/*! 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 21 Plot each point given in polar c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 21

Plot each point given in polar coordinates. $$ \left(3, \frac{\pi}{2}\right) $$

Short Answer

Expert verified
The point (3, \( \frac{\pi}{2} \)) is plotted 3 units up along the positive y-axis.

Step by step solution

01

- Understand Polar Coordinates

In polar coordinates, a point is represented as \(r, \theta\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.
02

- Identify the Given Coordinates

The given point in polar coordinates is \(3, \frac{\pi}{2}\). This means the radial distance \(r\) is 3 units, and the angle \(\theta\) is \(\frac{\pi}{2}\) radians.
03

- Interpret the Angle

Convert the angle \(\frac{\pi}{2}\) radians to degrees for simpler interpretation. \(\frac{\pi}{2} = 90^{\circ}\). This means the angle points straight up along the positive y-axis.
04

- Plot the Point

To plot the point \(3, \frac{\pi}{2}\), move 3 units from the origin along the angle \(90^{\circ}\). The point will lie on the positive y-axis, 3 units above the origin.

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.

Radial Distance
Polar coordinates use 'radial distance' to describe how far a point is from the origin. In the given example, \((3, \frac{\pi}{2})\), the radial distance is represented by \(r = 3\). This means the point is 3 units away from the origin. Think of radial distance as the length of a line from the center of a circle to any point on the edge.

  • Always make sure to measure the distance accurately.
  • If \(r < 0\), the point is in the opposite direction.
  • Radial distance determines 'how far,' not 'where.'
Remember, the radial distance can be any non-negative number that shows the length from the origin to the point.
Angle Measurement
In polar coordinates, the angle \( \theta \) plays a crucial role. The angle is measured from the positive x-axis and describes the direction of the point. For the point \( (3, \frac{\pi}{2}) \), \( \theta = \frac{\pi}{2} \) radians.

The angle informs us about orientation:
  • Angles between 0 and \( \pi \) are in quadrants I and II.
  • Angles between \( \pi \) and \( 2\pi \) are in quadrants III and IV.
  • Just like a clock, angles determine the 'how far around' direction.
Make sure to identify the correct quadrant when plotting points.
Radian to Degree Conversion
Angles in polar coordinates are often given in radians. It helps to convert radians to degrees for easier visualization. Use the conversion formula: \[\theta (degrees) = \theta (radians) \times \frac{180}{\pi}\] For \( \frac{\pi}{2} \) radians:
  • Multiply by \( \frac{180}{\pi} \)
  • \( \frac{\pi}{2} \times \frac{180}{\pi} = 90^{\circ} \)
  • So, \( \frac{\pi}{2} = 90^{\circ}\)
Converting gives a clear, visual interpretation and helps place the angle confidently in the coordinate system.
Coordinate System
The polar coordinate system is different from the Cartesian system. Instead of \(x, y\) coordinates, we use (r, \( \theta \)) to locate a point. The given point \( (3, \frac{\pi}{2}) \) can be plotted by:

  • Moving 3 units from the origin.
  • Along an angle of \( \frac{\pi}{2} = 90^{\circ} \).
  • This places the point on the y-axis, 3 units above the origin.
In this system, radial distance and angle together define the point's position. It simplifies plotting circular and rotational problems, making it a preferred choice in many applications.

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 letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ x=4 $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find all asymptotes of the graph of \(f(x)=\frac{2 x^{2}-5}{x^{2}-2 x-15}\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write the equation of the circle in standard form: $$ x^{2}+v^{2}-20 x+4 y+55=0 $$

Solve: \(\log _{4}(x+3)-\log _{4}(x-1)=2\).

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ 4 x^{2} y=1 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

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.