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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