NAME ________________________________
MATH 360:
DATE_________________________________
PERIOD _______
8.1 – 8.3 REVIEW
No calculators!
pö
æ
1) Let P be the point with polar coordinates ç 4, - ÷ . Rewrite the coordinates of P:
6ø
è
a) in polar form, with r > 0 and 0 < q < 2p .
b) in polar form, with r < 0 and 0 < q < 2p .
c) in polar form, with r < 0 and -2p < q < 0 .
d) in rectangular form.
2) Convert each set of the rectangular coordinates into polar form, using a positive coordinate
for r and an argument q Î [ 0, 2p ) . Give q exactly, as a value where possible or in terms of
inverse trigonometric functions when necessary.
a)
( -3,3)
b)
(3
3, - 3
)
c)
( -5, 0 )
d)
( -4, 2 )
3) Write each rectangular equation in polar form.
a)
( x - 1) + ( y + 1)
2
2
=2
b) x = 1
d) x + y = 6
c) y = x 3
4) Write each polar equation in rectangular form.
c) q =
b) r = 1 - sin q
a) r = cos 2q
3p
4
5) Use your calculator to help you graph each of the following polar curves. Label the exact
polar coordinates of at least three points on each curve.
æq ö
a) r = 5cos ( 3q )
b) r 2 = 4sin ( 2q )
c) r = cos ç ÷
è2ø
6) Determine whether the polar curve r = 1 + 2 cos q has symmetry about the pole, the polar axis,
or the line q =
p
2
.
p
æ pö
and P ç 3, ÷ lies on the
2
è 6ø
graph of r = f (q ) , determine which of the following points could represent the “mirror”
point of P on the graph.
7) Given a polar curve r = f (q ) is symmetric about the line q =
pö
æ
i) ç 3, - ÷
6ø
è
pö
æ
ii) ç -3, ÷
6ø
è
æ 5p ö
iii) ç 3,
÷
è 6 ø
8) True/False: Decide whether each statement is true or false.
1
æ1ö
a) If tan x = , then x = tan -1 ç ÷ + p k , k Î ! .
3
è3ø
2
æ2ö
b) If cos q = , then q = cos -1 ç ÷ .
3
è3ø
c) One possible set of polar coordinates for the rectangular point ( -3, - 4 ) is
æ
3 öö
-1 æ
ç 5, - cos ç - 5 ÷ ÷ .
è
øø
è
d) If r = f (q ) and f (q ) = f ( -q ) for all q , then the graph of r = f (q ) is symmetric
about the polar axis.
e) Replacing ( r , q ) with ( r , q + p ) in a polar equation is a test for graphical symmetry
across the pole.
f) If replacing r with -r (another test for symmetry across the pole) in a polar equation
does not produce an identical equation, then the graph is not symmetric across the pole.
g) The circle r = 4sin q is traced out exactly once on the interval q Î [ 0, 2p ] .
pö
æ
h) The graph of r = sin ç q - ÷ is the same as the graph of r = sin q rotated 30°
6ø
è
counterclockwise.
9) Match each polar graph (a)-(d) with its equation. (Some equations will not be used.)
I) r = 1 - cos q
a)
b)
c)
d)
e)
f)
II) r = 1 - sin q
III) r = 1 + 2 cos q
IV) r = 2 cos q
V) r = 1 - 2 cos q
VI) r = 2 + 2 cos q
VII) r = 2 cos 3q
VIII) r = 2sin 3q
IX) r = 2 cos 2q
10) The graphs of r = 1 + 2sin ( 2q ) are shown below—in both the Cartesian (x-y) rectangular
plane and in the polar plane. Points A, B, C, and D are shown on the Cartesian graph.
a) Plot the corresponding positions of points A, B, C, and D on the polar graph.
b) Determine the polar coordinates of point D.
11) The polar equation r = 8cos q - 6sin q represents a circle. Determine the center and radius
of the circle by rewriting the equation in rectangular form.
12) The graphs of r = 1 - sin q and r = sin q are shown in the plane below.
a) Determine a value of q where the graphs intersect each other in quadrant I.
b) Determine a value of q where the graphs intersect each other in quadrant II.
c) Note that the curves also intersect at the pole, but they reach the pole at different
values of q . Determine the first positive value of q when each curve reaches the
pole.
13) Convert each complex number into polar ( r cisq ) form. Give q exactly, as a value where
possible or in terms of inverse trigonometric functions when necessary.
b) -2 3 - 2i
a) 4 - 4i
æ 5p
14) Given z = 1 + i 3 and w = 2cis ç
è 6
c) 5 + 12i
d) -3 + 4i
ö
÷,
ø
a) write the product z × w in rectangular ( a + bi ) form.
b) write the quotient
w
in rectangular ( a + bi ) form.
z
c) evaluate z 9 , expressing your answer in both polar and rectangular forms.
d) evaluate
z9
, expressing your answer in both polar and rectangular forms.
w6
15) Given complex numbers z and w pictured below, graph each of the following in the
complex plane. Try to do these exercises geometrically, without determining the actual
Real and imaginary parts of z and w. (Note: The radial lines are equally spaced around the
circle.)
Im
a) 2z
b) w
c)
w
z
w
z
d) z 2
16) Solve each of the following equations for z. Give your answers in polar form, and graph
your solutions in the complex plane.
2
2
3 1
a) z 2 = -i
b) z 3 - 8i = 0
c) z 4 =
d) z 6 +
i
+ i=0
2
2
2 2
17) True/False: Determine whether each statement is true or false. You do not need to justify
your answers. In the statements below, z and w represent complex numbers.
a) z × z = z
b)
2
z = z
c) If z + z = 0 , then z is a Real number.
d) z + w = z + w
e) If z = w , then z 5 = w5 .
f) If z 5 = w5 , then z = w .
g) If z = rcisq , then z = rcis ( -q ) .
h) In problems #16(a), 16(b), 16(c), and 16(d), the sum of each set of solutions is 0.
18) Given z = 1 , prove that
1
= z.
z
**19) a) Using the sum formula for cosine, show that cos ( 3q ) = 4 cos3 q - 3cos q .
b) Let z = cos q + i sin q . Write z 3 :
i) in polar form, using DeMoivre’s Theorem.
ii) in a + bi form, using the Binomial Theorem.
Since (i) and (ii) represent the same complex number z 3 , their Real parts must be equal,
and their imaginary parts must be equal.
c) Show that by setting the Real parts equal to each other, we obtain the cos ( 3q ) identity
you found in part (a).
d) Show that by setting the imaginary parts equal to each other, we also obtain an
identity for sin ( 3q ) .
e) Use a similar process to the one outlined in parts (b)-(d) to discover identities for
cos ( 4q ) and sin ( 4q ) .