Complex Numbers Class Work - content.njctl.org
Transcript of Complex Numbers Class Work - content.njctl.org
Pre-Calc Polar & Complex #s ~1~ NJCTL.org
Complex Numbers – Class Work
Simplify using i.
1. √−16 2. √−36𝑏4 3. √−8𝑎2
4. √−32𝑥6𝑦7 5. √−16 ∙ √−25 6. √−8 ∙ √−10
7. 3𝑖 ∙ 4𝑖 ∙ 5𝑖 8. −2𝑖 ∙ 4𝑖 ∙ −6𝑖 ∙ 8𝑖 9. 𝑖9
10. 𝑖22 11. 𝑖75
Complex Numbers – Homework
Simplify using i.
12. √−81 13. √−121𝑏8 14. √−18𝑎6
15. √−48𝑥5𝑦6 16. √−9 ∙ √−4 17. √−12 ∙ √−75
18. 2𝑖 ∙ 5𝑖 ∙ 7𝑖 19. −𝑖 ∙ −3𝑖 ∙ −5𝑖 ∙ −7𝑖 20. 𝑖10
21. 𝑖23 22. 𝑖72
𝟒𝒊 𝟔𝒃𝟐𝒊 𝟐𝒂𝒊√𝟐
𝟒𝒙𝟑𝒚𝟑𝒊√𝟐𝒚 −𝟐𝟎 −𝟒√𝟓
−𝟔𝟎𝒊 𝟑𝟖𝟒 𝒊
−𝟏 −𝒊
𝟗𝒊 𝟏𝟏𝒃𝟒𝒊 𝟑𝒂𝟑𝒊√𝟐
𝟒𝒙𝟐𝒚𝟑𝒊√𝟑𝒙 −𝟔 −𝟑𝟎
−𝟕𝟎𝒊 𝟏𝟎𝟓 −𝟏
−𝒊 𝟏
Pre-Calc Polar & Complex #s ~2~ NJCTL.org
Adding, Subtracting, and Multiplying Complex Numbers – Class Work
Simplify
23. (6 + 5𝑖) + (4 + 3𝑖) 24. (7 + 4𝑖) + (−2 − 2𝑖)
25. (−3 − 2𝑖) + (3 − 𝑖) 26. (6 + 5𝑖) − (4 + 3𝑖)
27. (7 + 4𝑖) − (−2 − 2𝑖) 28. (−3 − 2𝑖) − (3 − 𝑖)
29. 5(4 − 2𝑖) 30. 2𝑖(−6 + 𝑖)
31. (6 + 5𝑖)(4 + 3𝑖) 32. (7 + 4𝑖)(−2 − 2𝑖)
33. (−3 − 2𝑖)(3 − 𝑖) 34. (8 − 3𝑖)(1 − 𝑖)
35. (4 − 2𝑖)2 36. (−6 + 𝑖)2
𝟏𝟎 + 𝟖𝒊 𝟓 + 𝟐𝒊
−𝟑𝒊 𝟐 + 𝟐𝒊
𝟗 + 𝟔𝒊 −𝟔 − 𝒊
𝟐𝟎 − 𝟏𝟎𝒊 −𝟐 − 𝟏𝟐𝒊
𝟗 + 𝟑𝟖𝒊 −𝟔 − 𝟐𝟐𝒊
−𝟏𝟏 − 𝟑𝒊 𝟓 − 𝟏𝟏𝒊
𝟏𝟐 − 𝟏𝟔𝒊 𝟑𝟓 − 𝟏𝟐𝒊
Pre-Calc Polar & Complex #s ~3~ NJCTL.org
Adding, Subtracting, and Multiplying Complex Numbers – Homework
Simplify
37. (2 + 3𝑖) + (8 + 2𝑖) 38. (4 + 9𝑖) + (−4 − 9𝑖)
39. (10 − 7𝑖) + (5 − 3𝑖) 40. (2 + 3𝑖) − (8 + 2𝑖)
41. (4 + 9𝑖) − (−4 − 9𝑖) 42. (10 − 7𝑖) − (5 − 3𝑖)
43. 6(5 − 6𝑖) 44. 2𝑖(4 − 3𝑖)
45. (2 + 3𝑖)(8 + 2𝑖) 46. (4 + 9𝑖)(−4 − 9𝑖)
47. (10 − 7𝑖)(5 − 3𝑖) 48. (−6 − 𝑖)(2 − 7𝑖)
49. (6 − 3𝑖)2 50. (−7 + 2𝑖)2
𝟏𝟎 + 𝟓𝒊 𝟎
𝟏𝟓 − 𝟏𝟎𝒊 −𝟔 + 𝒊
𝟖 + 𝟏𝟖𝒊 𝟓 − 𝟒𝒊
𝟑𝟎 − 𝟑𝟔𝒊 𝟔 + 𝟖𝒊
𝟏𝟎 + 𝟐𝟖𝒊 𝟔𝟓 − 𝟕𝟐𝒊
𝟐𝟗 − 𝟔𝟓𝒊 −𝟏𝟗 + 𝟒𝟎𝒊
𝟐𝟕 − 𝟑𝟔𝒊 𝟒𝟓 − 𝟐𝟖𝒊
Pre-Calc Polar & Complex #s ~4~ NJCTL.org
Dividing Complex Numbers – Class Work
Simplify
51. 2
𝑖 52.
