/*! 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 80 Convert the equation from polar ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 80

Convert the equation from polar coordinates into rectangular coordinates. $$ \theta=\frac{2 \pi}{3} $$

Short Answer

Expert verified
The polar equation \( \theta = \frac{2\pi}{3} \) converts to the rectangular line equation \( x = -\sqrt{3}y \).

Step by step solution

01

Understand Polar Coordinates

In polar coordinates, a point is represented by \[ (r, \theta) \] where \( r \) is the distance from the origin and \( \theta \) is the angle. The task is to convert the given polar equation into rectangular coordinates (\( x, y \)). We have \( \theta = \frac{2\pi}{3} \) without a specific \( r \) value, which means we have an angle direction but no specific radius.
02

Use Relationship Between Polar and Rectangular Coordinates

The relationships between polar and rectangular coordinates are given by:\[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]where \( x \) and \( y \) are the rectangular coordinates. Since no \( r \) is specified, focus on expressing \( \cos(\theta) \) and \( \sin(\theta) \) to find the line corresponding to \( \theta = \frac{2\pi}{3} \).
03

Calculate Trigonometric Values

First, find \( \cos(\theta) \) and \( \sin(\theta) \) for \( \theta = \frac{2\pi}{3} \). These are fundamental trigonometric positions:\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]\[ \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
04

Rewrite Equation in Rectangular Form

From the known relationships:- \( x = r \cos(\theta) \) becomes \( -x/2 \) when in terms of the given angle.- \( y = r \sin(\theta) \) becomes \( y\sqrt{3}/2 \).Since there鈥檚 no given \( r \), these variables relate as \[ y\sqrt{3} = -x \]This can be simplified as\[ \sqrt{3}y = -x \] or \[ x = -\sqrt{3}y \]representing the line through the origin with slope \(-\frac{1}{\sqrt{3}}\).
05

Conclusion: Rectangular Equation

The equation \( \theta = \frac{2\pi}{3} \) in polar coordinates is a line with direction angle only (angle without specific radius). It converts to the rectangular form: \[ x = -\sqrt{3}y \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a way to locate a point in the plane using a distance and an angle. Instead of the familiar
  • Cartesian system with axes and coordinates
  • polar coordinates are given as a pair
  • \((r, \theta)\) where:
    • \(r\) represents the radial distance from a fixed point called the origin.
    • \(\theta\) denotes the angle from a fixed direction, usually measured in radians from the positive x-axis.
This system is particularly useful for situations where length and angle measurement are more relevant than direct coordinates.
For example, when dealing with circular motion or spiral shapes.
In the exercise at hand, we have a given angle, \(\theta = \frac{2\pi}{3}\), but no specific radius.This implies we are focusing purely on the direction 鈥� representing a line rather than a specific point.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are defined by a pair
  • \((x, y)\) that describe a point's exact position on a plane using horizontal and vertical axes:
    • \(x\) is the horizontal distance from the origin.
    • \(y\) is the vertical distance.
These coordinates are straightforward, making them a standard choice for many mathematical and engineering problems.
They allow you to easily calculate distances and angles using algebra and geometry.
The exercise aims to transition from a polar representation where an angle is given, to this format. By doing so, we pinpoint how a direction defined in polar terms transforms into a line equation in rectangular terms.
Trigonometric Functions
Trigonometric functions are key to translating between polar and rectangular coordinates.
They relate the angles in polar coordinates to the respective x and y positions in rectangular coordinates.
  • \(\cos(\theta)\) helps determine the horizontal projection, and \(\sin(\theta)\) the vertical.
In our exercise, we're given \(\theta = \frac{2\pi}{3}\).
Calculating these gives:
  • \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\), indicating an acute negative horizontal component.
  • \(\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\), indicating a positive vertical component.
These values direct how we rewrite our direction as a straight line, bringing the angle into a rectangular context.
Coordinate Transformation
Coordinate transformation involves changing the description of a location from one coordinate system to another.
For problems like this exercise, it specifically addresses converting a point or line from polar to rectangular notation.
  • The process utilizes relationships:
    • \(x = r \cos(\theta)\)
    • \(y = r \sin(\theta)\)
Since \(r\) is not given in the original exercise, the interest lies in understanding directional components.
These relations transform into equations that describe the tilt of the line defined by the angle \(\theta = \frac{2\pi}{3}\).
This slide in representation gives equation \(x = -\sqrt{3}y\), a mathematical shift that helps visualize the directional nature of the original polar statement.

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

In Exercises \(41-50\), use set-builder notation to describe the polar region. Assume that the region contains its bounding curves. The region inside the circle \(r=4 \cos (\theta)\) which lies in Quadrant IV.

Convert the equation from polar coordinates into rectangular coordinates. $$ r^{2}=\sin (2 \theta) $$

In Exercises \(25-39\), find a parametric description for the given oriented curve. the circle \(x^{2}+y^{2}=25\), oriented counter-clockwise

In Exercises \(21-24\), plot the set of parametric equations with the help of a graphing utility. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=4 \cos ^{3}(t) \\ y=4 \sin ^{3}(t) \end{array} \text { for } 0 \leq t \leq 2 \pi\right. $$

The SS Bigfoot leaves Yeti Bay on a course of \(\mathrm{N} 37^{\circ} \mathrm{W}\) at a speed of 50 miles per hour. After traveling half an hour, the captain determines he is 30 miles from the bay and his bearing back to the bay is \(\mathrm{S} 40^{\circ} \mathrm{E}\). What is the speed and bearing of the ocean current? Round the speed to the nearest mile per hour and express the heading as a bearing, rounded to the nearest tenth of a degree.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.