Complex Numbers, calculus homework help

  

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For questions 1-3 the polar coordinates of a point are given. Find the rectangular coordinates of each
point.





2 
1.)  5, 
 4
2.)  2, 
6

3.)  1,
3 

For questions 4-6 the rectangular coordinates of a point are given. Find the polar coordinates of each
point.
4.)  4, 0 
5.)
 0, 3 
6.)
 2, 2
7.) Give two sets of polar coordinates that could be used to plot the given point.
a.)
b.)
8.) Graph each polar equation. Write the scale you are using for the polar axis.
(a) r  9cos 5
(b) r  2cos 
9.) Transform each polar equation to an equation in rectangular coordinates and identify its
shape.

r 

6
4
2cos   3 sin 
10.)
(a) 5
Compute the modulus and argument of each complex number.
(b) 5 – 5i
11.)
Write each complex number in rectangular form. Plot and label each point on the
polar axes below.

(a) 2 cos135  i sin135

3 cos240  i sin240


5
5 

5  cos
 i sin 
4
4 

3
3 

4  cos
 i sin 
2
2 




2  cos  i sin 
5
5



5 cos100  i sin100
12.)
Let z 

5 3 5
 i and w  1  3i .
2
2
(a) Convert z and w to polar form.
(b) Calculate zw.
(c) Calculate
z
.
w
For questions 13-12, let
  
  , z
z1  3 cos 60  i sin 60
2
  
  , z
 2 cos 35  i sin 35
3
  
  . Calculate the
 4 cos 45  i sin 45
following, keeping your answer in polar form.
13.)
z1  z2
14.)
z3  z2
15.)
z2 z3
16.)
z3
17.)
z1
z3
18.)
z1
; Use your answer to #15:
z2 z3
For questions 19-22, write each expression in the standard form for a complex number, a + bi.
    i sin  45 
3
  
5
19.)
 4 cos 45

20.)
1
 2 cos 72  i sin 72

 
; Use De Moivre’s Theorem
; Use De Moivre’s Theorem
21.)
The complex fourth roots of 5  5 3i .
22.)
The complex cube roots of 8  cos    i sin    .
 5 
 5 

23.)
Find all seventh roots of unity and sketch them on the axes below.

 4 
 4  
24.) Give two sets of coordinates that could be used to plot the given point.
1. Answer:
25.) The polar coordinates of a point are given. Find the rectangular coordinates of each point.
 
 5, 
(a)  4 


 2, 6 

b.) 
Answer:
26.)The rectangular coordinates of a point are given. Find the polar coordinates of each point.
a.)
 4, 0 
 0, 3 
Answer:
27.) Transform each polar equation to an equation in rectangular coordinates and identify its shape.
a.) r = 6
b.) r  2cos 
Answer:
28.) Identify, graph, and state the symmetries for each polar equation. Write the scale that you are
using for the polar axis.
r  1  2cos 
r 2  cos  2 
Answer:
29.) Compute the modulus and argument of each complex number. Plot and label each complex
number in the complex plane given.
a.) 1 + i
b.)
3 i
c.) 2i
d.)
5
e.) 5 – 5i
f.)
7 3  7i
g.)
3  4i
Answer:
a.)
b.)
c.)
d.)
e.)
f.)
g.)
30.) Let z 
5 3 5
 i and w  1  3i .
2
2
a.)Convert z and w to polar form.
b.) Calculate zw.
c.) Calculate
z
.
w
Answer:
31.) Write each expression in the standard form for a complex number, a + bi.
  
  
 
5
3 cos 27  i sin 27 


b.) 
6
2 cos 40  i sin 40 


c.)
 
Answer:

 4 
 4  
32.) Find the complex cube roots of 8  cos 
  i sin  5   .
5





Answer:
33.) Find all seventh roots of unity and sketch them on the axes below.
Answer:
…
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