UNIT TWO POLAR COORDINATES AND COMPLEX NUMBERS MATH 611B 15 HOURS

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1 UNIT TWO POLAR COORDINATES AND COMPLEX NUMBERS MATH 611B 15 HOURS Revised Dec 10, 02 38

2 SCO: By the end of grade 12, students will be expected to: C97 construct and examine graphs in the polar plane Elaborations - Instructional Strategies/Suggestions Polar Coordinates (9.1) Invite student groups to read and discuss AMC p Terms students should become familiar with are: < pole < polar axis < polar equation Student groups should be able to graph points and polar equations containing either r or 2. Students should recognize that the polar coordinates are not specific to only one point much like angles in standard position can be defined by more than one standard angle. If a point P has polar coordinates (r,2), then P can also be represented as (r, 2 + 2Bk) or (r, 2 + (2k + 1)B), where k is any integer. Student groups should recognize the distance formula for a Polar Plane 2 2 PP = r + r 2rr cos( θ θ ) as the cosine formula in the rectangular plane. 2 2 c = a + b 2ab cosθ Note to Teachers: Students will be expected to research the applications and advantages of the Polar Plane. Note to Students: When graphing polar equations, students should obtain a table of values and draw the graph on polar graph paper. Use the TI-83 to check your work. 39

3 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Research/Presentation Research the applications and advantages of polar coordinates and, as a group, present your group s findings to the class. Polar Coordinates Worksheet (9.1) (see Workbook) Research/Presentation Write a short paper on the life of Jakob Amsler-Laffon and his contributions to engineering. Pencil/Paper Graph each point in the polar plane: a) (3,30 ) b) (!2,90 ) c) (2,5B/6) d) (4,!60 ) e) (!3,!B/4) f) (2,270 ) Pencil/Paper Name three other pairs of polar coordinates for each point: a) (3,B/2) b) (2,65 ) Pencil/Paper Graph each polar equation: a) r = 2 b) 2 = 2B/3 c) r =!1 d) 2 =!150 Group Activity Find the distance between points (3,150 ) and (5,100 ). 40

4 SCO: By the end of grade 12, students will be expected to: C97 construct and examine graphs in the polar plane Elaborations - Instructional Strategies/Suggestions Graphs of Polar Equations (9.2) Student groups should read and discuss AMC p The classical curves using polar coordinates are described in the table on p.564. There are three major categories: < rose - of which a lemniscate is a special case. < limacon - of which a cardioid is a special case. < spiral Example: Graph r = 2 cos 32. The general equation for a rose is r = a cos n2; where n is even, then there are 2n petals where n is odd, then there is n petals General Equations: < rose - r = a cos n2 or r = a sin n2 < lemniscate of Bernoulli- r 2 = a 2 cos 22 or r = a 2 sin 22 < limacon of Pascal - r = a + b cos 2 or r = a + b sin 2 < cardioid - r = a + a cos 2 or r = a + a sin 2 < spiral of Archimedes - r = a 2 ( 2 must be in radians) Extension: < Cissoid of Diocles - r = 2a tan 2 sin 2 41

5 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Research/Presentation Research and present, as a group, to the class on the life and times of one of the following. Include in the presentation a description of the classical curve they were responsible for developing: a) Etienne Pascal b) Jacob Bernoulli c) Archimedes Group Activity Identify the type of curve each represents and graph each of the following polar equations: a) r = 2 cos 32 b) r = 32 c) r 2 = 4 cos 22 d) r = sin 2 e) r = cos 2 Polar Equation Worksheet (9.2) (see Workbook) Note to Teachers: The locus of a cardioid is a point on the circumference of a circle that is rolling around the circumference of a circle of equal radius. Communication Write an equation for a rose with 4 petals and describe your equation and graph to the class. 42

6 SCO: By the end of grade 12, students will be expected to: A27 translate between polar and rectangular coordinates Elaborations - Instructional Strategies/Suggestions Polar and Rectangular Coordinates (9.3) Student groups should read and discuss AMC p The trigonometric definitions: cos θ = x r x = sin θ = y r y = r cos θ r sin θ can be used to convert from polar to rectangular coordinates. To convert from rectangular to polar coordinates use: 2 2 r = x + y y tan θ = θ = x Arc tan y x Note to Teachers: At the end of the unit is an explanation of how to use the TI-83 for converting degrees to radians and vice versa. As well, the explanation shows how to convert from polar coordinates to rectangular coordinates and vice versa (See p. 61). 43

