Transforming a polar equation into a Cartesian equation?

Question

Transform the polar equation to a Cartesian (rectangular) equation: $$r= \frac5{5cosθ + 6sinθ}$$

These equations really stump me, so if you could be more "heavy-handed" with the explanation, I'd really appreciate it.

Answer 1

Use $$\frac x{\cos\theta}=\frac y{\sin\theta}=r$$

for $$5(r\cos\theta)+6(r\sin\theta)=5$$ which is too trivial

It would have been sweeter to have $$\frac1r=\frac5{5\cos\theta+6\sin\theta}$$

Answer 2

Most easily, try to form terms in form of $r\cos\theta$; $r\sin\theta$

In your question:

$$r= \frac5{5cosθ + 6sinθ}\implies 5(r\cos\theta)+6(r\sin\theta)=5$$ $$5x+6y=5$$

Your statement: These equations really stump me, so if you could be more "heavy-handed" with the explanation, I'd really appreciate it, requires too much explanation. Ask for specific hard-examples that bother you.

Answer 3

Here is how I would do things when I first learned polar coordinates.

First, draw a right triangle with horizontal leg $x$, vertical leg $y$, hypotenuse $r$, and angle $\theta$

Next try to solve for $x$ and $y$ in terms of $r$ and $\theta$. By the defintions of $\sin$ and $\cos$ we have

$$\cos\theta = \frac{x}{r}$$ $$\sin\theta = \frac{y}{r}$$

For translating cartesian coordinates into polar coordinates I would re express as

$$x=r\cos\theta$$ $$y=r\sin\theta$$

For translating the other way (from polar to cartesian) I would use

$$r^2=x^2+y^2$$ $$\tan\theta=\frac{y}{x}$$ So lets use this method on your example. We wish to translate from polar to cartesian coordinates. $$r=\frac{5}{5\cos\theta+6\sin\theta}$$ $$\sqrt{r^2}=\frac{\frac{5}{\cos\theta}}{\frac{5\cos\theta+6\sin\theta}{\cos\theta}}=\frac{5\sec\theta}{5+6\tan\theta}=\frac{5\sqrt{\tan^2\theta+1}}{5+6\tan\theta}$$ $$\sqrt{x^2+y^2}=\frac{5\sqrt{(\frac{y}{x})^2+1}}{5+6\frac{y}{x}}$$

Of course this could be simplified by multiplying the RHS by $\frac{x}{x}$.

$$\sqrt{x^2+y^2}=\frac{5\sqrt{y^2+x^2}}{5x+6y}$$ $$1=\frac{5}{5x+6y}$$ $$5x+6y=5$$

What I've just described is a general method that will always work, but keep in mind that it can get very messy and in some problems (like this one) you can take short-cuts so be looking for them. Here is the short cut.

$$r=\frac{5}{5\cos\theta+6\sin\theta}$$ $$r(5\cos\theta+6\sin\theta)=5$$ $$5r\cos\theta+6r\sin\theta=5$$ $$5x+6y=5$$