Math Analysis CP, 2017 Due Date 12/11/2017

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1 Math Analysis CP, 017 Due Date 1/11/017 Polar Project 100 points The purpose of the Polar Project is to familiarize students with polar coordinates and polar equations. You will do graphing calculator investigations, solve simultaneous polar equations, create your own polar design, and research historical polar curves. You will complete this project in groups of or 3 students. This project is worth 100 points and is due on Monday, December 11 at the beginning of the period. Ten points will be deducted for each day late. The polar project consists of the nine assignments listed below. Each assignment is worth 10 points. You will be graded on completeness, accuracy, neatness, and format. The assignments marked with a * must be completed by each individual student. The remaining assignments are to be completed as a group. You will turn in: a) A ½" rigid (non-flexible) binder with all assignments in the order listed b) A creative, original title page (This may be put inside a "view binder", but is not required) c) A table of contents matching the outline below. Assignments should be in this order. d) Each individual assignment * clearly labeled with student s name e) DO NOT put any pages into sheet protectors; include one copy of the rubric Assignments for Polar Project 1. A description of the polar coordinate system with a short paragraph about its history. Be sure to cite your sources. A description of conversions between polar and rectangular points. Include several examples of each, showing conversions in both directions.. Transitions Worksheet 3. Distance in Polar Coordinate System Worksheet 4. Graphing Investigation I Print and complete the Changing Values of n Worksheet from Website. 5. Graphing Investigation II Print and complete the Polar Curves Investigation from Website. 6. Systems of Polar Equations Solve the following... Choose 6 of the 8 to solve. Solve each system algebraically and show your work. Write your answer as an ordered pair. Sketch a graph of each system. Plot and label the points of intersection on your graph. r 3 r sin r sin r sin i. ii. iii. iv. r 6cos r 1sin r 3 cos r cos r sin r sin 1 r 1 4 v. vi. vii. r r sin r cos 1 r cos viii. cos r 4cos 7. Other Curves Choose three polar curves from the list below. Write a brief report (-3 paragraphs) on each curve. Sketch a graph of each curve (by hand) on graph paper and include its general equation. You must use at least two resources for your research and reference them using MLA format. This report must be typed and printed. 8. * Original Design Create an original design using at least three polar equations on Desmos. You must print the graph itself and the equations used. You may color or embellish your design after you have printed it. 9. * Polar Multiple Choice WS You must print and complete Worksheet from Website. Your answers must be written on the printed worksheet in the space provided. You must show your work on the back or separate sheet. The worksheet will be graded for accuracy. 1

2 List of Polar Curves Astroid Catenary Conchoid Double Folium Epitrochoid Freeth's Nephroid Kampyle of Eudoxus Lituus Pearls Rhodonea Curves Talbot's Curve Trifolium Bicorn Cayley's Sextic Conchoid of de Sluze Durer's Shell Curves Equiangular Spiral Hyperbolic Spiral Kappa Curve Neile's Parabola Plateau Curves Right Strophoid Tractrix Trisectrix of Maxlaurin Cartesian Oval Cissoid of Diocles Cycloid Eight Curve Fermat's Spiral Hypocycloid Lame Curves Nephroid Pursuit Curve Serpentine Tricuspoid Tschirnhaus' Cubic Cassinian Ovals Cochleoid Devil's Curve Epicycloid Folium of Descartes Hypotrochoid Lissajous Curves Pear-shaped Quartic Quadratix of Hippias Sinusoidal Spirals Trident of Newton Watt's Curve Resources posted on Website (You should print these as soon as possible): - Detailed grading rubric - Polar graph paper - Graphing Investigation I (Included in this document) - Graphing Investigation II (Included in this document) - Polar Multiple Choice Worksheet (Included in this document) - Transitions WS - Distance WS

3 WS- Graphing Investigation I Changing Values of n Graph the following equations on the polar grid provided using different colored pens or pencils. Graph r 1 in blue. Graph r 4 in green. Graph r 5 in red. Graph r in yellow. Verbally describe the changes to each curve as n changes: Graph the following equations on the polar grid provided using different colored pens or pencils. Graph 45 in blue. Graph 10 in green. Graph 150 in red. Graph 60 in yellow. Verbally describe the changes to each curve as n changes: 3

4 Graph the following equations on the polar grid provided using different colored pens or pencils. Graph r sin in blue. Graph r 3sin in green. Graph r sin in red. Verbally describe the changes to each curve as n changes: Graph the following equations on the polar grid provided using different colored pens or pencils. Graph r cos in blue. Graph r 4cos in green. Graph 3cos in red. Verbally describe the changes to each curve as n changes: 4

5 WS- Graphing Investigation II Polar Curves Investigation In this investigation, you will look at different equations and their graphs. Many of the pictures will look neat! Later, you will explore a design using your own equations. It will be helpful to have a grasp of the terms on your calculator. Switch your calculator mode to POL (Polar Mode). Pressing Y= now takes you to a menu containing r 1. Further, your calculator will no longer display an "X" when you press the button. Instead, it will display. So you will input an angle measurement and will get an output that is the length of a point to the origin. Thus, you are graphing the ordered pairs, r. Finally, your window will be extremely important. In general, your window settings should be as follows. Although you may use radians, it is recommended that you use degrees. Min: 0 or 0 X Min: -6 Y Min: -4 Max: 360 or X Max: 6 Y Max: 4 Step: or 0.03 Radians X Scl: 1 Y Scl: 1 For different problems, however, you may wish to change the max/min on the two axes to better see the graph. The Rose General Form: r asinb or r acosb Graph: r sin Graph: r sin Graph: r 4sin Dist. Between Circles: 0.5 Dist. Between Circles: 0.5 Dist. Between Circles: 1 Complete the table for r In general, what does the a in r asinb sin Complete the table for r 4sin r r r r So, what is the general shape of the Rose graph do 5

