Q. 66 - Calculus

Question

Express the formulas for converting from polar coordinates to rectangular coordinates found in Section 9.2 as functions of two variables. What is the domain of each function?

Step-by-Step Solution

Answer

The domain of functions of x and y are $$x=rcos\theta$$ and $$y=rsin\theta$$ respectively while the domain of functions of r and $$\theta$$ are $$r=\sqrt {x^{2}+y^{2}}$$ and $$\theta = tan^{-1}(\frac{y}{x})$$ respectively.

Step 1. Given Information

In Section 9.2, it was found that if a point is expressed as $$(r,\theta)$$ in polar coordinates, then it can be expressed in ectangular coordinates as $$(x,y)$$, where $$x=rcos\theta$$ and $$y=rsin\theta$$

Step 2. Explanation

We have, $$x=rcos\theta$$ and $$y=sin\theta$$

In general, $$x^{2}+y^{2}=r^{2}$$

So, the above equation can be expressed in terms of r as follows, $$r=\sqrt {x^{2}+y^{2}}$$

Now, rearranging the general equation in terms of $$\theta$$, we get

$$tan \theta=\frac{y}{x}$$

$$\Rightarrow \theta = tan^{-1}(\frac{y}{x})$$

Hence, the domains of each given functions are as follows,

$$x=rcos\theta$$ 

$$y=rsin\theta$$ 

$$r=\sqrt {x^{2}+y^{2}}$$ 

$$\theta = tan^{-1}(\frac{y}{x})$$