Topics in Mathematics 201-BNJ-05
Vincent Carrier
Complex Exercises (II)
C.3
Polar Form
Write in polar form.
1. − 2
3. − 1 +
2. 5 + 5i
√
6. − 2 3 − 2i
5. 7 − 7i
√
4. − 8i
3i
8. − 3 + 4i
7. 6i
Write in standard form.
9. 9eπi
10. 4e7πi/6
11. 6e−4πi/3
12. 5ei arctan(4/3)
13. 2e−5πi/4
14. 3ei[arctan(2)+π]
Write in standard form.
15. (1 − i)13
18. (1 + i)6 (1 −
16. (−1 +
√ 4
3i)
19.
√ 8
3i)
17. (−1 − i)−9
√
(− 3 + i)3
20.
(−1 − i)5
(−1 + i)6
√
( 3 + i)7
Use Euler’s identity eix = cos x + i sin x and the fact that ei(x+y) = eix eiy to prove the sine
and cosine addition formulas.
21. sin(x + y) = sin x cos y + cos x sin y
22. cos(x + y) = cos x cos y − sin x sin y
C.4
Roots
Find all complex numbers z (in polar form) that satisfy the following equations.
23. z 3 = 1
24. z = i1/2
26. z 2 = −i
27. z = 11/8
29. z = (−8i)1/3
30. z 2 = −1 +
25. z = (−1)1/6
√
28. z 4 = −1 − 3i
√
3i
31. z = (−16)1/4
Find all values of the following.
32. i2/3
33. (−1)3/4
34. (−i)3/2
Answers
1. 2eπi+2πki
√
2. 5 2eπi/4+2πki
k∈Z
3. 2e2πi/3+2πki
√
5. 7 2e−πi/4+2πki
7. 6eπi/2+2πki
4. 8e3πi/2+2πki
k∈Z
k∈Z
6. 4e−5πi/6+2πki
k∈Z
k∈Z
k∈Z
8. 5e[arctan(−4/3)+π]i+2πki
k∈Z
k∈Z
9. − 9
√
10. − 2 3 − 2i
√
11. − 3 + 3 3i
12. 3 + 4i
13. −
√
3
6
14. − √ − √ i
5
5
15. − 64 + 64i
√
18. 64 3 + 64i
2+
√
2i
√
16. − 128 − 128 3i
17. −
√
1
3
i
19. − −
32
32
20. 1 + i
1
1
+ i
32 32
23. e0 , e2πi/3 , e4πi/3
24. eπi/4 , e5πi/4
25. eπi/6 , eπi/2 , e5πi/6 , e7πi/6 , e3πi/2 , e11πi/6
26. e3πi/4 , e7πi/4
32. eπi/3 , e5πi/3 , eπi
27. 1, eπi/4 , eπi/2 , e3πi/4 , eπi , e5πi/4 , e3πi/2 , e7πi/4
33. eπi/4 , e3πi/4 , e5πi/4 , e7πi/4
28.
√
4
2eπi/3 ,
√
4
2e5πi/6 ,
√
4
2e4πi/3 ,
√
4
29. 2eπi/2 , 2e7πi/6 , 2e11πi/6
30.
√
2eπi/3 ,
√
2e4πi/3
31. 2eπi/4 , 2e3πi/4 , 2e5πi/4 , 2e7πi/4
2e11πi/6
34. eπi/4 , e5πi/4