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Name: ________________________ Class: ___________________ Date: __________
Review 4
Show All the steps to get full credit:
1. Write the complex number 14 +
−289 in standard form.
2. Write the complex number in standard form.
−36 +
16
3. Simplify and write the following complex number in standard form.
(3 + 7i) − (4 + 2i)
4. Simplify i − (6 − 2i) and write the answer in standard form.
5. Simplify (−3 + i) (7 − 12i) and write the answer in standard form.
6. Simplify and write the following complex number in standard form.
(−4 + 3i) (−1 + 2i)
7. Simplify and write the following complex number in standard form.
(5 − i) (5 + i)
8. Write the expression below as a complex number in standard form.
−
5
i
9. Simplify
2+i
and write the answer in standard form.
5 + 4i
10. Write the following expression as a complex number in standard form.
−1 + 2i
−6 − 4i
11. Evaluate the power of i.
i
216
1
ID: A
Name: ________________________
ID: A
12. Plot the complex number −2 − 3i.
13. Find the absolute value of the complex number −7 + 10i.
14. Find the trigonometric form of −17 + 17i.
15. Find the trigonometric form of −18
16. Find the trigonometric form of 4 − 4
3 − 18i.
3 i.
17. Find the standard form of the complex number 8 ÊÁË cos (150° ) + isin(150° ) ˆ˜¯ .
18. Represent the following complex number graphically.
4 2 ÁÊË cos (135° ) + isin(135° ) ˜ˆ¯
19. Multiply the complex numbers. Write the answer in trigonometric form.
ÊÁ
7π
7π
7 ÁÁÁ cos
+ isin
ÁË
9
9
ˆ˜
˜˜
˜˜
¯
ÊÁ
7π
7π
8 ÁÁÁ cos
+ i sin
ÁË
3
3
ˆ˜
˜˜
˜˜
¯
2
Name: ________________________
ID: A
ÁÊ
ÁË
20. Find the standard form of the complex number 8 ÁÁÁÁ cos
π
12
+ isin
π ˜ˆ˜˜
12 ˜˜¯
. Round values to the nearest
hundredth.
21. Find the standard form of the complex number 5 ÁÊË cos (21°41') + i sin(21°41') ˜ˆ¯ . Round values to the
nearest hundredth.
22. Multiply the complex numbers below and leave the result in trigonometric form.
ÍÈÍ ÁÊ
ÍÍÍ 5 ÁÁ cos 4π + isin 4π
ÍÍ ÁÁ
7
7
ÍÎ Ë
˘È
˜ˆ˜ ˙˙˙˙ ÍÍÍÍ ÁÊÁ
˜˜ ˙˙ ÍÍ 8 ÁÁ cos 8π + isin 8π
˜ ˙˙ ÍÍ Á
9
9
¯ ˚Î Ë
˘
˜ˆ˜ ˙˙˙˙
˜˜ ˙˙
˜ ˙˙
¯˚
23. Multiply the complex numbers below and leave the result in trigonometric form.
ÈÍÍ
˘È
˘
ÍÎ 0.4 (cos 110° + isin110° ) ˙˙˙˚ ÍÍÍÎ 0.8 (cos 100° + i sin100° ) ˙˙˙˚
24. Divide the complex numbers below and leave the result in trigonometric form.
5 (cos 140° + i sin140° )
6 (cos 170° + i sin170° )
25. Divide the complex numbers below and leave the result in trigonometric form.
10 ÁÊË cos (9π / 10) + i sin(9π / 10) ˜ˆ¯
6 ÊÁË cos (2π / 7) + i sin(2π / 7) ˆ˜¯
26. Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in
standard form.
Ê
10 ÁÁÁ
Ë
ˆ5
3 + i ˜˜˜
¯
27. Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in
standard form.
ÈÍÍ
˘ 10
ÍÎ 5 (cos 210° + isin210° ) ˙˙˙˚
28. Find the indicated power. Write the answer in standard form.
ÍÈÍÍ 2 (cos 240° + isin 240° ) ˙˘˙˙ 5
Î
˚
3
Name: ________________________
ID: A
29. Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in
standard form.
ÈÍ Ê
ÍÍ ÁÁ
ÍÍÍ −5 ÁÁ cos 5π + i sin 5π
ÍÍ ÁË
2
2
Î
ˆ˜ ˘˙˙˙ 5
˜˜ ˙˙
˜˜ ˙˙
¯ ˙˚
30. Transform the given coordinates to the indicated ordered pair.
ÊÁ
ˆ
ÁÁ 3, − 3 3 ˜˜˜ to (r, θ )
Ë
¯
2
2
31. Find a polar form of the equation x − y = 64.
32. Find a rectangular form of the equation r = 2 sinθ .
33. Find a rectangular form of the equation θ =
49π
.