7 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Polar and Rectangular Coordinates (9.3) Pencil/Paper Find the polar coordinates of the following points in rectangular form. Use 0 # 2 < 2B and r $ 0. a) (8,14) b) (!3,5) Polar and Rectangular Coordinates Worksheet (9.3) (see Workbook) Pencil/Paper Write each point using rectangular coordinates: a) (2,120 ) b) (!3,B/2) Pencil/Paper Write each polar equation in rectangular form: a) r = 3 cos 2 b) r =!4 c) 2 = B/4 Pencil/Paper Write each rectangular equation in polar form: a) x = 4 b) y =!2 c) x 2 + y 2 = 16 d) x 2 + y 2 = 6y Group Discussion tanθ Write r = cos θ in rectangular form. When converted to rectangular form we get y = x 2 ". 44

8 SCO: By the end of grade 12, students will be expected to: B43 simplify and perform operations on complex numbers Elaborations - Instructional Strategies/Suggestions Simplifying Complex Numbers (9.5) Invite student groups to read and discuss AMC p Students should review complex numbers in rectangular form. A complex number of the form a + bi, where a is called the real part and b is called the imaginary part. 1. If b = 0, the complex number is a real number (part of 2). 2. If b 0, the complex number is an imaginary number. 3. If a = 0 & b 0, the complex number is a pure imaginary number. (part of 2). Note: All previous sets of numbers are subsumed by the set of complex numbers. Note to Teachers: Complex numbers were covered in Math 521A, Unit 2 MathPower 11 p.185 Students will be expected to be able to perform operations on complex numbers. They should as well be exposed to the concept of complex conjugates. Note to Teachers: At the end of the unit is a short essay on The Development of Number Systems which may be useful (See p.64). 45

9 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Simplifying Complex Numbers (9.5) Pencil/Paper Simplify: Simplifying Complex Numbers Worksheet (9.5) (see Workbook) a) i 3 b) i 12 c) ( 3 + 5i) + ( 7 + i) d) ( 3 + 2i) 2 2i e) 1 5i f) ( 2 5i)( 4 + i) Journal Write to explain how to simplify any integral power of i. 46

10 SCO: By the end of grade 12, students will be expected to: A26 translate between polar and rectangular coordinates on the complex plane C88 represent complex numbers in a variety of ways Note to Teachers: Replacing 2 with B yields Euler s Equation; π i e = cosπ + isinπ π i e = 1 π i e + 1= 0 Elaborations - Instructional Strategies/Suggestions Complex Numbers in Polar Form (9.6) Invite student groups to read and discuss AMC p Students will be expected to convert complex numbers from rectangular to polar form and vice versa. Students should be familiar with the following concepts: For complex numbers in rectangular form: < Argand Plane < real axis, imaginary axis < absolute value of a complex number. 2 2 If z = a + bi, then z = a + b This absolute value represents the distance from zero on the complex plane. For complex numbers in polar form: < modulus, r (absolute value of the complex number) < argument, 2 (amplitude of the complex number or the angle between r and the zero line) θi Euler s Formula states that e = cos θ + i sin θ θ < z = r(cos θ + i sin θ) = re i Note to Teachers: If a complex number is in rectangular form, then plot it on a rectangular coordinate plane. Ex 4 p.588: Express!3 + 4i in polar form. Using the TI-83: enter!3 + 4i math < < CPX 7: < Polar enter If it is in polar form, graph it on a polar coordinate plane. enter or 5 cis 2.21 To convert to rectangular form press math < < CPX 6: < Rect enter... 47

11 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Complex Numbers in Polar Form (9.6) Journal Write to explain how to find the absolute value of a complex number. Complex Numbers in Polar Form Worksheet (9.6) (see Workbook) Pencil/Paper Solve for x and y: 2x + y + 3xi + 5yi = 7 + 4i Pencil/Paper Graph!3! 2i on the Argand Plane and find its absolute value. Pencil/Paper Express 4! 3i in polar form. Pencil/Paper Graph 3(cos 2 + i sin 2) and express in rectangular form. Group Research/Group Presentation Write a short paper on the contributions of Jean Robert Argand to mathematics. 48

