/*! 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 67 Convert to a rectangular equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 67

Convert to a rectangular equation. $$r=5 \sec \theta$$

Short Answer

Expert verified
The rectangular equation is \( x = 5 \).

Step by step solution

01

Understand the given polar equation

The given polar equation is: \[ r = 5 \sec \theta \] This equation involves the polar coordinates where \( r \) is the radius and \( \theta \) is the angle.
02

Recall the definition of secant

Recall that \( \sec \theta \) is the reciprocal of \( \cos \theta \). So, \[ \sec \theta = \frac{1}{\cos \theta} \] We can rewrite the given equation to express \( r \) in terms of \( \cos \theta \) instead.
03

Substitute \( \sec \theta \)

Substitute \( \sec \theta \) with \( \frac{1}{\cos \theta} \). So the equation becomes: \[ r = \frac{5}{\cos \theta} \] Multiply both sides by \( \cos \theta \) to clear the fraction: \[ r \cos \theta = 5 \]
04

Use the polar-to-rectangular conversion formulas

Recall the polar-to-rectangular conversion formulas: \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \] From the equation obtained after Step 3, we have \( r \cos \theta = x \). Thus, \[ x = 5 \]
05

State the rectangular equation

The final rectangular equation, obtained from converting the given polar equation, is: \[ x = 5 \]

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 provide a way to locate a point in the plane using a radius and an angle rather than traditional \(x\) and \(y\) coordinates. In polar coordinates, each point is defined by two values:
  • **\[r\]**: The distance from the origin to the point.
  • **\[\theta\]**: The angle measured from the positive x-axis in a counter-clockwise direction.
Imagine a circle centered at the origin. Any point on this circle can be described by how far it is from the center (\[r\]) and what angle it is at (\[\theta\]). Converting between polar and rectangular coordinates can help in solving many kinds of problems where angles and distances are crucial.
Rectangular Coordinates
Rectangular, or Cartesian, coordinates are the most common way to describe a point in the plane. Here, each point is defined by two orthogonal values:
  • **\[x\]**: The horizontal distance from the origin.
  • **\[y\]**: The vertical distance from the origin.
. The standard formulas to convert from polar coordinates (\[r,\theta\]) to rectangular coordinates (\[x,y\]) are:
  • \[ x = r \cos \theta \]
  • \[ y = r \sin \theta \]
The exercise you're working on involves these conversion formulas. In our problem, we used the formula \[ x = r \cos \theta \] to transform our given polar equation into its rectangular form.
Secant Function
The secant function, written as \[ \sec \theta \], is closely related to the cosine function. Specifically, it's the reciprocal of cosine:
  • **\[\sec \theta = \frac{1}{\cos \theta} \]**
Understanding this relationship is key to converting polar equations that involve \[ \sec \theta \] into a form involving cosine, which can then be transformed into rectangular coordinates. In the exercise, we converted \[ \sec \theta \] into \[ \frac{1}{\cos \theta} \], and then substituted it into the polar equation to facilitate the conversion to rectangular form. This process is a fundamental part of working with trigonometric identities in coordinate transformations.

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

Find the angle between the given vectors, to the nearest tenth of a degree. $$\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}, \mathbf{v}=-\mathbf{i}+4 \mathbf{j}$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$y-3=17 x$$

Multiply: \((1-i)(1+i)\)

Solve.$$2 x-4=x+8$$

Find the magnitude and the direction angle \(\theta\) of the vector. $$\mathbf{v}=\left\langle-\frac{1}{2}, \frac{\sqrt{3}}{2}\right\rangle$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure 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.