pre-calc unit 14: polar assignment...

Pre-Calc Unit 14: Polar Assignment Sheet

April 27th to May 7

th 2015

Date Objective/ Topic Assignment Did it

Monday

April 27th

Polar Discovery Activity pp. 4 - 5

Tuesday

April 28th

Converting between Polar and

Rectangular systems.

Notes pp. 6 - 8

pp. 9 - 10

Wednesday

April 29th

Graphing Polar Equations

Notes p. 11

pp. 12 - 13

Thursday

April 30th

Writing Equations from Graphs

Notes p. 14

p. 15 review

p.16

Friday

May 1st

Unit 14 Review pp. 17 -18 study

Monday

May 4th

Unit 14 test

Tuesday

May 5th

Work on Polar Project Work on project

Wednesday

May 6th

Work on Polar Project Work on project

Thursday

May 7th

Work on then Turn in Polar Project

Print out last unit

POLAR GRAPHS DISCOVERY ACTIVITY

Put your graphing calculator in POLAR mode and RADIAN mode. Graph the following equation on your calculator,

sketch the graphs on this sheet, and answer the questions.

1. r 2cos 2. r 3cos 3. r 3cos

4. r 2sin 5. r 3sin 6. r 3sin

7. What is similar about the graphs of #1-3? 8. How are they different?


11. r 2 2cos 12. r 1 2cos 13. r 2 cos

14. r 2 2sin 15. r 1 2sin 16. r 2 sin



Put your graphing calculator in POLAR mode and RADIAN mode. Graph the following equation on your calculator,

sketch the graphs on this sheet, and answer the questions.

1. r 2cos3 2. r 3cos5 3. r 4cos7

4. r 2sin3 5. r 3sin5 6. r 4sin7

7. How does the coefficient affect the graphs?

8. How does the coefficient of the affect the graphs?

9. r 3cos2 10. r 2cos 4 11. r 4cos6

12. r 3sin2 13. r 2sin4 14. r 4sin6

15. How does the coefficient affect the graphs?

16. How does the coefficient of the affect the graphs?

Polar Coordinates Notes

The Polar Coordinate System is an alternative to the Cartesian system of rectangular coordinates for locating points in a plane. It consists

of a fixed point O, called the pole or origin and a fixed ray ,OA called the polar axis with O as its initial point.

The polar coordinates of a fixed point P in the polar coordinate system consist of an ordered pair (r, θ).

The directed distance from the pole to P is R, and the measure of the angle from the polar axis to OP is θ.

P (r, θ)

O A

Both r and θ can be either positive or negative.

When r is positive, the polar distance is measured from O along the terminal side of the angle θ, and when r is negative, it is measured

from O on the opposite the terminal side of θ.

When θ is positive, the polar angle is obtained by rotating OP counterclockwise from the polar axis, and when θ is negative, the rotation

is clockwise.

rθ- plane is a plane where polar coordinates (r, θ) are used to identify its points.

Examples. Graph:

1) P ( 5, 60° )

2) Q ( 5, -60° )

3) W ( -5, 60° )

4) V ( -5, -60° )

5) A ( 3 150º)

6) B (-3, -150º)

Rotations of θ and θ + 2nπ or θ + 360°n produce the same angle so there are infinitely many ways to represent the

same angle.

Examples:

1) Plot the point P (2, 45°) and find 3 other 2) Plot the point P (1, π) and find 3 other

polar representations of the point. polar representations of the point.

Polar Equation: an equation with polar coordinates

Polar Graph: a graph of the set of all points (r, θ) that satisfy a given polar equation.

The two most basic polar equations are:

r = c a circle of radius c

θ = a line through the origin that forms an angle θ with the polar axis

Examples.

1) Sketch r = 3. 2) Sketch r = –2. 3) Sketch θ = 30°. 4) Sketch θ = – 45°.

If you superimpose a Rectangular Coordinate system over a Polar Coordinate system:

y 222 yxr so r =

22 yx

P(x, y) r

xcos so cosrx

x r

ysin so sinry

polar axis x

ytan

Convert from Rectangular to Polar Coordinates.

1) ( 3, 3) 2) (2, 2 3 ) 3) (0, -2) 4) ( – 4 3 , 4)

Convert from Polar to Rectangular Coordinates.

1) (-2, π) 2) (3, 135°) 3) ( -5, 240°) 4) (4, 6

)

Convert the Polar Equations to Rectangular form. Identify.

1) r = 1 2) θ = 45° 3) sec5r 4) csc4r

5) sin3r 6) sin3cos2

6

r 7)

cos2

2

r

Convert the rectangular Equations to Polar form. Identify.

