/*! 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 15 Plot the points whose polar coor... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 15

Plot the points whose polar coordinates are given. $$(2,-\pi / 4)$$

Short Answer

Expert verified
Plot the point 2 units from the origin at an angle of \(-\frac{\pi}{4}\).

Step by step solution

01

Understand Polar Coordinates

Polar coordinates are given in the form \(r, \theta\), where \(r\) represents the radial distance from the origin, and \(\theta\) represents the angle measured from the positive x-axis.
02

Identify the Given Values

In this case, we have \(r = 2\) and \(+\theta = -\frac{\pi}{4}\). So the radial distance is 2 units and the angle is \(-\frac{\pi}{4}\).
03

Convert the Angle to Degrees (Optional)

Though not necessary, converting \(-\frac{\pi}{4}\) to degrees helps visualize the angle. \(-\frac{\pi}{4} = -45^{\circ}\). This means the point is 45 degrees below the positive x-axis.
04

Plot the Radial Distance

From the origin, mark 2 units along the direction corresponding to \(-45^{\circ} \, or\) \(-\frac{\pi}{4}\) radians.
05

Plot the Point

Place a point 2 units away from the origin in the direction of \(-\frac{\pi}{4}\).

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
In polar coordinates, the radial distance, denoted as \( r \), is the distance from the origin to the point. It's similar to the radius of a circle.
For example, in the exercise with coordinates \( (2, -\pi / 4) \), the value of \( r \) is 2. This means the point is located 2 units away from the origin.
Think of it as if you are drawing a line from the origin to the point. The length of this line is the radial distance.
angle conversion
Angles in polar coordinates are measured from the positive x-axis. The angle, denoted as \( \theta \), can be in radians or degrees.
Sometimes, converting the angle into degrees can make it easier to visualize.
For instance, in the exercise \( \theta = - \frac{\pi}{4} \). Converting this into degrees, it becomes -45掳. This tells us the direction of the point is 45掳 below the positive x-axis.
The conversion between radians and degrees is given by the formula:
\[ \theta (degrees) = \theta (radians) \times \frac{180}{\pi} \]
coordinate plotting
Once you understand the radial distance and the angle, you can plot the point on a polar grid.
Follow these steps for plotting:
  • Start from the origin.
  • Move according to the radial distance. For our example, it's 2 units.
  • Use the angle you have. Here, \(-\pi/4\) (-45掳).
You then place your point 2 units from the origin in the \(-\pi/4\) direction.
This way, you accurately plot your point in the polar coordinate system.
negative angle
Negative angles in polar coordinates are measured in the clockwise direction from the positive x-axis.
In the exercise, the angle is \(-\frac{\pi}{4} \), which means we start from the positive x-axis and rotate 45掳 clockwise.
Remember, positive angles rotate counterclockwise, while negative angles rotate clockwise.
This is important for correctly plotting the points as per the given angles.

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

For each given complex number, determine its complex conjugate in trigonometric form. $$3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$

Write each vector as a linear combination of the unit vectors i and \(\mathbf{j}\). $$\langle- 1 / 2,0\rangle$$

In each case, find the magnitude of the resultant force and the angle between the resultant and each force. Forces of 4.2 newtons (a unit of force from physics) and 10.3 newtons act at an angle of \(130^{\circ}\) to each other.

Given that \(\mathbf{A}=\langle 3,1\rangle\) and \(\mathbf{B}=\langle- 2,3\rangle,\) find the magnitude and direction angle for each of the following vectors. $$\mathbf{A}+\mathbf{B}$$

Find all solutions to the equation \(4 \sin ^{2}(3 x)-3=0\) in the interval \((0, \pi)\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

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.