Chapter 9: Q. 76 (page 568)

The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
$r = \sin θ + 1$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

We are given an equation $r = \sin θ + 1$ .

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. Modify the relations

As $x^{2} + y^{2} = r^{2}$ , so we can take square root both sides and see that

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

Now substitute $\sqrt{x^{2} + y^{2}}$ for $r$ in the rectangular coordinates formula

$\begin{array}{rcl} x & = & r \cos θ \\ x & = & \sqrt{x^{2} + y^{2}} \cos θ \\ \frac{x}{\sqrt{x^{2} + y^{2}}} & = & \cos θ \end{array}$

and

$\begin{array}{rcl} y & = & r \sin θ \\ y & = & \sqrt{x^{2} + y^{2}} \sin θ \\ \frac{y}{\sqrt{x^{2} + y^{2}}} & = & \sin θ \end{array}$

Step 4. Convert into rectangular coordinates

In the given equation $r = \sin θ + 1$ , substitute $\sqrt{x^{2} + y^{2}}$ for $r$ and $\frac{y}{\sqrt{x^{2} + y^{2}}}$ for $\sin θ$ we get

$\begin{array}{rcl} r & = & \sin θ + 1 \\ \sqrt{x^{2} + y^{2}} & = & \frac{y}{\sqrt{x^{2} + y^{2}}} + 1 \\ \sqrt{x^{2} + y^{2}} & = & \frac{y + \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}} \\ {(\sqrt{x^{2} + y^{2}})}^{2} & = & y + \sqrt{x^{2} + y^{2}} \\ x^{2} + y^{2} - y - \sqrt{x^{2} + y^{2}} & = & 0 \end{array}$

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The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
$r = \sin θ + 1$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Modify the relations

Step 4. Convert into rectangular coordinates

One App. One Place for Learning.

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The letters r and θrepresent polar coordinates. Write each equation using rectangular coordinates x,y.r=sinθ+1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Modify the relations

Step 4. Convert into rectangular coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Logic and Functions

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
$r = \sin θ + 1$