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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 71 (page 568)

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, 胃)$ .
$2 x y = 1$

Short Answer

Expert verified

The equation is written using polar coordinates as $r^{2} \sin (2 胃) = 1$ .

Step by step solution

01

Step 1. Concept used

If P is a point with polar coordinates $(r, 胃)$ , the rectangular coordinates $(x, y)$ of P are given by

$x = r \cos 胃 y = r \sin 胃$

02

Step 2. Convert into polar coordinates

In the given equation $2 x y = 1$ , we substitute $r \cos 胃$ for $x$ and $r \sin 胃$ for $y$ we get

role="math" localid="1646741958517" $\begin{array}{rcl} 2 x y & = & 1 \\ 2 (r \cos 胃) (r \sin 胃) & = & 1 \\ 2 r^{2} \cos 胃 \sin 胃 & = & 1 \end{array}$

Now using the double angle formula $\sin (2 胃) = 2 \sin 胃 \cos 胃$ the equation can be written as

$\begin{array}{rcl} 2 r^{2} \cos 胃 \sin 胃 & = & 1 \\ r^{2} (2 \sin 胃 \cos 胃) & = & 1 \\ r^{2} \sin (2 胃) & = & 1 \end{array}$

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