3
4𝑖 53.
−2
3𝑖
54. 2+𝑖
𝑖 55.
2
1+𝑖 56.
3
2−𝑖
57. 2+𝑖
3+𝑖 58.
4−𝑖
3−2𝑖
Dividing Complex Numbers – Homework
Simplify
59. 3
𝑖 60.
2
5𝑖 61.
−4
7𝑖
62. 4−𝑖
𝑖 63.
8
3+𝑖 64.
2𝑖
4−𝑖
65. 2−𝑖
2+3𝑖 66.
5−𝑖
4−3𝑖
−𝟐𝒊 −𝟑
𝟒𝒊
𝟐
𝟑𝒊
𝟏 − 𝟐𝒊 𝟏 − 𝒊 𝟔+𝟑𝒊
𝟓
𝟕+𝒊
𝟏𝟎
𝟏𝟒+𝟓𝒊
𝟏𝟑
−𝟑𝒊 −𝟐
𝟓𝒊
𝟒
𝟕𝒊
−𝟏 − 𝟒𝒊 𝟏𝟐−𝟒𝒊
𝟓
−𝟐+𝟖𝒊
𝟏𝟕
𝟏−𝟖𝒊
𝟏𝟑
𝟐𝟑+𝟏𝟏𝒊
𝟐𝟓
Pre-Calc Polar & Complex #s ~5~ NJCTL.org
Graphing Complex Numbers – Class Work
Determine the quadrant of each of the following.
67. 9 – 3i 68. -2 + 4i
69. (5 + 4i) – (6 – 3i) 70. -3i(4 – 5i)
71. (2 + 3i)2 72. 3−i
i
73. 2
4+i 74.
5−3i
2+4i
Homework
Determine the quadrant of each of the following.
75. -7 – 3i 76. 5 - 4i
77. (3 + 2i) – (-5 + 4i) 78. (3 – i)(-4 + 5i)
79. (-1 + 5i)2 80. −2−i
3i
81. 4
3−i 82.
−6+2i
3−2i
IV II
II III
II III
IV III
III IV
IV II
III II
I III
Pre-Calc Polar & Complex #s ~6~ NJCTL.org
Polar Properties – Class Work
Name the point three other ways using polar coordinates.
83. [5,π
2] 84. [−4,
2π
3]
85. [3,−4π
7] 86. [−6,0]
Convert the point to rectangular form.
87. [5,π
2] 88. [−4,
2π
3]
89. [3,−4π
7] 90. [−6,0]
Convert the point to polar form.
91. ( 3, 6) 92. (-4, 2)
93. (1, 0) 94. (7, 7)
(−𝟓,𝟑𝝅
𝟐) , (𝟓, −
𝟑𝝅
𝟐) , (−𝟓, −
𝝅
𝟐) (𝟒,
𝟓𝝅
𝟑) , (𝟒, −
𝝅
𝟑) , (−𝟒, −
𝟒𝝅
𝟑)
(−𝟑, −𝟏𝟏𝝅
𝟕) , (−𝟑,
𝟑𝝅
𝟕) , (𝟑,
𝟏𝟎𝝅
𝟕) (𝟔, 𝝅), (𝟔, −𝝅), (−𝟔, 𝟐𝝅)
(𝟎, 𝟓) (𝟐, −𝟐√𝟑)
(−𝟎. 𝟔𝟔𝟕𝟔, −𝟐. 𝟗𝟐𝟒𝟖) (−𝟔, 𝟎)
(𝟑√𝟓, 𝟔𝟑. 𝟒𝒐) (𝟐√𝟓, 𝟏𝟓𝟑. 𝟒𝒐)
(𝟏, 𝟎𝒐) (𝟕√𝟐, 𝟒𝟓𝒐)
Pre-Calc Polar & Complex #s ~7~ NJCTL.org
Polar Properties – Homework
Name the point three other ways using polar coordinates.
95. [7,π
3] 96. [−6,
2π
5]
97. [2,−3π
5] 98. [3, 𝜋]
Convert the point to rectangular form.
