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Chapter 7: Problem 92

Explain how to convert a point from rectangular to polar coordinates. Provide an example with your explanation.

Short Answer

Expert verified
To convert a point from rectangular to polar coordinates: find r using \( r = \sqrt{x^2 + y^2} \), and find 胃 using \( 胃 = tan^{-1}(y/x) \). For example, the rectangular coordinates (3, 4) convert to polar coordinates as approximately (5, 53.13掳).

Step by step solution

01

Understand the formulas

Two main formulas relate rectangular and polar coordinates. They are \( r = \sqrt{x^2 + y^2} \) and \( 胃 = tan^{-1}(y/x) \). r represents the distance of the point from the origin and 胃 is the angle made with the positive x-axis.
02

Example - Convert the point (3, 4) from rectangular to polar coordinates

Firstly, to find r, substitute x as 3 and y as 4 in the formula for r. That comes out as \( r = \sqrt{3^2 + 4^2} = 5 \). Secondly, to find 胃, substitute y as 4 and x as 3 in the formula for 胃. That results in \( 胃 = tan^{-1}(4/3) \), the value of which depends on the unit (degrees or radians) one needs the answer in. Assuming 胃 is required in degrees, \( 胃 = tan^{-1}(4/3) \approx 53.13掳 \).
03

Present the result

After converting, the rectangular coordinates (3, 4) have the polar coordinates approximately as (5, 53.13掳).

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Most popular questions from this chapter

Exercises \(116-118\) will help you prepare for the material covered in the next section. Simplify: \(4(5 x+4 y)-2(6 x-9 y)\)

Explaining the Concepts Describe the test for symmetry with respect to the line \(\theta=\frac{\pi}{2}\)

In Exercises \(77-80,\) convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$ i(2+2 i)(-\sqrt{3}+i) $$

Use a graphing utility to graph the polar equation. $$r=\frac{3}{\cos \theta}$$

Use a graphing utility to graph the polar equation. $$r=\frac{1}{1-\sin \theta}$$
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