Question

How do you convert the polar equation \[r = 8\csc \theta \] into rectangular form?

Answer 1

Hint: In converting the system to rectangular form we have to take the take the cosine of \[\theta \] in order to find the corresponding Cartesian \[x\] coordinate and sine of \[\theta \] in order to find \[y\], and we will use the formulas \[{r^2} = {x^2} + {y^2}\], where \[x = r\cos \theta \], and \[y = r\sin \theta \].

Complete step-by-step solution:
To convert an equation given in polar form (in the variables\[r\]and\[\theta \]) into rectangular form (in\[x\]and\[y\]) you use the transformation relationships between the two sets of coordinates:
\[{r^2} = {x^2} + {y^2}\], where \[x = r\cos \theta \], and\[y = r\sin \theta \].
Polar coordinates are a complementary system to Cartesian coordinates, which are located by moving across an x-axis and up and down the y-axis in a rectangular fashion. While Cartesian coordinates are written as\[\left( {x,y} \right)\], polar coordinates are written as\[\left( {r,\theta } \right)\].
Now given polar equation is,
\[r = 8\csc \theta - - - - (1)\],
Now using the trigonometric identity \[\csc \theta = \dfrac{1}{{\sin \theta }}\],
The equation becomes,
\[r = \dfrac{8}{{\sin \theta }} - - - - (2)\],
Using the polar coordinates, i.e,\[y = r\sin \theta \],
\[ \Rightarrow \sin \theta = \dfrac{y}{r}\],
Substituting the value in (2) we get,
\[ \Rightarrow r = \dfrac{8}{{\dfrac{y}{r}}}\],
Now simplifying we get,
\[ \Rightarrow r = \dfrac{{8 \cdot r}}{y}\],
Now eliminating the like terms we get,
\[ \Rightarrow 1 = \dfrac{8}{y}\],
Now taking y to Left hand side we gte,
\[ \Rightarrow y = 8\],
This the required rectangular form.

\[\therefore \]The required rectangular form of polar form \[r = 8\csc \theta \] is \[y = 8\].

Note: In polar coordinates, a point in the plane is determined by its distance \[r\] from the origin and the angle \[\theta \] (in radians) between the line from the origin to the point and the x-axisThe polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction
In polar coordinates the origin is often called the pole. Because we aren't actually moving away from the origin/pole we know that. However, we can still rotate around the system by any angle we want and so the coordinates of the origin/pole are.