99. [7,π
3] 100. [−6,
2π
5]
101. [2,−3π
5] 102. [3, π]
Convert the point to polar form.
103. ( -3, 2) 104. (-7, -8)
105. (5, 10) 106. (-7, 0)
(𝟕, −𝟓𝝅
𝟑) , (−𝟕,
𝟒𝝅
𝟑) , (−𝟕, −
𝟐𝝅
𝟑) (𝟔,
𝟕𝝅
𝟓) , (−𝟔, −
𝟖𝝅
𝟓) , (𝟔, −
𝟑𝝅
𝟓)
(𝟐,𝟕𝝅
𝟓) , (−𝟐,
𝟐𝝅
𝟓) , (−𝟐, −
𝟖𝝅
𝟓) (−𝟑, 𝟎), (−𝟑, 𝟐𝝅), (𝟑, −𝝅)
(𝟑. 𝟓, 𝟔. 𝟎𝟔𝟐) (−𝟏. 𝟖𝟓𝟒, −𝟓. 𝟕𝟎𝟔)
(−𝟎. 𝟔𝟏𝟖, −𝟏. 𝟗𝟎𝟐) (−𝟑, 𝟎)
(√𝟏𝟑, 𝟏𝟒𝟔. 𝟑𝒐)
(𝟓√𝟓, 𝟔𝟑. 𝟒𝒐)
(√𝟏𝟏𝟑, 𝟐𝟐𝟖. 𝟖𝒐)
(𝟕, 𝟏𝟖𝟎𝒐)
Pre-Calc Polar & Complex #s ~8~ NJCTL.org
Geometry of Complex Numbers – Class Work
Let a =3 + 4i and b= -2 + 5i, perform the operation and write the answer in complex, rectangular,
polar, and trigonometric forms.
107. a + b 108. b – a
109. ab 110. a2
111. b2 112. 3a2b
113. 𝑎 = 4(𝑐𝑜𝑠𝜋
4+ 𝑖𝑠𝑖𝑛
𝜋
4) and 𝑏 = 3(𝑐𝑜𝑠
7𝜋
6+ 𝑖𝑠𝑖𝑛
7𝜋
6), find ab.
114. 𝑐 = [5,2𝜋
5] and 𝑑 = [3,
4𝜋
6], find cd. 115. Find z if z[10, 80°]= [15, 140°]
𝟏 + 𝟗𝒊
(𝟏, 𝟗)
(√𝟖𝟐, 𝟖𝟑. 𝟕𝒐)
√𝟖𝟐(𝐜𝐨𝐬 𝟖𝟑. 𝟕 + 𝒊 𝐬𝐢𝐧 𝟖𝟑. 𝟕)
−𝟓 + 𝒊
(−𝟓, 𝟏)
(√𝟐𝟔, 𝟏𝟔𝟖. 𝟕𝒐)
√𝟐𝟔(𝐜𝐨𝐬 𝟏𝟔𝟖. 𝟕 + 𝒊 𝐬𝐢𝐧 𝟏𝟔𝟖. 𝟕)
−𝟐𝟔 + 𝟕𝒊
(−𝟐𝟔, 𝟕)
(𝟓√𝟐𝟗, 𝟏𝟔𝟒. 𝟗𝒐)
𝟓√𝟐𝟗(𝐜𝐨𝐬 𝟏𝟔𝟒. 𝟗 + 𝒊 𝐬𝐢𝐧 𝟏𝟔𝟒. 𝟗)
−𝟕 + 𝟐𝟒𝒊
(−𝟕, 𝟐𝟒)
(𝟐𝟓, 𝟏𝟎𝟔. 𝟑𝒐)
𝟐𝟓(𝐜𝐨𝐬 𝟏𝟎𝟔. 𝟑 + 𝒊 𝐬𝐢𝐧 𝟏𝟎𝟔. 𝟑)
−𝟐𝟏 − 𝟐𝟎𝒊
(−𝟐𝟏, −𝟐𝟎)
(𝟐𝟗, 𝟐𝟐𝟑. 𝟔𝒐)
𝟐𝟗(𝐜𝐨𝐬 𝟐𝟐𝟑. 𝟔 + 𝒊 𝐬𝐢𝐧 𝟐𝟐𝟑. 𝟔)
−𝟑𝟏𝟖 − 𝟐𝟒𝟗𝒊
(−𝟑𝟏𝟖, −𝟐𝟒𝟗)
(𝟕𝟓√𝟐𝟗, 𝟐𝟏𝟖. 𝟏𝒐)
𝟕𝟓√𝟐𝟗(𝐜𝐨𝐬 𝟐𝟏𝟖. 𝟏 + 𝒊 𝐬𝐢𝐧 𝟐𝟏𝟖. 𝟏)
𝟏𝟐(𝐜𝐨𝐬𝟏𝟕𝝅
𝟏𝟐+ 𝒊 𝐬𝐢𝐧
𝟏𝟕𝝅
𝟏𝟐)
(𝟏𝟓,𝟏𝟔𝝅
𝟏𝟓) 𝒛 = (
𝟑
𝟐, 𝟔𝟎𝒐)
Pre-Calc Polar & Complex #s ~9~ NJCTL.org
Geometry of Complex Numbers – Homework
Let a =7 - 3i and b= -3 - 8i, perform the operation and write the answer in complex, rectangular,
polar, and trigonometric forms.
