Chapter 9: Q. 78 (page 568)

The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
$r = \sin θ - \cos θ$

Short Answer

Expert verified

The equation is written using rectangular coordinates as $x^{2} + y^{2} + x - y = 0$ .

Step by step solution

Step 1. Given Information

We are given an equation $r = \sin θ - \cos θ$ .

We need to write the equation using rectangular coordinates.

Step 2. Concept used

If P is a point with polar coordinates $(r, θ)$ , the rectangular coordinates $(x, y)$ of P are given by

$x = r \cos θ y = r \sin θ$

and it implies that

$x^{2} + y^{2} = r^{2}$

Step 3. Convert into rectangular coordinates

In the given equation $r = \sin θ - \cos θ$ multiply both sides by $r$ and we get

$\begin{array}{rcl} r & = & \sin θ - \cos θ \\ r \cdot r & = & r (\sin θ - \cos θ) \\ r^{2} & = & r \sin θ - r \cos θ \end{array}$

Now using the relations substitute $x^{2} + y^{2}$ for $r^{2}$ , $y$ for $r \sin θ$ , and $x$ for $r \cos θ$ we get

$\begin{array}{rcl} x^{2} + y^{2} & = & y - x \\ x^{2} + y^{2} + x - y & = & 0 \end{array}$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
$r = \sin θ - \cos θ$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept used

Step 3. Convert into rectangular coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Calculus

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

The letters r and θrepresent polar coordinates. Write each equation using rectangular coordinates (x,y).r=sinθ-cosθ

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept used

Step 3. Convert into rectangular coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Calculus

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
$r = \sin θ - \cos θ$