/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 Convert each polar equation to a... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 74

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r^{2} \sin 2 \theta=4$$

Short Answer

Expert verified
The rectangular equation is \(xy = 2\) and it represents a hyperbola opening towards the first and third quadrants.

Step by step solution

01

Convert polar to rectangular coordinates

The conversion of polar coordinates (r, 胃) into rectangular coordinates (x, y) involves the use of the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). Additionally, we can use the Pythagorean relation \(r^{2} = x^{2} + y^{2}\) and the double angle relation \(\sin 2\theta = 2 \sin \theta \cos \theta\). Using these relations, our given equation transforms to \((x^{2} + y^{2}) \cdot 2 \sin \theta \cos \theta = 4\).
02

Convert \( \sin \theta \) and \( \cos \theta \)

Now replace \(\sin \theta = \frac{y}{r}\) and \(\cos \theta = \frac{x}{r}\) in equation obtained in step 1. We obtain: \((x^{2} + y^{2}) \cdot 2 \frac{y}{r} \frac{x}{r} = 4.\) After cancelling r, we end up with \( 2 x y = 4 \).
03

Simplify the rectangular equation

Our goal now is to simplify this expression, which will result in our rectangular equation. To achieve this, we divide both sides by 2 to isolate \(xy\) on one side, giving us the rectangular equation \(xy = 2\).
04

Plot the rectangular function

This is a hyperbola equation of the form \(xy = c\), and it opens towards the first and the third quadrant. For a detailed view, plot x in the range of -3 to 3 and plot y as \( \frac{2}{x} \).

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