/*! 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 37 Use algebraic methods to find as... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 37

Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points. $$r=3 \sin \theta \text { and } r=3 \cos \theta$$

Short Answer

Expert verified
Answer: The intersection points of the curves given by $$r=3 \sin \theta$$ and $$r=3 \cos \theta$$ are $$\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$$ and $$\left(-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2}\right)$$.

Step by step solution

01

Set the equations equal to each other

First, in order to find where the two curves intersect, we must set the two equations equal to each other: $$3 \sin \theta = 3 \cos \theta$$.
02

Isolate the trigonometric functions

To continue solving, we want to isolate the trigonometric functions on one side of the equation. So, we can simply divide both sides by 3 and get the equation: $$\sin \theta = \cos \theta$$.
03

Use the identity for sine and cosine

We know that the sine and cosine functions are related by the identity $$\sin \left(\frac{\pi}{2} - \theta\right) = \cos \theta$$. So, we can rewrite our equation as $$\sin \theta = \sin \left(\frac{\pi}{2} - \theta\right)$$.
04

Find the angles where the functions are equal

By the identity, when $$\theta = \frac{\pi}{4}$$, the sine function will be equal to the cosine function. Using the identity $$\sin \theta = \sin \left(\frac{\pi}{2} - \theta\right)$$, we identify $$\theta = \frac{\pi}{4}$$ as one of the angles for which the two polar curves intersect.
05

Graph the curves

To find the remaining intersection points, we must graph the curves in a polar coordinate system. We know that $$\theta = \frac{\pi}{4}$$ is one of the intersecting angles, and after graphing the two curves, we can also see that they intersect at $$\theta = -\frac{3\pi}{4}$$. So, our intersection points are: $$\left(3\sin(\frac{\pi}{4}), \frac{\pi}{4}\right)$$ and $$\left(3\sin(-\frac{3\pi}{4}), -\frac{3\pi}{4}\right)$$.
06

Convert polar coordinates to rectangular coordinates

To fully understand the intersection points, we will convert the polar coordinates to rectangular coordinates. $$\left(3\sin(\frac{\pi}{4}), \frac{\pi}{4}\right) = (\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2})$$ and $$\left(3\sin(-\frac{3\pi}{4}), -\frac{3\pi}{4}\right) = (-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2})$$. The intersection points of the curves given by $$r=3 \sin \theta$$ and $$r=3 \cos \theta$$ are $$\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$$ and $$\left(-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2}\right)$$.

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

Use a graphing utility to graph the parabolas \(y^{2}=4 p x,\) for \(p=-5,-2,-1,1,2,\) and 5 on the same set of axes. Explain how the shapes of the curves vary as \(p\) changes.

Consider the following sequence of problems related to grazing goats tied to a rope. A circular concrete slab of unit radius is surrounded by grass. A goat is tied to the edge of the slab with a rope of length \(0 \leq a \leq 2\) (see figure). What is the area of the grassy region that the goat can graze? Note that the rope can extend over the concrete slab. Check your answer with the special cases \(a=0\) and \(a=2\)

Water flows in a shallow semicircular channel with inner and outer radii of \(1 \mathrm{m}\) and \(2 \mathrm{m}\) (see figure). At a point \(P(r, \theta)\) in the channel, the flow is in the tangential direction (counterclockwise along circles), and it depends only on \(r\), the distance from the center of the semicircles. a. Express the region formed by the channel as a set in polar coordinates. b. Express the inflow and outflow regions of the channel as sets in polar coordinates. c. Suppose the tangential velocity of the water in \(\mathrm{m} / \mathrm{s}\) is given by \(v(r)=10 r,\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.5, \frac{\pi}{4}\right)\) or \(\left(1.2, \frac{3 \pi}{4}\right) ?\) Explain. d. Suppose the tangential velocity of the water is given by \(v(r)=\frac{20}{r},\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.8, \frac{\pi}{6}\right)\) or \(\left(1.3, \frac{2 \pi}{3}\right) ?\) Explain. e. The total amount of water that flows through the channel (across a cross section of the channel \(\theta=\theta_{0}\) ) is proportional to \(\int_{1}^{2} v(r) d r .\) Is the total flow through the channel greater for the flow in part (c) or (d)?

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of the parabola \(y^{2}=4 p x\) or \(x^{2}=4 p y\) is \(4|p|\)

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1+\sin \theta}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

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.