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Chapter 12: Q. 66 (page 917)

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?

Short Answer

Expert verified

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 by step solution

01

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 $$y=rsin\theta$$

02

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,

$$x=rcos\theta$$

$$y=rsin\theta$$

$$r=\sqrt {x^{2}+y^{2}}$$

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

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