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Chapter 7: Problem 94

In converting \(r=5\) from a polar equation to a rectangular equation, describe what should be done to both sides of the equation and why this should be done.

Short Answer

Expert verified
The polar equation \(r=5\) is converted to the rectangular equation \(x^2 + y^2 = 25\). This is done by squaring both sides of the given equation and replacing \(r^2\) with \(x^2 + y^2\). This converts the equation to rectangular form and the result represents a circle of radius 5, centered at the origin.

Step by step solution

01

Understand the Conversion Formula

In order to convert polar coordinates to rectangular coordinates, it's important to understand the relationship between them. If we consider a point with polar coordinates (r, 胃), where r is the distance from the origin and 胃 is the angle of the point from the positive x-axis, in a Cartesian system the coordinates of this point would be \((x, y)\), and the relationship between \((r, 胃)\) and \((x, y)\) can be represented by \(x^2 + y^2 = r^2\).
02

Apply the Conversion Formula

To start the conversion process, the given equation \(r = 5\) should be squared on both sides. This gives \(r^2 = 25\). Now, using the conversion formula previously mentioned, replace \(r^2\) with \(x^2 + y^2\). This gives \(x^2 + y^2 = 25\).
03

Interpret the Result

The result, \(x^2 + y^2 = 25\), represents a circle in rectangular coordinates with a radius of 5 (since the radius r is the square root of 25) centered at the origin (0,0). This is the rectangular form of the polar equation \(r=5\).

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Most popular questions from this chapter

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 64

What is the polar form of a complex number?

$$ \text { Solve: } \quad \tan ^{2} x-\sec x-1=0,0 \leq x<2 \pi $$

Exercises \(99-101\) will help you prepare for the material covered in the next section. Refer to Section 1.4 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i .\) $$ (1+i)(2+2 i) $$

Use a graphing utility to graph the polar equation. $$r=2+2 \cos \theta$$
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