Precalculus Review Sheet #11
Name __________________
This should be completed without the aid of a calculator, except for number 12.
4
5
3
3,
1. Change to rectangular coordinates: a.
b. 2,
c. 5,
3
6
2
2. Change to polar coordinates (one with r>0 and one with r<0):
a. (3 3, 3)
b. (0, 7)
c. 5 2, 5 2
3. Plot each of the following on a polar coordinate plane:
a.
b. ( 3,150o )
2,
4
4. Change to an equation with polar coordinates:
a. x 2 y 2 2 x 4 y 0
5. Change to an equation with rectangular coordinates:
a. r 6sin
b. r = 4
6. Change each of the following complex numbers to trigonometric form 0
a. 3 3i
b.
3 i
2 i 2
c.
d.
1
2
i 3
2
7. Change each of the following complex numbers to standard form:
5
5
i sin
a. 3(cos 315o i sin 315o )
b. 2 3 cos
6
6
2
e. 2i
c.
2
cis315o
3
8. Perform the indicated operation. All answers should be in trigonometric form. 0
a. z1z 2 where z1
b.
z1
where z1
z2
c. z1z 2 where z1
d.
z1
where z1
z2
2 cos
2 cos
5
9
5
9
i sin
i sin
5
9
2
cis45o and z 2
3
2
cis45o and z 2
3
5
9
and z 2
and z 2
5
5
i sin
36
36
5
5
3 cos
i sin
36
36
3 cos
3
cis60
4
3
cis60
4
9. Find the indicated power (Leave answers in indicated form):
a. [2cis40o ]5 (Trigonometric)
b. (2cis30o ) 6 (Standard)
c. (2 2i 3)5 (Trigonometric)
10. Find the indicated roots of the following complex numbers:
a. 4th roots of z 3 cos100 i sin100
b. 5th roots of z
1 i
1
:
2
:
11. Solve each of the following over 0,3
a. 2 cos 2 (4x) cos(4x) 1 0
2
:
b. tan 2
3
3 0
12. The monthly sales (in thousands of units) of a seasonal product are approximated by
s 74.50 43.75sin
t
where t is the time in months, with t=1 corresponding to January, t=2
6
corresponding to February, etc. Determine the months when sales exceed 100,000 units.
Answer: t = 1.19 to t = 4.81 so Early January through late April
Test Notes: The following are topics that will be covered on the test:
1. Rectangular and Polar Coordinates of ordered pairs of real numbers
Change between the different forms
Change between the equations
Plot points on polar coordinate plane
2. Trigonometric and Standard Form of Complex Numbers
Change between the forms
Multiply and divide complex numbers (Must be in trig form to perform the operations)
DeMoivre’s Theorem to raise a complex number to a power (Must be in trig form; one solution)
Find the roots of a complex number (Must be in trig form; Multiple solutions)
1. Solve trig equations over an interval
Argument of the trig expression is x
Argument of the trig expression is a multiple of x
2
Answers:
3 3 3
,
2 2
(1)
(a)
(2)
(a) 6,
(3)
(a)
(4)
(a) r
(5)
(a) x 2
(6)
(a) 3 2cis
11
6
(b)
&
6,
5
6
(b) 7,
3
2
&
7,
(c) 10,
2
5
4
&
10,
(b)
4sin
2cos
y2 6 y
(e) 2cis
7
4
(b) x 2
0
(b) 2cis
3 2
2
3i 2
2
(a)
(8)
(a) 6cis
(9)
(a) 32cis 200
25
36
(c) 2cis
5
4
(b) 3 i 3
(b)
4
3cis 25
4
3cis115
4
3cis 205
4
3cis 295
(11) (a) x
16
5
6
(d) 1cis
1
64
(b)
(12) Between January and end of April
2
3
5
3
i 2
3
(d)
(c) 1024cis
3
20
11
10
2cis
20
19
(b) 10 2cis
20
27
10
2cis
20
35
10
2cis
20
,
1
cis105
2
(c)
2cis
2 5 7 4
, ,
,
and x
6 3 3 6 6 3
,
(c)
2
5
cis
3
12
10
(a)
y2
2
(7)
(10)
(c) 0, 5
3, 1
0,
2
10
2cis
,
7
4
(b)
[ t 1.19 and t
3
4.81 ]
8
cis345
9
3
4