116. a + b 117. a – b
118. b – a 119. ab
120. a2 121. b2
122. 3a 123. 3a2b
124. 𝑎 = 7(𝑐𝑜𝑠𝜋
3+ 𝑖𝑠𝑖𝑛
𝜋
3) and 𝑏 = 2(𝑐𝑜𝑠
5𝜋
6+ 𝑖𝑠𝑖𝑛
5𝜋
6), find ab.
125. 𝑐 = [12,7𝜋
4] and 𝑑 = [. 5,
5𝜋
3], find cd. 126. Find z if z[20, 100°]= [15, 140°]
𝟒 − 𝟏𝟏𝒊
(𝟒, −𝟏𝟏)
(√𝟏𝟑𝟕, 𝟐𝟗𝟎𝒐)
√𝟏𝟑𝟕(𝐜𝐨𝐬 𝟐𝟗𝟎 + 𝒊 𝐬𝐢𝐧 𝟐𝟗𝟎)
𝟏𝟎 + 𝟓𝒊
(𝟏𝟎, 𝟓)
(𝟓√𝟓, 𝟐𝟔. 𝟔𝒐)
𝟓√𝟓(𝐜𝐨𝐬 𝟐𝟔. 𝟔 + 𝒊 𝐬𝐢𝐧 𝟐𝟔. 𝟔)
−𝟏𝟎 − 𝟓𝒊
(−𝟏𝟎, −𝟓)
(𝟓√𝟓, 𝟐𝟎𝟔. 𝟔𝒐)
𝟓√𝟓(𝐜𝐨𝐬 𝟐𝟎𝟔. 𝟔 + 𝒊 𝐬𝐢𝐧 𝟐𝟎𝟔. 𝟔)
−𝟒𝟓 − 𝟒𝟕𝒊
(−𝟒𝟓, −𝟒𝟕)
(𝟔𝟓. 𝟏, 𝟐𝟐𝟔. 𝟐𝒐)
𝟔𝟓. 𝟏(𝐜𝐨𝐬 𝟐𝟐𝟔. 𝟐 + 𝒊 𝐬𝐢𝐧 𝟐𝟐𝟔. 𝟐)
𝟒𝟎 − 𝟒𝟐𝒊
(𝟒𝟎, −𝟒𝟐)
(𝟓𝟖, 𝟑𝟏𝟑. 𝟔𝒐)
𝟓𝟖(𝐜𝐨𝐬 𝟑𝟏𝟑. 𝟔 + 𝒊 𝐬𝐢𝐧 𝟑𝟏𝟑. 𝟔)
−𝟓𝟓 + 𝟒𝟖𝒊
(−𝟓𝟓, 𝟒𝟖)
(𝟕𝟑, 𝟏𝟑𝟖. 𝟗𝒐)
𝟕𝟑(𝐜𝐨𝐬 𝟏𝟑𝟖. 𝟗 + 𝒊 𝐬𝐢𝐧 𝟏𝟑𝟖. 𝟗)
𝟐𝟏 − 𝟗𝒊
(𝟐𝟏, −𝟗)
(𝟑√𝟓𝟖, 𝟑𝟑𝟔. 𝟖𝒐)
𝟑√𝟓𝟖(𝐜𝐨𝐬 𝟑𝟑𝟔. 𝟖 + 𝒊 𝐬𝐢𝐧 𝟑𝟑𝟔. 𝟖)
−𝟏𝟑𝟔𝟖 − 𝟓𝟖𝟐𝒊
(−𝟏𝟑𝟔𝟖, −𝟓𝟖𝟐)
(𝟏𝟕𝟒√𝟕𝟑, 𝟐𝟎𝟑𝒐)
𝟏𝟕𝟒√𝟕𝟑(𝐜𝐨𝐬 𝟐𝟎𝟑 + 𝒊 𝐬𝐢𝐧 𝟐𝟎𝟑)
𝟏𝟒(𝐜𝐨𝐬𝟕𝝅
𝟔+ 𝒊 𝐬𝐢𝐧
𝟕𝝅
𝟔)
(𝟔,𝟒𝟏𝝅
𝟏𝟐) (
𝟑
𝟒, 𝟒𝟎𝒐)
