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
π
0
6
π
3
π
2
π
2
3
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