/*! 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 63 Sketch a graph of the polar equa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 63

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=2 \cos 3 \theta $$

Short Answer

Expert verified
The graph of the polar equation \( r = 2 \cos 3 \theta \) resembles a three-leaved flower. The tangents at the pole occur whenever the curve passes through the pole, that is, for the cases when \( \theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6} \). Radiating lines from the origin to points where \( \theta \) has these values act as potential tangents.

Step by step solution

01

Graphing the Polar Equation

Plot different points by taking various values for \( \theta \) between 0 and \( \pi \) and then find the corresponding \( r \) using the given equation. Start with \( \theta = 0 \) and continue with small increments. Remember that in polar coordinates a point is expressed as (r, \( \theta \)), where \( r \) is the distance from the origin and \( \theta \) the angle from the positive x-axis. This should provide a clear graph of the given polar equation.
02

Identifying Pole Locations

The pole of a graph in polar coordinates is the origin of the coordinate system, where the value of \( r \) is zero. In this case, evaluate the equation \( r = 2 \cos 3 \theta \) to find for which values of \( \theta \), if any, \( r \) becomes zero.
03

Find the Tangents at the Pole

For those \( \theta \) values where \( r \) is zero, the curve passes through the pole. For such points, multiple tangents can be formed; with the pole being the origin, radiating lines (or rays) from the origin to points where \( \theta = \text{(constant)} \) function as potential tangents. These lines represent the different directions in which the curve leaves the pole.

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

The curve represented by the equation \(r=a \theta,\) where \(a\) is a constant, is called the spiral of Archimedes. (a) Use a graphing utility to graph \(r=\theta,\) where \(\theta \geq 0\). What happens to the graph of \(r=a \theta\) as \(a\) increases? What happens if \(\theta \leq 0 ?\) (b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0)\) where the curve crosses the polar axis. (c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\). (d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\).

Use the integration capabilities of a graphing utility to approximate to two decimal places the area of the surface formed by revolving the curve about the polar axis. $$ r=\theta, \quad 0 \leq \theta \leq \pi $$

Consider a projectile launched at a height \(h\) feet above the ground and at an angle \(\theta\) with the horizontal. If the initial velocity is \(v_{0}\) feet per second, the path of the projectile is modeled by the parametric equations \(x=\left(v_{0} \cos \theta\right) t\) and \(y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}\) The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of \(\theta\) degrees with the horizontal at a speed of 100 miles per hour (see figure). (a) Write a set of parametric equations for the path of the ball. (b) Use a graphing utility to graph the path of the ball when \(\theta=15^{\circ} .\) Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when \(\theta=23^{\circ} .\) Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run.

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=4 \sin 2 \theta, y=2 \cos 2 \theta $$

Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.