Pre-Calc Polar & Complex #s ~10~ NJCTL.org
Polar Equations and Graphs – Class Work
127. Draw the graph of 𝑟 = sin 𝜃 128. Draw the graph of 𝑟 = 3 + 𝑐𝑜𝑠𝜃
129. Draw the graph of 𝑟 = 5 130. Draw the graph of 𝜃 =2𝜋
3
131. Draw the graph of 𝑟𝑐𝑜𝑠𝜃 = 6
Pre-Calc Polar & Complex #s ~11~ NJCTL.org
Polar Equations and Graphs – Homework
132. Draw the graph of 𝑟 = 𝑐𝑜𝑠𝜃 133. Draw the graph of 𝑟 = 4 + 𝑠𝑖𝑛𝜃
134. Draw the graph of 𝑟 = −5 135. Draw the graph of 𝜃 =3𝜋
4
136. Draw the graph of 𝑟𝑠𝑖𝑛𝜃 = −6
Pre-Calc Polar & Complex #s ~12~ NJCTL.org
Rose Curves and Spirals – Class Work
137. How many petals and what is a petals length for 𝑟 = 4𝑐𝑜𝑠3𝜃? Draw the graph.
138. How many petals and what is a petals length for 𝑟 = 5𝑠𝑖𝑛6𝜃? Draw the graph.
139. How many petals and what is a petals length for 𝑟 = 2𝑐𝑜𝑠4𝜃? Draw the graph.
140. How many petals and what is a petals length for 𝑟 = 7𝑐𝑜𝑠5𝜃? Draw the graph.
141. What kind of spiral is 𝑟 = 3𝜃? 142. What kind of spiral is 𝑟 = 2𝜃 + 2?
3 petals
length: 4
12 petals
length: 5
8 petals
length: 2
5 petals
length: 7
Logarithmic Archimedes
Pre-Calc Polar & Complex #s ~13~ NJCTL.org
Rose Curves and Spirals – Homework
143. How many petals and what is a petals length for 𝑟 = 6𝑐𝑜𝑠2𝜃? Draw the graph.
144. How many petals and what is a petals length for 𝑟 = 4𝑠𝑖𝑛7𝜃? Draw the graph.
145. How many petals and what is a petals length for 𝑟 = 3𝑐𝑜𝑠6𝜃? Draw the graph.
146. How many petals and what is a petals length for 𝑟 = 5𝑐𝑜𝑠3𝜃? Draw the graph.
147. What kind of spiral is 𝑟 = 2𝜃? 148. What kind of spiral is 𝑟 = 3𝜃 + 1?
4 petals
length: 6
7 petals
length: 4
12 petals
length: 3
3 petals
length: 5
Logarithmic Archimedes
Pre-Calc Polar & Complex #s ~14~ NJCTL.org
Powers of Complex Numbers – Class Work
Compute the given power and write your answer in the original form.
149. ([3,60°])5 150. (4 (𝑐𝑜𝑠𝜋
5+ 𝑖𝑠𝑖𝑛
𝜋
5))
7
151. (5 − 6𝑖)6 152. (−5,9)8
153. If a tenth root of w is (3,8) what is w?
Homework
Compute the given power and write your answer in the original form.
154. ([9,80°])7 155. (5 (𝑐𝑜𝑠4𝜋
3+ 𝑖𝑠𝑖𝑛
4𝜋
3))
9
156. (−4 + 7𝑖)8 157. (−7, −3)10
158. If a sixth root of w is 7(𝑐𝑜𝑠0 + 𝑖𝑠𝑖𝑛0) what is w?
(𝟐𝟒𝟑, 𝟑𝟎𝟎𝒐) 𝟏𝟔𝟑𝟖𝟒( 𝐜𝐨𝐬𝟕𝝅
𝟓+ 𝒊 𝐬𝐢𝐧
𝟕𝝅
𝟓)
𝟏𝟏𝟕𝟒𝟔𝟗 + 𝟏𝟗𝟒𝟐𝟐𝟎𝒊 (−𝟕𝟔𝟗𝟔𝟓𝟏𝟎𝟒, −𝟏𝟎𝟎𝟎𝟕𝟒𝟐𝟒𝟎)
(𝟏𝟖𝟕𝟎𝟏𝟖𝟏𝟐𝟐𝟓, −𝟖𝟗𝟒𝟒𝟓𝟒𝟎𝟑𝟐)
(𝟒𝟕𝟖𝟐𝟗𝟔𝟗, 𝟐𝟎𝟎𝒐) 𝟏𝟗𝟓𝟑𝟏𝟐𝟓( 𝐜𝐨𝐬 𝟐𝝅 + 𝒊 𝐬𝐢𝐧 𝟐𝝅)
−𝟗𝟒𝟕𝟎𝟐𝟎𝟕 − 𝟏𝟓𝟏𝟑𝟏𝟒𝟐𝟒𝒊 (−𝟒𝟎𝟒𝟐𝟐𝟎𝟖𝟎𝟎, −𝟓𝟏𝟕𝟏𝟏𝟔𝟕𝟔𝟖)
𝟏𝟏𝟕𝟔𝟒𝟗(𝐜𝐨𝐬 𝟎 + 𝒊 𝐬𝐢𝐧 𝟎)
Pre-Calc Polar & Complex #s ~15~ NJCTL.org
Roots of Complex Numbers – Class Work
Find the given roots and write the answer in the same form as the original.
