Download 6.5- Trigonometric Form of Complex Numbers

Precalculus Honors Review 6.5, 9.6 through 9.8
Name: __________________________________ Date: ___________________
6.5- Trigonometric Form of Complex Numbers
 Absolute Value of Complex Numbers
o |𝑎 + 𝑏𝑖| = √𝑎2 + 𝑏 2
 Trigonometric Form of a Complex Number
o 𝑧 = 𝑎 + 𝑏𝑖
o 𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃)
o 𝑎 = 𝑟 cos 𝜃
o 𝑏 = 𝑟 sin 𝜃
o 𝑟 = √𝑎2 + 𝑏 2
𝑏
o 𝜃 = tan−1 𝑎
 Product of Two Complex Numbers
o 𝑧1 = 𝑟1 (cos 𝜃1 + 𝑖 sin 𝜃1 )
o 𝑧2 = 𝑟2 (cos 𝜃2 + 𝑖 sin 𝜃2 )
o 𝑧1 𝑧2 = 𝑟1 𝑟2 (cos 𝜃1 + 𝜃2 + 𝑖 sin 𝜃1 + 𝜃2 )
 Quotient of Two Complex Numbers
o 𝑧1 = 𝑟1 (cos 𝜃1 + 𝑖 sin 𝜃1 )
o 𝑧2 = 𝑟2 (cos 𝜃2 + 𝑖 sin 𝜃2 )
𝑧
𝑟
o 𝑧1 = 𝑟1 (cos 𝜃1 − 𝜃2 + 𝑖 sin 𝜃1 − 𝜃2 ), 𝑧2 ≠ 0
2
2
 DeMoivre’s Theorem
o 𝑧 𝑛 = [𝑟(cos 𝜃 + 𝑖 sin 𝜃)]𝑛 = 𝑟 𝑛 (cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃)
 nth Root of a Complex Number
o
𝑛
√𝑟 (cos
𝜃+2𝜋𝑘
𝑛
9.6- Polar Coordinates
 Polar Coordinates
o (𝑟, 𝜃)
o 𝑥 = 𝑟 cos 𝜃
o 𝑦 = 𝑟 sin 𝜃
𝑏
o 𝜃 = tan−1 𝑎
o 𝑟2 = 𝑥2 + 𝑦2
+ 𝑖 sin
𝜃+2𝜋𝑘
𝑛
) , 𝑘 = 0, 1, 2, … . , 𝑛 − 1
Block: _________
9.7- Graphs of Polar Equations
 Testing for Symmetry
𝜋
o The line 𝜃 = 2 : Replace (𝑟, 𝜃) 𝑏𝑦 (𝑟, 𝜋 − 𝜃) 𝑜𝑟 (−𝑟, −𝜃)
o The polar axis: Replace (𝑟, 𝜃) 𝑏𝑦 (𝑟, −𝜃) 𝑜𝑟 (−𝑟, 𝜋 − 𝜃)
o The pole: Replace (𝑟, 𝜃) 𝑏𝑦 (𝑟, 𝜋 + 𝜃) 𝑜𝑟 (−𝑟, 𝜃)
 Special Polar Graphs
o Limacons: 𝑟 = 𝑎 ± 𝑏 cos 𝜃 𝑜𝑟 𝑟 = 𝑎 ± 𝑏 sin 𝜃 , (𝑎 > 0, 𝑏 > 0)
𝑎

Inner Loop if 𝑏 < 1

Cardioid (Heart Shaped): 𝑏 = 1

Dimpled: 1 <

Convex: 𝑏 > 2
𝑎
𝑎
𝑏
𝑎
<2
o Rose Curves: 𝑟 = 𝑎 cos 𝑛𝜃 𝑜𝑟 𝑟 = 𝑎 sin 𝑛𝜃
 n petals if n is odd
 2n petals if n is even (𝑛 ≥ 2)
o Circles: 𝑟 = 𝑎 cos 𝜃 𝑜𝑟 𝑟 = 𝑎 sin 𝜃
o Lemniscates: 𝑟 2 = 𝑎2 cos 2𝜃 𝑜𝑟 𝑟 2 = 𝑎2 sin 2𝜃
9.8- Conics
 Equations of Conics
𝑒𝑝
o 𝑟=
𝑜𝑟 𝑟 =
1±𝑒 cos 𝜃
o Ellipse: 0 < 𝑒 < 1
o Parabola: 𝑒 = 1
o Hyperbola: 𝑒 > 1
𝑒𝑝
1±𝑒 sin 𝜃
, 𝑤ℎ𝑒𝑟𝑒 𝑒 > 0
Directions: Write the following in the indicated form.
1. 𝑟 = −2 csc 𝜃 in rectangular form
1
2. 𝑟 = 2 sin 𝜃 in rectangular form
3. 5𝑦 − 𝑦 2 = 𝑥 2 in polar form
Directions: Plot the following points and given an alternate form for the points.
4. A(2, −
7𝜋
6
)
𝜋
5. B(−3, 6 )

𝜋
6. C(4, 3 )
𝜋
7. D(−5, − 4 )
Directions: Convert the given points as indicated.
8. Give rectangular coordinates for (−3, 150°)
9. Give polar coordinates for (2√3, −2), 𝑘𝑒𝑒𝑝 𝜃 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠




Directions: Keep all the angles in terms of π and all other values in simplest radical form where
appropriate. Write the values in the indicated form.
10. Write in trigonometric form: 𝑧1 = −5𝑖
11. Write in trigonometric form: 𝑧2 = √3 + 3𝑖
12. Calculate 𝑧1 𝑧2 using polar form (use 𝑧1 and 𝑧2 from problems 10 and 11).
13. Convert #12 into rectangular form.
14. Use DeMoivre’s Theorem to find (𝑧2 )3 (use 𝑧2 from problem 11).
15. Express your answer to 14 in rectangular form.
Directions: Keep all the angles in degrees and round all answers to the nearest hundredth.
16. Find the fourth roots of -81 using trigonometric form.
17. Convert your answer to 16 into rectangular form.
18. Use DeMoivre’s Theorem to show that 2 cos 45° is a fifth root of −16√2 − 16√2𝑖.
19. Find the other fifth roots of −16√2 − 16√2𝑖 in trigonometric form.
20. Perform the indicated operation. Leave your answer in trigonometric form.
12(cos 37° + 𝑖 sin 37°)
42(cos 12° + 𝑖 sin 12°)
Directions: Identify the type of graph given the equation. If it is a rose curve, identify the
number of petals. If it is a limacon, identify the specific limacon.
21. 𝑟 = 3 cos 2𝜃
22. 𝑟 = 5 − 5 sin 𝜃
23. 𝑟 2 = 9 cos 2𝜃
4
27. 𝑟 = 1−cos 𝜃
3
28. 𝑟 = 2−cos 𝜃
3
29. 𝑟 = 2+cos 𝜃
4
24. 𝑟 = 3 cos 𝜃
30. 𝑟 = 1−3sin 𝜃
25. 𝑟 = 6 sin 2𝜃
31. 𝑟 = 1+2 sin 𝜃
26. 𝑟 = 1 + 4 cos 𝜃
3
4
32. 𝑟 = 1+sin 𝜃