartesian formula to find the required equation for the above polar equation. On doing some simplification we get the required answer.

Formula used:
When we think about plotting points in the plane, we usually think of rectangular coordinates $(x,y)$in the Cartesian coordinate plane.
However, there are other ways of writing a coordinate pair and other types of grid system.
Polar coordinates are points labelled $(r,\theta )$ and plotted on a polar grid.
The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
To convert from polar to rectangular (Cartesian) coordinates use the following formulas (derived from their trigonometric function definitions):
So, we can write that: $\cos \theta = \dfrac{x}{r}$.
Using cross multiplication, we get:
$\Rightarrow x = r\cos \theta ..............(1)$.
Again, we can write that:
$\Rightarrow \sin \theta = \dfrac{y}{r}$.
Using cross multiplication, we get:
$\Rightarrow y = r\sin \theta .............(2)$.
So,
After squaring the equation$(1)$, we get:
$\Rightarrow {x^2} = {r^2}{\cos ^2}\theta ..............(3)$.
After squaring the both sides of the equation$(2)$, we get:
$\Rightarrow {y^2} = {r^2}{\sin ^2}\theta ..............(4)$.
Now, add equation $(3)$and $(4)$, we get:
$\Rightarrow {x^2} + {y^2} = {r^2}{\cos ^2}\theta + {r^2}{\sin ^2}\theta .$
After re-arranging, we get:
$\Rightarrow {x^2} + {y^2} = {r^2}({\cos ^2}\theta + {\sin ^2}\theta ).$
But we know that $({\cos ^2}\theta + {\sin ^2}\theta ) = 1.$
So, if we put this into the above equation, we get:
$\Rightarrow {x^2} + {y^2} = {r^2} \times 1$.
Or, we can write the following form:
$\Rightarrow {x^2} + {y^2} = {r^2}.$
Another formula:
if we divide equation$(2)$by equation$(1)$, we get:
$\Rightarrow \dfrac{{r\sin \theta }}{{r\cos \theta }} = \dfrac{y}{x}$.
Or, we can re-write it as following:
$\Rightarrow \tan \theta = \dfrac{y}{x}$.

Complete step by step answer:
To find the rectangular form of the polar equation$r = - 3$, we will use the Cartesian form of the rectangular coordinates.
So, from the formula:
$\Rightarrow {x^2} + {y^2} = {r^2}.$
Now, taking the square root on the both sides, we get:
$\Rightarrow r = \sqrt {{x^2} + {y^2}}$.
So, if we put the value of $r = - 3$, we get:
$\Rightarrow - 3 = \sqrt {{x^2} + {y^2}}$.
Now, taking the square on the both sides, we get:
$\Rightarrow {( - 3)^2} = {\left( {\sqrt {{x^2} + {y^2}} } \right)^2}$.
Now, we can write it as:
$\Rightarrow {x^2} + {y^2} = 9.$

Therefore, the require answer is ${x^2} + {y^2} = 9.$

Note: Points to remember:
Polar coordinates are the coordinates, those can be plotted in the circular grid.
Polar coordinates can have numeric numbers as well as the angular values made by the axis or the radius inside the grid.
On the other hand, rectangular coordinates are the coordinates, those can be plotted in the $X - Y$ plane.
Rectangular coordinates can only have numeric values as there is only horizontal axis and vertical axis exists.
So, polar coordinates are written in the form of $(r\cos \theta ,r\sin \theta )$, while the rectangular coordinates are written in the form of $(x,y)$.