159. fifth root of [3,60°] 160. fourth root of 4 (𝑐𝑜𝑠𝜋
5+ 𝑖𝑠𝑖𝑛
𝜋
5)
161. sixth root of 5 − 6𝑖 162. eighth root of (−5,9)
163. a to the fourth is √3(cos 20° + 𝑖𝑠𝑖𝑛 20°), find a
(𝟏. 𝟐𝟒𝟔, 𝟏𝟐𝒐)
(𝟏. 𝟐𝟒𝟔, 𝟖𝟒𝒐)
(𝟏. 𝟐𝟒𝟔, 𝟏𝟓𝟔𝒐)
(𝟏. 𝟐𝟒𝟔, 𝟐𝟐𝟖𝒐)
(𝟏. 𝟐𝟒𝟔, 𝟑𝟎𝟎𝒐)
𝟏. 𝟒𝟏𝟒 (𝐜𝐨𝐬𝝅
𝟐𝟎+ 𝒊 𝐬𝐢𝐧
𝝅
𝟐𝟎)
𝟏. 𝟒𝟏𝟒 (𝐜𝐨𝐬𝟏𝟏𝝅
𝟐𝟎+ 𝒊 𝐬𝐢𝐧
𝟏𝟏𝝅
𝟐𝟎)
𝟏. 𝟒𝟏𝟒 (𝐜𝐨𝐬𝟐𝟏𝝅
𝟐𝟎+ 𝒊 𝐬𝐢𝐧
𝟐𝟏𝝅
𝟐𝟎)
𝟏. 𝟒𝟏𝟒 (𝐜𝐨𝐬𝟑𝟏𝝅
𝟐𝟎+ 𝒊 𝐬𝐢𝐧
𝟑𝟏𝝅
𝟐𝟎)
. 𝟖𝟕𝟓 + 𝟏. 𝟏𝟎𝟓𝒊
−. 𝟓𝟏𝟗 + 𝟏. 𝟑𝟏𝒊
−𝟏. 𝟑𝟗𝟒+. 𝟐𝟎𝟓𝒊
−. 𝟖𝟕𝟓 − 𝟏. 𝟏𝟎𝟓𝒊
. 𝟓𝟏𝟗 − 𝟏. 𝟑𝟏𝒊
𝟏. 𝟑𝟗𝟒−. 𝟐𝟎𝟓𝒊
(𝟏. 𝟐𝟗𝟑, 𝟎. 𝟑𝟒𝟒)
(. 𝟔𝟕𝟏, 𝟏. 𝟏𝟓𝟕)
(−. 𝟑𝟒𝟒, 𝟏. 𝟐𝟗𝟑)
(−𝟏. 𝟏𝟓𝟕, 𝟎. 𝟔𝟕𝟏)
(−𝟏. 𝟐𝟗𝟑, −𝟎. 𝟑𝟒𝟒)
(−. 𝟔𝟕𝟏, −𝟏. 𝟏𝟓𝟕)
(. 𝟑𝟒𝟒, −𝟏. 𝟐𝟗𝟑)
(−. 𝟔𝟕𝟏, 𝟏. 𝟏𝟓𝟕)
𝟏. 𝟏𝟒𝟕(𝐜𝐨𝐬 𝟓° + 𝒊 𝐬𝐢𝐧 𝟓°)
𝟏. 𝟏𝟒𝟕(𝐜𝐨𝐬 𝟗𝟓° + 𝒊 𝐬𝐢𝐧 𝟗𝟓°)
𝟏. 𝟏𝟒𝟕(𝐜𝐨𝐬 𝟏𝟖𝟓° + 𝒊 𝐬𝐢𝐧 𝟏𝟖𝟓°)
𝟏. 𝟏𝟒𝟕(𝐜𝐨𝐬 𝟐𝟕𝟓° + 𝒊 𝐬𝐢𝐧 𝟐𝟕𝟓°)
Pre-Calc Polar & Complex #s ~16~ NJCTL.org
Homework
Find the given roots and write the answer in the same form as the original.