1) 1275 yx 2) x = 11 3) y = 6

4) 922 yx 5) (y – 2)2 + x

2 = 4

Polar Coordinates Homework April 28th

Convert from Rectangular to Polar Coordinates then graph

A ( –3, 3 3 ) B (4, – 4 3 ) C (0, – 5) D ( – 3 , 1 ) E (5, – 5)

Graph then, Convert from Polar to Rectangular Coordinates.

F (1, 2

) G (6, 120°) H ( 4, –270°) I (2,

4

) J (3, π)

Give 3 additional coordinates for the points given.

1) ( 1, 45º) 2) ( 2, 210°)

Convert the Polar Equations to Rectangular form. Identify.

1) r = 3 2) θ = 30° 3) sec7r 4) csc8r

5) sin2r 6) sin2cos4

5

r 7)

sin1

3

r

Convert the rectangular Equations to Polar form. Identify.

1) 853 yx 2) x = 4 3) y = 9

4) 1622 yx 5) y2 + (x – 3)2 = 9

Notes: Graphing Polar Equations

Circles: cosar and sinar where a is the diameter of the graph

The form cosar is symmetrical about the polar ( horizontal ) axis

The form sinar is symmetrical about the line 2

.

Limacons: sinbar and cosbar

Cardiod: ba Heart Inner Loop: ba Indentation: ba one side also may appear flat

Roses: bar cos and bar sin If b is odd there are b petals. If b is even there are 2b petals.

a = the length of the petal

Make and Fill in a table then Graph each polar equation. Identify.

1. 3sin4r 2. sin22r

3. cos21r 4. cos23r

Homework: Graphing Polar Equations April 29th

For each equation, make a table then graph. Identify.

1. 4cos4r 2. 3cos4r

3. cos5r 4. sin4r

5. 3cos6r 6. 3sin5r

7. cos33r 8. cos23r

9. cos32r 10. sin23r

11. sin42r 12. sin22r

13. sin41r 14. cos22r

Notes: Writing Polar Equations

1) 2) 3)

4) 5) 6)

7) 8) 9)

Homework: Writing Polar Equations April 30th

1) 2) 3)

4) 5) 6)

7) 8) 9)

Review Polar Test Homework April 30th

1. Graph each point on the Polar grid on page 18.

)2,4( A )65,3( B )225,2( C )300,1( D

)270,5( E )43,3( F )30,4( G )240,2( H

2. Convert from Rectangular to Polar Coordinates.

a. ( – 5, –5) b. )3,33( c. (–4, 0) d.

4

3,

4

1

3. Convert from Polar to Rectangular Coordinates.

a. (–3, 60°) b. )2,2( c. )32,5( d. (4, –210°)

4. Give 3 additional coordinates for each of the points.

a. )6,4( b. (3, 50°) c. (2, 220°) d. (5, 90°)

5. Convert to Polar form.

a. 2x + 4y = 7 b. x = –5 c. y = 3 d. (x – 3)2 + y

2 = 9

6. Convert to Rectangular form.

a. r = 3cscθ b. r = 5cosθ c. r – 5cosθ = 7sinθ d. sin3cos

2

r

7. Graph on a Polar grids on page 18.

a. r = 2secθ b. r = 3cosθ c. r = 5sin3θ d. r = 4cos2θ

e. r = 3 + 2cosθ f. r = 2 – 2sinθ g. r = 2 – 3sinθ

8. Matching: Use each letter twice.

a. horizontal line b. vertical line c. oblique line d. circle with center (0, 0)

e. circle: center on y-axis f. circle: center on x-axis g. limacon with inner loop

h. limacon with indentation i. cardiod j. rose

_____ 1. r = 3secθ _____ 2. r = 4cosθ _____ 3. θ = –60° _____ 4. r = 2sinθ

_____ 5. r = 6cscθ _____ 6. r = 3 + 3sinθ _____ 7. r = 3 _____ 8. r = 3 – 2sinθ

_____ 9. r = –7secθ _____ 10. r = 2 – 3sinθ _____ 11. r = 5cos3θ _____ 12. r = 6cosθ

_____ 13. r = 4 + 2sinθ _____ 14. r = –3cscθ _____ 15. r = 4 + 5cosθ _____ 16. r = 7

_____ 17. r = 5 – 5cosθ _____ 18. r = 3sin2θ _____ 19. 5sin-cos2

3r _____ 20. r = –6sinθ

9. Write the polar equation.

a) b) c)

Use for #1 8a.

8b. 8c.

8d. 8e.

8f. 8g.

pre-calc unit 14: polar assignment...

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