12 SCO: By the end of grade 12, students will be expected to: B42 multiply and divide complex numbers in polar form Elaborations - Instructional Strategies/Suggestions Products and Quotients of Complex Numbers in Polar Form (9.7) Invite student groups to read and discuss AMC p Product of complex numbers in polar form: r1(cosθ1+ isin θ1) r2(cosθ2 + isin θ2) iθ iθ = re 1 re 2 i( θ1+ θ2) = rre 1 2 = rr 1 2[cos( θ1+ θ2) + isin( θ1+ θ2)] = rrcis( θ + θ ) The modulus (r 1 r 2 ) of the product of 2 complex numbers is the product of their modulii. The argument or amplitude ( ) of the product of 2 complex numbers is the sum of their arguments. Thus rcisθ rcisθ 2 2 = rrcis( θ + θ ) Ex. o 4cis45 5cis15 = 20cis60 o o Quotient of complex numbers in polar form: r1(cosθ1+ isin θ1) r (cosθ + isin θ ) r1 e i( θ1 θ2) = r 2 r [cos( ) i sin( )] 1 = θ1 θ2 + θ1 θ2 r2 r1 = cis ( θ1 θ2) r2 The modulus of the quotient of 2 complex numbers is the quotient of their modulii. The argument is the difference of their arguments. 49

13 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Products and Quotients of Complex Numbers in Polar Form (9.7) Pencil/Paper Find each product or quotient. Convert the answer into rectangular form. a) 4 cis 3 cis 20 b) 8 cis 2 cis 40 o 10cis90 c) o 5cis25 o 12cis40 d) o 2cis60 π π e) 4cis 9cis 3 4 2π 25cis f) 3 π 5cis 6 Products and Quotients of Complex Numbers in Polar Form Worksheet (9.7) (see Workbook) 50

14 SCO: By the end of grade 12, students will be expected to: B44 derive and apply De Moivre s Theorem for powers and roots Elaborations - Instructional Strategies/Suggestions Powers and Roots of Complex Numbers (9.8) Invite student groups to read and discuss AMC p De Moivre s Theorem n n z = [(cos r θ + isin θ )] iθ n = [ re ] n inθ = re n = r [cos nθ + isin nθ ] n = rcisnθ Looking at examples 1&2 on p.599 & 600 we see the answer to be It is important for students to see the process by which the answer was arrived at but they should also be aware that the calculator can do the work as well. A useful application of De Moivre s Theorem is in finding the roots of a complex number. The theorem can be re-written as: p p p p θ + 2kπ z = [ rcisθ] = [ rcis( θ + 2 kπ)] = r cis p where k = 0,1,...,p!1 For instance we would normally think of the answer of in fact there are two other cube roots ( 1 + 3i) and ( 1 3i). 3 8 to be 2 but In the essay, The Development of Number Systems,at the end of the unit we will solve this example and others in detail. 51

15 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Powers and Roots of Complex Numbers (9.8) Pencil/Paper/Technology Find ( 3 5i) 4. Express the result in both rectangular and polar form. Powers and Roots of Complex Numbers Worksheet (9.8) (see Workbook) Using the TI-83: note the mode Technology Use the TI-83 to find all roots for x 4! 1 = 0. Presentation Present to the class a summary of the life and contributions of Benoit Mandlebrot. Research/Presentation Investigate the life of Abraham De Moivre and his contributions to mathematics. 52

16 Polar Earth Map (9.1) 53

17 Hodographs (9.1) A hodograph is a plot representing the vertical distribution of horizontal winds using polar coordinates. A hodograph is obtained using data from a radiosonde balloon. By plotting the end points of the wind vectors (wind speed and direction) at various altitudes and connecting these points in order of increasing height. Interpretation of a hodograph can help in forecasting the subsequent evolution of thunderstorms. 54

18 Polar Planimeter (9.1) Planimeters don t calculate in the sense of allowing a user to enter some numbers and producing a result. They do, however, allow the user to calculate the area of any closed shape. They are, in essence, an integration machine. Visit for more details. 55

19 Polar Equation Worksheet (9.2) 1. Graph the following classical curves. Inductively state the pattern that evolves. a) r = cos 2 b) r = 2 cos 2 c) r = 3 cos 2 d) What effect does a have on the graph? e) Make a conjecture as to how the r = sin 2 graph compares with the r = cos 2 graph. 2. Verify your conjecture by graphing the following: a) r = sin 2 b) r = 2 sin 2 c) r = 3 sin 2 3. Graph the following classical curves. Inductively state the pattern that evolves. a) r = 2 cos 2 b) r = 2 cos 22 c) r = 2 cos 32 d) What effect does n have on the graph? e) Make a prognostication as to what r = 2 cos 52 would look like. Verify your prognostication by graphing. 4. Graph the following classical curves. Inductively state the pattern that evolves. a) r = cos 2 b) r = cos 2 c) r = cos 2 d) r = cos 2 e) What effect does a have on the graph? f) Which of these curves is a cardioid? g) Predict what the graph of r = sin 2 would look like. Verify your prediction by graphing. h) Take part (c) above and investigate what effect changing the sign of a and/or b has on the basic graph. Check your work by trying a few more examples. Finally make a statement about the effects of the size and sign of these coefficients on the graph and convey this to the class. 5. Graph the following classical curves: a) r 2 = 4 cos 22 b) r 2 = 4 sin 22 c) What is the name for this particular classical curve? 6. Graph the following classical curves. Inductively state the pattern that evolves a) r = 2 b) r = 22 c) r = 32 d) r =!2 e) r =!22 f) r =!32 g) What effect does the a coefficient have on the graph? 56