164. fifth root of [9,80°] 165. fourth root of 5 (𝑐𝑜𝑠4𝜋
3+ 𝑖𝑠𝑖𝑛
4𝜋
3)
166. sixth root of (−4 + 7𝑖) 167. eighth root of (−7, −3)
168. a to the sixth is √3(cos 30° + 𝑖𝑠𝑖𝑛 30°), find a
(𝟏. 𝟓𝟓, 𝟏𝟔𝒐)
(𝟏. 𝟓𝟓, 𝟖𝟖𝒐)
(𝟏. 𝟓𝟓, 𝟏𝟔𝟎𝒐)
(𝟏. 𝟓𝟓, 𝟐𝟑𝟐𝒐)
(𝟏. 𝟓𝟓, 𝟑𝟎𝟒𝒐)
𝟏. 𝟒𝟗𝟓 (𝐜𝐨𝐬𝝅
𝟑+ 𝒊 𝐬𝐢𝐧
𝝅
𝟑)
𝟏. 𝟒𝟗𝟓 (𝐜𝐨𝐬𝟓𝝅
𝟔+ 𝒊 𝐬𝐢𝐧
𝟓𝝅
𝟔)
𝟏. 𝟒𝟗𝟓 (𝐜𝐨𝐬𝟒𝝅
𝟑+ 𝒊 𝐬𝐢𝐧
𝟒𝝅
𝟑)
𝟏. 𝟒𝟗𝟓 (𝐜𝐨𝐬𝟏𝟏𝝅
𝟔+ 𝒊 𝐬𝐢𝐧
𝟏𝟏𝝅
𝟔)
(𝟏. 𝟑𝟑𝟏 + 𝟎. 𝟒𝟖𝟑𝒊)
(𝟎. 𝟐𝟒𝟕 + 𝟏. 𝟑𝟗𝟒𝒊)
(−𝟏. 𝟎𝟖𝟒 + 𝟎. 𝟗𝟏𝟏𝒊)
(−𝟏. 𝟑𝟑𝟏 − 𝟎. 𝟒𝟖𝟑𝒊)
(−𝟎. 𝟐𝟒𝟕 − 𝟏. 𝟑𝟗𝟒𝒊)
(𝟏. 𝟎𝟖𝟒 − 𝟎. 𝟗𝟏𝟏𝒊)
(𝟏. 𝟐𝟖𝟕, 𝟎. 𝟎𝟔𝟓)
(𝟎. 𝟖𝟔𝟒, 𝟎. 𝟗𝟓𝟔)
(−𝟎. 𝟎𝟔𝟓, 𝟏. 𝟐𝟖𝟕)
(−𝟎. 𝟗𝟓𝟔, 𝟎. 𝟖𝟔𝟒)
(−𝟏. 𝟐𝟖𝟕, −𝟎. 𝟎𝟔𝟓)
(−𝟎. 𝟖𝟔𝟒, −𝟎. 𝟗𝟓𝟔)
(𝟎. 𝟎𝟔𝟓, −𝟏. 𝟐𝟖𝟕)
(𝟎. 𝟗𝟓𝟔, −𝟎. 𝟖𝟔𝟒)
𝟏. 𝟎𝟗𝟔(𝐜𝐨𝐬 𝟓° + 𝒊 𝐬𝐢𝐧 𝟓°)
𝟏. 𝟎𝟗𝟔(𝐜𝐨𝐬 𝟔𝟓° + 𝐢 𝐬𝐢𝐧 𝟔𝟓°)
𝟏. 𝟎𝟗𝟔(𝐜𝐨𝐬 𝟏𝟐𝟓° + 𝐢 𝐬𝐢𝐧 𝟏𝟐𝟓°)
𝟏. 𝟎𝟗𝟔(𝐜𝐨𝐬 𝟏𝟖𝟓° + 𝐢 𝐬𝐢𝐧 𝟏𝟖𝟓°)
𝟏. 𝟎𝟗𝟔(𝐜𝐨𝐬 𝟐𝟒𝟓° + 𝐢 𝐬𝐢𝐧 𝟐𝟒𝟓°)
𝟏. 𝟎𝟗𝟔(𝐜𝐨𝐬 𝟑𝟎𝟓° + 𝐢 𝐬𝐢𝐧 𝟑𝟎𝟓°)
Pre-Calc Polar & Complex #s ~17~ NJCTL.org
Polar and Complex Numbers Unit Review
Multiple Choice
1. Simplify: −4𝑖 ∙ 6𝑖 ∙ −2𝑖 ∙ −𝑖
a. -48i
b. 48i
c. -48
d. 48
2. Simplify: (6 − 𝑖)2
a. 35 + 12i
b. 35 - 12i
c. 37 - 12i
d. 37 + 12i
3. Simplify: 3−𝑖
4−2𝑖
a. 7
10+
1
10i
b. 7
6+
1
6i
c. 7
10−
1
10i
d. 7
6−
1
6i
4. What quadrant is (6 + 2i) – (7 – 4i) in?
a. I
b. II
c. III
d. IV
5. What quadrant is (3 - 5i)2 in?
a. I
b. II
c. III
d. IV
6. What quadrant is 3−𝑖
4−2𝑖 in?
