Question

How do you convert $r = \dfrac{6}{{2\cos (\theta ) - 3\sin (\theta )}}$ into Cartesian form?

Answer 1

Hint: The given trigonometric is $r = \dfrac{6}{{2\cos (\theta ) - 3\sin (\theta )}}$. First we identify the form or your equation. A glance at your equation should tell you what form it is in. It contains $r$ and $\theta $, it is in polar form. It contains $x$ and $y$, it is a rectangular form
If your equation is in polar form, your goal is to convert it in such a way that you are only left with $x$ and $y$. If it is in rectangular form, your goal is to only have $r$ and $\theta $.
Examine your equation. Now, take a moment to examine your equation. Here are some key components you should be looking for.
Simplify your equation by combining the terms.

Complete step-by-step solution:
The given trigonometric is $r = \dfrac{6}{{2\cos (\theta ) - 3\sin (\theta )}}$
We know our polar conversions:
${r^2} = {x^2} + {y^2}$
We know that
$r\cos \theta = x$
$r\sin \theta = y$
Hence the given equation is;
$ \Rightarrow r = \dfrac{6}{{2\cos (\theta ) - 3\sin (\theta )}}$
Multiply by $2\cos (\theta ) - 3\sin (\theta )$ on both sides, hence we get
$ \Rightarrow r(2\cos (\theta ) - 3\sin (\theta )) = \dfrac{6}{{{{2\cos (\theta ) - 3\sin (\theta )}}}}{{(2\cos (\theta ) - 3\sin (\theta ))}}$
Multiply $2\cos (\theta ) - 3\sin (\theta )$ by $r$, hence we get
$ \Rightarrow 2r\cos (\theta ) - 3r\sin (\theta ) = 6$
Now we substitute $r\cos \theta = x$ and $r\sin \theta = y$ in the equation, hence we get
$ \Rightarrow 2x - 3y = 6$

Hence the Cartesian form is $2x - 3y = 6$

Note: The Cartesian coordinate and the polar coordinate system concept given below:
The Cartesian coordinate system is a two-dimensional coordinate system using a rectilinear grid. The $x$ and $y$ the coordinates of a point measures the respective distances from the point to a pair of perpendicular lines in the plane called the coordinate axes, which meet at the origin.
The polar coordinate system is a two-dimensional coordinate system using a polar grid. The $r$ and $\theta $of a point $P$ measure respectively the distance from $P$ to the origin $O$ and the angle between the line $OP$ and the polar axis.
Points in the Cartesian coordinate system and points in the polar coordinate system can be converted into each other via the formulae:
$r\cos \theta = x$
$r\sin \theta = y$
${r^2} = {x^2} + {y^2}$