/*! 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 64 Are based on material learned ea... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 64

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. Plot the point given by the polar coordinates \(\left(-1, \frac{5 \pi}{4}\right)\) and find its rectangular coordinates.

Short Answer

Expert verified
The rectangular coordinates are \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\).

Step by step solution

01

- Understanding Polar Coordinates

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

- Plotting the Point

To plot the point given in polar coordinates \(\left(-1, \frac{5 \pi}{4}\right)\), first locate the angle \(\frac{5 \pi}{4}\). This angle is 225 degrees, which is in the third quadrant. Since the radius \(r\) is \(-1\), move 1 unit in the opposite direction (toward the first quadrant).
03

- Finding Rectangular Coordinates

Rectangular coordinates \((x, y)\) can be found using the formulas: x = r \cos(\theta)\ and y = r \sin(\theta)\. Substitute \(-1\) for \(r\) and \(\frac{5 \pi}{4}\) for \(\theta\).
04

- Calculate the x-coordinate

\(x = -1 \cos\left( \frac{5 \pi}{4} \right) = -1 \left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}\)
05

- Calculate the y-coordinate

\(y = -1 \sin\left( \frac{5 \pi}{4} \right) = -1 \left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}\)
06

- Combine the Coordinates

Therefore, the rectangular coordinates for the point \(\left(-1, \frac{5 \pi}{4}\right)\) are \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\).

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!

Key Concepts

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

Polar Coordinates
Polar coordinates are a way of representing points in a plane using two values: the radial distance from a reference point (usually the origin) and the angle from a reference direction (usually the positive x-axis). This system is particularly useful for problems involving circular and rotational symmetry.
  • The radial distance, denoted by \(r\), indicates how far from the origin the point is.
  • The angle, denoted by \(\theta\), specifies the direction of the point relative to the positive x-axis, measured counterclockwise.
For example, in the given exercise, the point \((-1, \frac{5 \pi}{4})\) has:
  • \(r = -1\): The point is 1 unit away from the origin, but the negative sign indicates it is in the direction opposite to \(\theta\).
  • \( \theta = \frac{5 \pi}{4} \) (or 225 degrees): This angle places the direction in the third quadrant.
By understanding these two components, you can locate any point on a plane using polar coordinates.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, represent points in a plane using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point is described by an ordered pair \((x, y)\), where
  • \(x\) is the horizontal distance from the y-axis, or how far left or right the point is.
  • \(y\) is the vertical distance from the x-axis, or how far up or down the point is.
The transformation between polar and rectangular coordinates involves trigonometric functions:
  • \(x = r \cos(\theta)\)
  • \(y = r \sin(\theta)\)
For the given exercise:
  • \(r = -1\)
  • \( \theta = \frac{5 \pi}{4} \)
  • Using the trigonometric functions, we calculate: \( x = -1 \cos\left(\frac{5 \pi}{4} \right) = \frac{\sqrt{2}}{2}\) and \(y = -1 \sin\left(\frac{5 \pi}{4} \right) = \frac{\sqrt{2}}{2}\).
So, the rectangular coordinates are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)\).
Trigonometric Functions
Trigonometric functions are essential in converting between polar and rectangular coordinates. These functions relate the angles and sides of right triangles to each other and are fundamental in various areas of mathematics and engineering.
  • \( \cos(\theta)\) and \( \sin(\theta)\) are the primary trigonometric functions used in such conversions.
  • \( \cos(\theta)\) represents the ratio of the adjacent side to the hypotenuse in a right triangle.
  • \( \sin(\theta)\) represents the ratio of the opposite side to the hypotenuse in a right triangle.
For the exercise:
  • \( \theta = \frac{5 \pi}{4} \) places the terminal side of the angle in the third quadrant, where both sine and cosine values are negative.
  • Given \(r = -1\), we effectively take the opposite direction, resulting in positive coordinates when using the trigonometric functions.
  • \( \cos\left( \frac{5 \pi}{4} \right) = -\frac{\sqrt{2}}{2} \) and \( \sin\left( \frac{5 \pi}{4} \right) = -\frac{\sqrt{2}}{2} \). Multiplying by \(-1\) gives us \( \frac{\sqrt{2}}{2} \) for both x and y coordinates.
Understanding trigonometric functions is key to mastering conversions between different coordinate systems.

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

Write a brief paragraph outlining your strategy for solving a system of two linear equations containing two variables.

A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, 6 min for filling, and 1 min for labeling. Each case of grapefruit juice requires 10 min for sterilizing, 4 min for filling, and 2 min for labeling. Each case of tomato juice requires 12 min for sterilizing, 4 min for filling, and 1 min for labeling. If the company runs the sterilizing machine for 398 min, the filling machine for 164 min, and the labeling machine for 58 min, how many cases of each type of juice are prepared?

Write \(\cos \left(\csc ^{-1} u\right)\) as an algebraic expression in \(u\)

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 the area of the triangle with vertices at \((0,5),(3,9),\) and (12,0)

A landscape company is hired to plant trees in three new subdivisions. The company charges the developer for each tree planted, an hourly rate to plant the trees, and a fixed delivery charge. In one subdivision it took 166 labor hours to plant 250 trees for a cost of \(\$ 7520 .\) In a second subdivision it took 124 labor hours to plant 200 trees for a cost of \(\$ 5945 .\) In the final subdivision it took 200 labor hours to plant 300 trees for a cost of \(\$ 8985 .\) Determine the cost for each tree, the hourly labor charge, and the fixed delivery charge.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.