4
34. Find a rectangular form of the equation r (9 cos θ + sin θ ) = 1.
35. Sketch the graph of the polar equation.
r = 4 cos 2θ
36. Sketch the graph of the polar equation.
r = 4 (1 − cos θ )
37. Sketch the graph of the polar equation.
r = 3 + 3 sin θ
38. Simplify: i
451
39. Test for symmetry with respect to the line, polar axis and the pole
r = 1- 2 cos
40. r = 3 cos2
4
ID: A
Review 4
Answer Section
SHORT ANSWER
1. ANS:
14 + 17i
PTS: 1
2. ANS:
OBJ: Write complex numbers in standard form
PTS: 1
NOT: Skill
3. ANS:
OBJ: Write the complex number in standard form
PTS: 1
4. ANS:
OBJ: Simplify complex numbers
PTS: 1
5. ANS:
OBJ: Simplify and write in standard form
PTS: 1
6. ANS:
OBJ: Simplify and write in standard form
PTS: 1
7. ANS:
OBJ: Simplify complex numbers
NOT: Skill
OBJ: Simplify complex numbers
NOT: Skill
4 + 6i
−1 + 5i
NOT: Skill
−6 + 3i
−9 + 43i
−2 − 11i
26
PTS: 1
8. ANS:
5i
PTS: 1
NOT: Skill
9. ANS:
OBJ: Write expressions as complex numbers in standard form
14 3
−
i
41 41
PTS: 1
OBJ: Simplify and write in standard form
1
ID: A
10. ANS:
−
1 4
– i
26 13
PTS: 1
NOT: Skill
11. ANS:
OBJ: Write expressions as complex numbers in standard form
1
PTS: 1
12. ANS:
OBJ: Evaluate the power of i
PTS: 1
13. ANS:
149
PTS: 1
14. ANS:
ÊÁ
3π
3π
17 2 ÁÁÁÁ cos
+ isin
4
4
Ë
PTS: 1
15. ANS:
ÊÁ
7π
7π
36 ÁÁÁ cos
+ isin
ÁË
6
6
PTS: 1
16. ANS:
ÊÁ
5π
5π
8 ÁÁÁ cos
+ isin
ÁË
3
3
ˆ˜
˜˜
˜˜
¯
ˆ˜
˜˜
˜˜
¯
ˆ˜
˜˜
˜˜
¯
PTS: 1
2
NOT: Skill
ID: A
17. ANS:
−4 3 + 4i
PTS: 1
18. ANS:
PTS: 1
19. ANS:
56 cis
28π
9
PTS: 1
20. ANS:
OBJ: Multiply complex numbers
7.73 + 2.07i
PTS: 1
21. ANS:
4.65 + 1.85i
PTS: 1
22. ANS:
ÁÊ
92
92
40 ÁÁÁÁ cos π + isin π
63
63
Ë
˜ˆ˜
˜˜˜
¯
PTS: 1
23. ANS:
0.32 (cos 210° + isin210° )
PTS: 1
24. ANS:
5
(cos 330° + isin330° )
6
PTS: 1
3
NOT: Skill
ID: A
25. ANS:
5
3
ÊÁ
ÁÁ cos 43 π + i sin 43 π
ÁÁ
70
70
Ë
ˆ˜
˜˜
˜˜
¯
PTS: 1
26. ANS:
−160 3 + 160i
PTS: 1
27. ANS:
9765625 9765625 3
−
i
2
2
PTS: 1
28. ANS:
−16 + 16i 3
PTS: 1
NOT: Skill
29. ANS:
OBJ: Use DeMoivre's Theorem to find a power of a trig equation
−3125i
PTS: 1
30. ANS:
(6, − 60° )
PTS: 1
NOT: Skill
31. ANS:
OBJ: Transform rectangular coordinates to polar coordinates
r (cos 2θ ) = 64
2
PTS: 1
NOT: Skill
32. ANS:
2
OBJ: Find the polar form of a rectangular equation
2
x + y − 2y = 0
PTS: 1
NOT: Skill
33. ANS:
OBJ: Find rectangular forms of polar equations
PTS: 1
NOT: Skill
OBJ: Find rectangular forms of polar equations
y=x
4
ID: A
34. ANS:
y = −9x + 1
PTS: 1
NOT: Skill
35. ANS:
OBJ: Find rectangular forms of polar equations
PTS: 1
36. ANS:
OBJ: Graph polar equations
NOT: Skill
PTS: 1
OBJ: Graph polar equations
NOT: Skill
5
ID: A
37. ANS:
PTS: 1
38. ANS:
OBJ: Graph polar equations
–i
PTS: 1
OBJ: Simplify and write in standard form
39. ANS:
Symmetry with respect to the polar axis
PTS: 1
40. ANS:
Symmetry with respect to polar axis, the line and the pole
PTS: 1
6
NOT: Skill