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

How do you convert r=62cos⁡(θ)−3sin⁡(θ)r = \dfrac{6}{{2\cos (\theta ) - 3\sin (\theta )}} into Cartesian form?

Explanation

Solution

The given trigonometric is r=62cos⁡(θ)−3sin⁡(θ)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 rr and θ\theta , it is in polar form. It contains xx and yy, 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 xx and yy. If it is in rectangular form, your goal is to only have rr and θ\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=62cos⁡(θ)−3sin⁡(θ)r = \dfrac{6}{{2\cos (\theta ) - 3\sin (\theta )}}
We know our polar conversions:
r2=x2+y2{r^2} = {x^2} + {y^2}
We know that
rcos⁡θ=xr\cos \theta = x
rsin⁡θ=yr\sin \theta = y
Hence the given equation is;
⇒r=62cos⁡(θ)−3sin⁡(θ)\Rightarrow r = \dfrac{6}{{2\cos (\theta ) - 3\sin (\theta )}}
Multiply by 2cos⁡(θ)−3sin⁡(θ)2\cos (\theta ) - 3\sin (\theta ) on both sides, hence we get
⇒r(2cos⁡(θ)−3sin⁡(θ))=62cos⁡(θ)−3sin⁡(θ)(2cos⁡(θ)−3sin⁡(θ))\Rightarrow r(2\cos (\theta ) - 3\sin (\theta )) = \dfrac{6}{{{{2\cos (\theta ) - 3\sin (\theta )}}}}{{(2\cos (\theta ) - 3\sin (\theta ))}}
Multiply 2cos⁡(θ)−3sin⁡(θ)2\cos (\theta ) - 3\sin (\theta ) by rr, hence we get
⇒2rcos⁡(θ)−3rsin⁡(θ)=6\Rightarrow 2r\cos (\theta ) - 3r\sin (\theta ) = 6
Now we substitute rcos⁡θ=xr\cos \theta = x and rsin⁡θ=yr\sin \theta = y in the equation, hence we get
⇒2x−3y=6\Rightarrow 2x - 3y = 6

Hence the Cartesian form is 2x−3y=62x - 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 xx and yy 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 rr and θ\theta of a point PP measure respectively the distance from PP to the origin OO and the angle between the line OPOP 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:
rcos⁡θ=xr\cos \theta = x
rsin⁡θ=yr\sin \theta = y
r2=x2+y2{r^2} = {x^2} + {y^2}