a. I
b. II
c. III
d. IV
7. Which of the point choices listed are not equal to: [5,π
2]
a. (0,5)
b. 5(𝑐𝑜𝑠𝜋
2+ 𝑖𝑠𝑖𝑛
𝜋
2)
c. [−5,3π
2]
d. they are all equivalent
C
C
A
B
C
A
D
Pre-Calc Polar & Complex #s ~18~ NJCTL.org
8. Convert the point to rectangular form: [4,π
3]
a. (2,√3
2)
b. (√3
2, 2)
c. (2,√3)
d. (2,2√3)
9. Convert the point to polar form: ( 2.5 , 6)
a. (6.5, 0.395)
b. (6.5 , 1.176°)
c. (6.5 , 22.620°)
d. (6.5, 67.380°)
10. Let a =8 - 2i and b= -5 - 7i, which of the following is not a + b?
a. (3,-9)
b. [3√10, −71.565]
c. 10(cos 288.435° + i sin 288.435°)
d. –( -3 + 9i)
11. 𝑎 = 6(𝑐𝑜𝑠𝜋
4+ 𝑖𝑠𝑖𝑛
𝜋
4) and 𝑏 = −3(𝑐𝑜𝑠
5𝜋
3+ 𝑖𝑠𝑖𝑛
5𝜋
3), find ab.
a. −18(𝑐𝑜𝑠6𝜋
7+ 𝑖𝑠𝑖𝑛
6𝜋
7)
b. −18(𝑐𝑜𝑠5𝜋
12+ 𝑖𝑠𝑖𝑛
5𝜋
12)
c. −18(𝑐𝑜𝑠17𝜋
12+ 𝑖𝑠𝑖𝑛
17𝜋
12)
d. −18(𝑐𝑜𝑠23𝜋
12+ 𝑖𝑠𝑖𝑛
23𝜋
12)
12. How many petals and what is a petals length for 𝑟 = 4𝑐𝑜𝑠8𝜃?
a. 4 petals, length 8
b. 8 petals, length 4
c. 8 petals, length 8
d. 16 petals, length 4
13. Compute: (7 − 3𝑖)6
a. ( 195112, 220.809°)
b. ( 45.694, 220.809°)
c. ( 195112, 1.871𝜋)
d. ( 45.694, 1.871𝜋)
14. If a tenth root of w is [5,2𝜋
3], what is w?
a. [50,20π
3]
b. [9765625,20π
3]
c. [50,4π
3]
d. [9765625,4π
3]
D
D
C
D
D
A
B
Pre-Calc Polar & Complex #s ~19~ NJCTL.org
15. Find the third root of 27 (𝑐𝑜𝑠𝜋
2− 𝑖𝑠𝑖𝑛
𝜋
2)
a. [3,π
6]
b. [3,π+4kπ
6] for k ∈ {1,2}
c. [3,4+kπ
6] for k ∈ {1,2,3}
d. [3,π+4kπ
6] for k ∈ {0,1,2}
Extended Response
16. Let a =8 - 2i and b= -5 - 7i.
a. Find 3a2b.
b. How far from the origin is a + b?
c. What is the angle of rotation of a+b?
17. Write an equation
a. for a rose curve with 8 petals of length 5
b. for a rose curve with 5 petals of length 6
c. a Spiral of Archimedes with 6𝜋 between the spirals
D
−𝟏𝟓𝟕𝟐 − 𝟕𝟖𝟎𝒊
𝟑√𝟏𝟎
𝟐𝟖𝟖. 𝟒𝟑𝒐
𝒓 = 𝟓 𝐜𝐨𝐬 𝟒𝜽 𝒐𝒓 𝒓 = 𝟓 𝐬𝐢𝐧 𝟒𝜽
𝒓 = 𝟔 𝐜𝐨𝐬 𝟓𝜽 𝒐𝒓 𝒓 = 𝟔 𝐬𝐢𝐧 𝟓𝜽
𝒓 = 𝟑𝜽 + 𝒌 𝒇𝒐𝒓 𝒌 ≥ 𝟏