Chapter 9 – Polar Coordinates (r, θ)
Chapter 9 – Polar Coordinates (r, θ) Name ___________________
Date 4/22 - Period ____
|Sec |Topics |Assignments |Due Date ) | |
|9.1 |Polar Coordinates |p.558 # 11 – 39 odds, | | |
|9.2 |Graphs of Polar Equations |p.565 # 14 (table 0 < x < 360o in increments | | |
| | |of 15o ) | | |
|9.3 |Polar and Rect. Coordinates |p.572 #14 - 38 E | | |
|9.4 |Polar Form of a Lin Equations. |p.578 #12 – 23 | | |
| | |#24 - 29 | | |
|9.5 |Simp. Comp. Numb. |p.583 # 13 - 35 | | |
|9.6 |Comp. Plane and Polar |p.590 # 16-40 | | |
|9.7 |Pro. & Quo. Comp Num. |p.596 # 10-18 | | |
|9.8 |Powers and Root |p.605 #13-31 | | |
| |Pre Test | | | |
|Quizzes Test |
Chapter 9 – Polar Coordinates (r, θ)
Examples: Plot the following points
1. P(5, 30°)
2. Q(3, 125°)
3. R(-4, 60°)
4. S(-1,-30°)
5. T(7, -45°)
6. U(-2, -150°)
Other representations of a polar coordinate
P(2, 60°)
Q(-1, 45°)
Graphing:
a) θ = [pic] b) r = 3
c) [pic] d) r = - 5
Graphing Polar Equations
The Rose:
Graph each of the following families:
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
Extra graphs
Limaçon – shaped like _______________
Examples: graph the following
a. [pic] [pic] b. [pic] [pic]
Lemniscate – sounds like combined “lemon & biscuit”
Examples: graph the following
a. [pic] b. [pic]
Cardioid – looks like a _____________
Examples: graph the following
a. [pic] b. [pic]
Spiral of Archimedes
Examples: graph the following
a. [pic] b. [pic]
Final notes about polar graphing:
|Graphing |Table |
|Look at ________________________ |Look at ___________________________ |
Then…..decide:
| | |
Solving Polar Equations—graphically vs. algebraically
Remember: we are finding _____________________________________
GRAPHICALLY ALGEBRAICALLY
1. [pic]
[pic]
2. [pic]
[pic]
Changing Polar and Rectangular Coordinates
A. Polar to Rectangular
Rectangular coordinates (x, y) can be
expressed from polar coordinates (r, θ) using
[pic]________(this is horizontal component)
[pic]________(this is vertical component)
Examples:
1. Find the rectangular coordinates given the polar coordinates:
a) C(3, 270°) b) D[pic]
B. Rectangular to Polar
Polar coordinates (r, θ) can be expressed from rectangular coordinates (x, y) by
Examples:
1. Find the polar coordinate given the rectangular coordinate:
a) A(2, 5) b) B(-3,1)
c) C(0, 5) d) D(-4,0)
e) E(-4, -3)
Equations in Rectangular/Polar form: if the equation is in rectangular form, change to polar form and vice-versa.
Examples:
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic] 9. [pic]
Polar form of a Linear Equation:
p = length of normal
[pic] = angle from 0°
_______________ are variables while _______________ are constants
Examples:
1. Write 2x + 3y – 1 = 0 in polar form
2. Write x + 2y – 3 = 0 in polar form
Polar form to Rectangular Form
1. Write [pic] in rectangular form
2. Write [pic]in rectangular form
3. Write [pic][pic] in rectangular form 4. Write [pic][pic] in rectangular form
Simplifying Complex Numbers:
[pic] Alg. 2 version [pic] Pre-Calc version
[pic] Find:
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic]
[pic]
[pic]
Simplify:
1. [pic] 2. [pic]
3. [pic] 4.[pic]
Solve:
5. [pic] 6.[pic]
7. [pic] 8. [pic]
Polar Form of Complex Numbers:
______________ Form _______________Form (a.k.a Trig form)
_____________ [pic]+[pic] _________________OR ____________
(by substitution) (Factor to get) abbreviated form
Graph & Write in Polar Form:
1. [pic] 2. -3 – 2i
3. [pic] 4. [pic]
Product and Quotients of Complex Numbers
Product: [pic]= ___________________________
Examples: Find the product. Write in rectangular form.
a) [pic]
b) [pic]
c) [pic]
Quotient: [pic]=_________________________________
Examples: Find the quotient. Write in rectangular form.
a) [pic]
b) [pic]
Powers of Complex Numbers
De Moivre’s Theorem:
[pic]=[pic] [pic][pic]+[pic][pic][pic]
1.) expand [pic] Algebraic 2 method:
De Moivre’s method: r = ______, [pic]_____
Use De Moivre’s method to expand the following binomials:
2.) [pic]
3.)[pic] 4.)[pic]
Finding “p” roots of a Complex number
[pic][pic]
Example:
1. Find the five “fifth roots” of -243 2. Solve [pic]
-----------------------
R(r, ¸) can be expressed by:
General form:
[pic]
General form:
General form:
General form:
General form:
vs.
r
y
x
Reminder: [pic]
AND [pic] is measured in radians.
[pic]= ________________ when x>0
[pic] = _______________ when x ................
................
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