QUIZ SOLUTIONS:

1.! Convert the following equations from rectangular to polar form.

•! 2x + 7y = 0

2rcos! + 7rsin! = 0 x = r cos!, y = r sin!

r (2cos! + 7sin!) = 0 Factor out r.

2.! Convert the following equations from polar to rectangular form.

•! r(-2 sin! + 3 cos!) = 2

r (-2sin! + 3cos!) = 2 Expand the left side of the given

equation.

-2rsin t + 3rcos t = 2 Distribute r.

-2y + 3x = 2 y = r sin! and x = r cos!

-2y = -3x + 2 Separate out the y variable.

y = 3x/2 – 1 Equation of a line.

3.! Convert the following rectangular coordinates to polar coordinates.

a.)! (1, 3)

! = (3/1) = (3) = 1.25

r = = = = 3.16

(r, !) = ( , 1.25)

b.)! (2.35, -4.81)

! = (-4.81/2.35) = -1.11

r = = 3.35

(r, !) = (3.35, -1.11)

4.! Convert the following polar coordinates to rectangular coordinates.

a.)! (4.27, 14"/15)

x = (4.27) cos(14"/15) = -4.18

y = (4.27) sin(14"/15) = 0.89

(x, y) = (-4.18, 0.89)

b.)! (-2, 3"/4)

x = (-2) cos(3"/4) = 1.414

y = (-2) sin(3"/4) = -1.414

(x, y) = (1.414, -1.414)

5.! Find the intersection point(s) between 0 # ! # 2" of the sets of polar equations

by hand.

a.)! r = cos!

r = 1 - cos!

cos! = 1 - cos!

1 + 2cos! = 0

cos! = -1/2 ! = -2$/3 , -4$/3

b.)! r = 2 - cos!

r = 3cos!

2 – cos! = 3cos!

4cos! = 2

cos! = % ! = $/3 , 5$/3

6.! Sketch the following polar equation by converting the equation into

rectangular form. Write the name of the shape sketched. Include arrows in

your sketch.

r = 5 + 5sin(!) Shape: limaçon

x = rcos!

y = rsin!

x = (5 + 5 sin!)cos!

y = (5 + 5sin!)sin!

7.! Sketch the following polar equation by completing the table. Write the name

of the shape sketched. Include arrows in your sketch.

r = 8cos2θ Shape: rose

Ө r

8

4

-4

-8

-4

4

8

6

π

3

π

2

π

2

3

π

5

6

π

π

0

6

π

3

π

2

π

2

3

π

5

6

π

π

0

Ө x y

5 0

6.496 3.75

4.665 9.080

0 10

-4.665 9.080

-6.496 3.75

-5 0