Write Equation Using Rectangular Coordinates

[greatwhiteshark]

For the following 3 questions, the letters r = radius and t = theta represent polar coordinates. Write each equation using rectangular coordinates.

1) r = sint + 1

2) r = 2

3) r = 4/1 - cost

 

[soroban]

Hello, greatwhiteshark!

For the following 3 questions, the letters r and θ represent polar coordinates.
Write each equation using rectangular coordinates.

The conversions are:
. . . r cos θ .= .x
. . . r sin θ .= .y . . . . . . . . . . . . . . _____
. . . . . . .r² .= .x² + y² . or: .r .= .√x² + y²

Moral: we want to have r cos θ or r sin θ, but usually we don't have them.
. . . . . . Often, we must <u>create</u> them. .(See #1 below.)

1) r = sin θ + 1

Multiply through by r: . r² .= .r sin θ + r
. . . . . . . . . . . . . . . . . . . . . . . ______
Substitute: . x² + y² .= .y + √x² + y²

Basically, we can stop here.
We can juggle those terms around, get rid of the square root, etc.
. . but it doesn't really simplify very much.

2) r = 2

Not much here to work with . . . just substitute fore the r.
. . . . ______
. . . √x² + y² .= .2 . . ---> . . x² + y² .= .4

[A circle, center at the origin, radius 2.]

3) r = 4/(1 - cos θ)

Clear the denominator: . r(1 - cos θ) .= .4

And we have: . r - r cos θ .= .4
. . . . . . . . . . . ______
Substitute: . √x² + y² - x .= .4

This one can be simplified a bit . . .
. . . . ______
. . . √x² + y² . = . x + 4

Square: . x² + y² . = . x² + 8x + 16

And we have: . y² . = . 8(x + 2)


You might recognize this as a parabola (or not) . . .
. . . vertex at (-2,0), opens to the right, with p = 4.

 

[greatwhiteshark]

ok

Your explanations are divine.