Name ______Precalculus
Polar Unit
Polar Coordinates
Section / Assignment1. Polar Coordinates
2. Graphs of Polar Equations
3. Conversions of Equations
4. The Complex Plane
5. Product / Quotient
6. DeMoivre’s Theorem / p. 747 – 748: 1 – 23 odd
p. 432 - 434: 5, 15, 25, 45
p. 748: 31 – 35, 37, 39
p. 432 - 434: 7, 19, 43, 27
p. 748 – 749: 25, 27, 29, 55, 57, 59, 61
p. 432 - 434: 6, 16, 51, 50
p. 733: 21 – 27 odd, 33 – 39 odd, 47 – 51 odd
p. 432 – 434: 8, 20, 47, 48
p. 733: 57, 59, 63, 65
p. 739: 3, 5, 9, 13, 23, 27
p. 435: 1, 5, 7
Answers to evens
p. 432 – 434:
1
6.
16.
50.
8.
20.
48.
1
Graphing Fun With Polar Functions
Calculator set up
MODE
Make sure your calculator is set to RADIAN mode and POL mode.
WINDOW
You will notice two things:
when you press it will now say
when you press the variable key you will see a instead of an X.
Graphing
Press , enter a 5, then press GRAPH. What do you see? Where did it start graphing? What direction does it take?
Now try graphing . Where did this one start graphing and in which direction?
Try typing in other constants. What graphs do you get?
Now for some graphs other than circles with center at the origin. For each of the following try values of . Start with an a value of 1. Some of the graphs may not be seen with some of the a values and the window we have set up. The name of the graph is given in parentheses. Sketch an example of each and indicate the a value.
(Circle)(Circle)
(Cardioid)(Limaçon of Pascal)
(Spiral of Archimedes) (Bifolium)
(Rose Curves) try n values that are even and n values that are odd – also try using sine
(Conchoid of Nicomedes)(Cissoid of Diocles)
Polar Coordinate System
r: distance of point from the pole
: angle formed by the polar axis and ray from pole through point
Rename the coordinate
Such that
Such that
Such that
Rename the coordinate
Such that
Such that
Such that
Polar to Rectangular
Rectangular to Polar
1
1
Conversions of Equations
Rectangular Polar
Lines
Circles
Parabolas
Polar Rectangular
Lines
Circles
Parabolas
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Review of Complex Numbers
Write each expression in the form .
- 2.
3. 4.
5. 6.
7. 8.
Plot each of the following Complex Numbers on the Complex Plane
9. 10. 11.
Example #1:/
Example #2:
/
Example #3:
/
Example #1:
/
Example #2:
/
Example #3:
/
Product & Quotient of Complex Numbers
Product:
Quotient: If ,
Examples
Given . Given .
Find zw.Find .
Find the product and quotient of the following complex numbers.
Simplify the following:
DeMoivre’s Theorem
Examples
Write in standard form.
Write in standard form.
Write in standard form.
Finding the nth root of a complex number
Find the complex cube roots of.
Basically, the question is asking us to solve the equation: ______
By DeMoivre’s Theorem:
Thus,
1. Find the fifth roots of . / 2. Find the cube roots of .3. Find the fourth roots of . / 4. Find the cube roots of .
Practice
Graph each of the following.
1.
2.
3.Rename the point where:
- r < 0,
b.
c.
4. Convert from polar to rectangular form. Round to the nearest hundredth where necessary.
a. b.
5. Convert from rectangular to polar form. Round to the nearest hundredth where necessary.
a. b.
6. Write a polar equation given the rectangular equation.
a. b.
7. Write a rectangular equation given the polar equation.
a. b.
Find the rectangular coordinates for the point.
Find the polar coordinates for the point.
Change the equation to polar form.
Change the equation to rectangular form.
REVIEW - Polar Coordinates
Convert each coordinate. (Polar to rectangular or rectangular to polar.
1. 2. 3.
4. 5. 6.
Convert each equation from polar to rectangular.
7. 8. 9.
Convert each equation from rectangular to polar.
10. 11. 12.
Convert each of the following from rectangular to polar (complex numbers).
13. 14.
15. 16.
Convert each of the following from polar to rectangular (complex numbers).
17. 18.
19.20.
GRAPH each of the following equations on graph paper.
21.r = 22.r = 23.r =
Perform the given operations. Give answers in both polar and rectangular form.
24.25.
26.27.
28.29.
Polar Coordinates – More Review
Convert each equation from polar to rectangular form.
1.2.
Convert each equation from rectangular to polar form.
3.4.
Convert each of the following complex numbers into polar form.
5. 6. 7.
Convert each of the following polar form of a complex number into rectangular form.
8. 9.
Sketch each of the following equations without using your calculator.
10.11.
Perform the given operations. Give answers in both polar and rectangular form.
12.13.
14. 15.
16.
Find allsolutions to the given equations.
17. 18.19.
1