/*! 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 Rectangular-to-Polar Conversion ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 21

Rectangular-to-Polar Conversion In Exercises 15- 24 , the rectangular coordinates of a point are given. Plot the point and find two sets. of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$(\sqrt{7},-\sqrt{7})$$

Short Answer

Expert verified
The polar coordinates for the point given by the rectangular coordinates \((\sqrt{7},-\sqrt{7})\) are \((\sqrt{14}, 胃)\). Here, 胃 is obtained as mentioned in step 2, providing two distinct values of 胃 for \(0 \leq 胃 < 2蟺\)

Step by step solution

01

Compute for r

The distance, r, from the origin to the point can be computed using the Pythagorean theorem. For the point given by the rectangular coordinates \((\sqrt{7},-\sqrt{7})\), we compute r as: \( r = \sqrt{x^2 + y^2} = \sqrt{(\sqrt{7})^2 + (-\sqrt{7})^2} = \sqrt{14} \) So, the radius or distance from the origin to the point is \(\sqrt{14}\).
02

Compute for 胃

Next, calculate the angle 胃 from the positive x-axis to the point, which can be found using the formula: \( 胃 = \text{atan2}(y, x) \) Here, atan2 is a two-argument variant of the arc tangent (atan) that takes x, y values rather than a single ratio. Plug in the given x and y values into this formula to find 胃: \( 胃 = \text{atan2}(-\sqrt{7},\sqrt{7}) \) The resulting angle falls in the fourth quadrant. However, we need to express 胃 such that it falls within \(0 \leq 胃 < 2蟺\). If the calculated angle is negative, add \(2蟺\) to bring it into the required range: \( 胃 = \text{atan2}(-\sqrt{7},\sqrt{7}) + 2蟺 \) The second set of polar coordinates can then be obtained by subtracting \(2蟺\) from the angle.

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.

Rectangular-to-Polar Conversion
Converting rectangular coordinates to polar coordinates involves representing a point in the plane by its distance from the origin and the angle it makes with the positive x-axis. This process helps to express points in a way that's often more natural for certain applications. In rectangular coordinates, a point is described by
  • its x-coordinate (horizontal position)
  • its y-coordinate (vertical position).
To convert these to polar coordinates, we look for two values:
  • the radius (distance from the origin) \(r\)
  • the angle \(\theta\) that describes the direction of the point.
Polar coordinates are typically written as \((r, \theta)\). The radius \(r\) is always positive, and \(\theta\) ranges from \(0\) to \(2\pi\). By understanding this conversion, we can easily switch between the two systems depending on what is required for the problem.
Angle Calculation
Calculating the angle \(\theta\) in polar coordinates involves using the rectangular coordinates to determine the direction of the point from the origin. The angle tells us how far the point is rotated about the origin, starting from the positive x-axis. Calculating \(\theta\) involves using trigonometric functions:
  • If both x and y are positive, the angle is straightforward as it falls in the first quadrant.
  • If either x or y is negative, the point lies in one of the other quadrants, and the angle reflects this.
Special care must be taken to ensure the angle falls within the required range of \(0 \leq \theta < 2\pi\). Often, if the computed angle from functions such as \(\text{atan2}\) is negative, adding \(2\pi\) rectifies this, providing us with a valid polar coordinate angle.
Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry that helps us find the distance \(r\) from the origin to any given point in the plane. For a point with coordinates \((x, y)\), \(r\) can be calculated as \[ r = \sqrt{x^2 + y^2} \] This formula arises from the relationship between the sides of a right triangle: the hypotenuse \(r\) is equal to the square root of the sum of the squares of the other two sides \(x\) and \(y\). This theorem not only helps convert rectangular coordinates to polar but also reinforces the understanding of distance as an extension of triangle properties. It remains essential for various applications like physics and engineering.
Atan2 Function
The \(\text{atan2}\) function is an an advanced computation tool that offers more control over angle calculations than the basic arc tangent \(\text{atan}\). This function takes two arguments, \(y\) and \(x\), representing the y-coordinate and x-coordinate of a point. Using \(\text{atan2}\), we can:
  • Determine the correct quadrant for the angle, making it more reliable for computing polar angles.
  • Avoid undefined values that can occur when y or x is zero, which is a limitation of \(\text{atan}\).
This method is especially useful because it computes angles in a range of \(-\pi\) to \(\pi\), simplifying the conversions in multidirectional calculations. For converting negative angles into the positive range \(0\) to \(2\pi\), simply add \(2\pi\). It enhances accuracy and simplifies challenges in polar coordinate conversions.

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

Finding an Angle In Exercises \(107-112,\) use the result of Exercise 106 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\) . Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta .\) Identify the angle \(\psi\) . \(r=2(1-\cos \theta) \quad \theta=\pi\)

Approximating Area Consider the circle \(r=8 \cos \theta\) (a) Find the area of the circle. (b) Complete the table for the areas \(A\) of the sectors of the circle between \(\theta=0\) and the values of \(\theta\) in the table. (c) Use the table in part (b) to approximate the values of \(\theta\) for which the sector of the circle composes \(\frac{1}{4}, \frac{1}{2}\) and \(\frac{3}{4}\) of the total area of the circle. (d) Use a graphing utility to approximate, to two decimal places, the angles \(\theta\) for which the sector of the circle composes \(\frac{1}{4}, \frac{1}{2},\) and \(\frac{3}{4}\) of the total area of the circle (e) Do the results of part (d) depend on the radius of the circle? Explain..

Finding the Area of a Surface of Revolution In Exercises \(65-68\) , find the area of the surface formed by revolving the polar equation over the given interval about the given line. $$\begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\\ {r=a(1+\cos \theta)} & {0 \leq \theta \leq \pi}\end{array} \begin{array}{ll}{\text { Axis of Revolution }} \\ {\text Polar \quad axis}\end{array}$$

Finding the Arc Length of a Polar Curve In Exercises \(59-64\) , use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve. $$r=\frac{1}{\theta}, \quad[\pi, 2 \pi]$$

Finding the Area of a Polar Region Between Two Curves In Exercises \(45-48\) , find the area of the region. $$Inside r=a(1+\cos \theta)\( and outside \)r=a \cos \theta$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.