/*! 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 60 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 60

Use a graphing utility to graph the polar equation. $$r=4 \cos 6 \theta$$

Short Answer

Expert verified
When the polar equation \( r=4 \cos 6 \theta \) is graphed using a graphing utility, it results in a rose pattern with 12 petals. The graph oscillates around the origin, with 12 clear 'petal' shapes that form due to the cosine function.

Step by step solution

01

Understand the equation

The equation for this particular exercise is \( r=4 \cos 6 \theta \) which is a polar equation. Here, \( r \) represents the distance from the origin and \( \theta \) is the angle from the positive x-axis. The equation shows how \( r \) varies with \( \theta \), and this forms the pattern for the graph. It is a cosine function, which means that the graph will have an oscillating pattern that repeats every \( 2\pi \) radians.
02

Use the graphing utility

There are many free and paid graphing utilities available online such as Desmos, Grapher on Mac, or the TI graphing calculators. In any of these utilities, ensure the mode is set to polar instead of the default Cartesian (Rectangular). In the utility, enter the equation \( r=4 \cos 6 \theta \) into the appropriate input field to generate the graph.
03

Interpret the graph

Upon entering the equation, the utility will generate the graph for the polar equation. This graph is a rose pattern with 12 petals. This is because the number of petals for a polar function represented by \( r=a \cos (n\theta) \) or \( r=a \sin (n\theta) \) is determined by the constant \( n \). If \( n \) is even, the graph will have \( 2n \) petals, and if \( n \) is odd it will have \( n \) petals. In our equation, we have \( n=6 \), so we get \( 2*6=12 \) petals in the graph.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j},\) find a vector orthogonal to \(\mathbf{v}\)

Determine the amplitude, period, and phase shift of \(y=3 \cos (2 x+\pi) .\) Then graph one period of the function.

A force of 80 pounds on a rope is used to pull a box up a ramp inclined at \(10^{\circ}\) from the horizontal. The rope forms an angle of \(33^{\circ}\) with the horizontal. How much work is done pulling the box 25 feet along the ramp?

What are parallel vectors?

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}-\mathbf{w})$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.