6 Graph: r 4sin (Use the table below) Graph: r 4sin3 Graph: r 4sin4 Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 So in general, what does the b in r asinb do How does the graph change when the b value is even vs. odd Graph: r cos Graph: r cos Graph: r 4cos3 (Use the table) Dist. Between Circles: 0.5 Dist. Between Circles: 0.5 Dist. Between Circles: 1 How do the cosine Rose graphs differ from the sine graphs Complete the table for r 4sin Complete the table for r 4cos3 r r r r

7 The Leminscate r a cos General Form: or r a sin NOTE: You must take the square root of both sides since it is Graph: r 1cos (Use the table) Graph: r 4cos Graph: r 4sin Dist. Between Circles: 0.5 Dist. Between Circles: 1 Dist. Between Circles: 1 r. What is the general shape of the Leminscate How do the sine Leminscates differ from the cosine graphs Now let s see what happens when a is imaginary Graph: r 1cos (Use the table) Graph: r 4cos Graph: r 4sin Dist. Between Circles: 0.5 Dist. Between Circles: 1 Dist. Between Circles: 1 So what does the a do in r a cos or r a sin Be sure to discuss both real and imaginary values. Complete the table for r 1cos Complete the table for r 1cos r r r r

8 The Limacon General Form: Graph: r cos or r a bsin r a b 1 cos Graph: r 1 cos Graph: r 1 3cos Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 Graph: r 1 sin Graph: r sin Graph: r 3 cos Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 So, what is the basic shape (s) of the Limacon graph How do the sine and cosine graphs differ In general, what do a and b in cos or r a bsin r a b do What is/are the requirements to have a loop in the middle versus just an indent Starting with r 3 cos can you find a value for a that does not make a loop or an indentation Create a cosine equation where there is a loop in the middle: Create a sine equation where there is only an indentation: 8

9 The Cardioid (A Special Limacon) General Form: r a acos r a asin Graph: r or 1 cos (Use the table) Graph: r sin Graph: r sin (Use the table) Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1 Why do you suppose we call this graph a Cardoid In general, what does the value of a in cos or r a asin r a a do What, if anything, changes when the "+" is changed to a "-" Complete the table for r 1 cos Complete the table for r sin r r r r

10 Worksheet #01- Polar Project Review Write the letter of the correct answer on the blank. YOU MUST INCLUDE YOUR WORK!!! 1. Which is a value for if the points 3, 300 and 3, are equivalent A. 30 B. 60 C. 10 D What is the graph of A. a point B. a vertical line C. a horizontal line D. 3. Which is the polar equation for a circle with center at the pole and radius A. r cos B. C. r cos D. r 4. Which of the following is an equation of a spiral of Archimedes A. r B. r cos C. r sin D. r cos 5. How many petals does the rose given by r cos4 have A. B. 8 C. 16 D What is the name of the classical curve represented by r 4sin A. rose B. cardiod C. leminscate D. limacon 7. What is the solution of this system of equations r cos r 1cos A. 0, B., 0 C. 0, 0 &, 0 D. 0, &, 0 8. What are the polar coordinates of 3, 1 A. 4, 300 B.,10 C. 9. What are the rectangular coordinates of 3,135 A. 6 6, B. 3, 3 C., 330 D. 3, 6 6, D. 3, What is the polar form of 7 7i A cos i sin C. 7sin i cos What is the rectangular form of 6cos i sin 3 3 A. B. cos i sin 4 4 D cos i sin i B. 3 3i 3 C i D What is the product of cos i sin and 3cos i sin 4 4 A. C. 5 cos i sin cos i sin cos isin 4cos i sin What is A. 8cos6 i sin6 B. B. 5 5 C. 3cos i sin What is 6 6i i 3 3 6cos i sin D. 5cos i sin cos i sin D. 8cos i sin 6 6 A. 0 B. 3 3i C. 3 3i D What is 3 i 5 3 i 10. What is the polar form of x 4x y 0 A. r r B. r 4x 0 C. r 4cos D. 4sin r 0 A i B. 3 3i 3 C i D i 11. Which of the following is equivalent to i 18 A. i B. 1 C. i D What is the simplest form of A. 1 4i i i 1 B. i C. i 3 3 D i 19. What is 1 i A. 1 i B. i C. i D. i 0. Which of the following is a root of the equation x 1 0 A. 1 3 i B. 1 3 i C. 1 D. i 6 10

(b) ( 1, s3 ) and Figure 18 shows the resulting curve. Notice that this rose has 16 loops.

(b) ( 1, s3 ) and Figure 18 shows the resulting curve. Notice that this rose has 16 loops. SECTIN. PLAR CRDINATES 67 _ and so we require that 6n5 be an even multiple of. This will first occur when n 5. Therefore we will graph the entire curve if we specify that. Switching from to t, we have