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