20 Polar Graph Paper (9.2)

21 Polar Graph Paper (9.2)

22 Spiral Graph Example (9.2) 59

23 Use of TI-83 (9.3) Convert degrees to radians Convert 50 to radians. Have calculator in radian mode. Press 50 2 nd Angle 1: enter Convert radians to degrees Convert 2.3 to degrees. Have calculator in degree mode. Press nd Angle 3: enter Convert rectangular to polar coordinates Convert (!8,!12) to polar coordinates. Press 2 nd Angle 5:R < P r (!8,!12) enter Press 2 nd Angle 6:R < P 2 (!8,!12) enter This illustrates the limitations of calculators. The answer should be If the calculator was in radian mode the answer would be given in radians. The answer in polar coordinates is (14.4, 236.3) or (14.4, 4.1). Convert polar to rectangular coordinates Convert (2,80 ) to rectangular coordinates. is in degrees. Calculator must be in degree mode when angle in question Press 2 nd Angle 7:P < R x (2, 80) enter Press 2 nd Angle 8:P < R y (2, 80) enter The answer in rectangular coordinates is (.35, 1.97). 60

24 Number Systems (9.5) 61

25 Graphic of Powers of Complex Numbers (9.8) 62

26 The Development of Number Systems The Egyptians invented the number system we use today replacing the Roman numeral system making computations much simpler. Once people started borrowing gardening implements from their neighbours, the need arose for negative numbers. When a hunter tried to share with his 5 friends the 3 geese he had bagged, then a need for fractions was born (either that or he didn t tell some of his friends about his good fortune). Once algebra came into play, problems like x 2! 5 = 0 needed to be dealt with using irrational numbers. These number systems taken together completely filled the number line (the real number line ú). All seemed right with the world until the Renaissance when some troublemaker looked for a solution to x = 0. This created a problem because any real number when squared gives a result greater than or equal to zero. To a chorus of protests from many of the prominent mathematicians of the day, the number i = 1 was defined. These objecting mathematicians coined the phrase imaginary number to voice their opposition. An important property of this number is that when squared it yields a negative result; i 2 = 1. It was evident that numbers like 2i, 5! 4i, etc. were very useful, but there was no way of representing these numbers on the real number line. The solution to this dilemma was put forth by a Swiss mathematician, Jean Robert Argand. He placed an imaginary number line at right angles to the real number line. This is called the Argand Plane. For all real numbers a,b the number a + bi is a complex number. The letter, C, is used to represent the set of complex numbers. We can represent a complex number z = a + bi as a vector on the Cartesian coordinate plane or the Argand plane. T he conjugate of a complex number z = a + bi is z = a bi. Thus 4! 3i is the conjugate of 4 + 3i. The modulus(magnitude), r, or absolute value of a complex number z = a + bi is 2 2 r = z = a + b. This is a measure of the length of the vector z = a + bi. 63

27 The argument, 2, or amplitude of a complex number z = a + bi is Arg( a bi) θ tan This is the angle the vector makes with the positive side of the horizontal axis. b a F + = = H G I K J 1 From trigonometry we know that: a cos θ = or a = r cos θ r b and sin θ = or b = r sin θ. r Thus z = a + bi = rcos 2 + irsin 2 = r(cos 2 + isin 2) = rcis 2 Multiplication of complex numbers The modulus of the product of two complex numbers is the product of their modulii. The argument of the product of two complex numbers is the sum of their arguments. iθ1 iθ2 i( θ1+ θ2) re re rre rcis rcis rrcis = = θ θ = ( θ + θ ) Ex. 4cis 5cis 15 = 20cis 60 Division of complex numbers The modulus of the quotient of two complex numbers is the quotient of their modulii. The argument of the quotient of two complex numbers is the difference of their modulii. rcis 1 θ 1 rcisθ 2 2 r1 = r cis ( θ1 θ2) 2 Ex. o 8cis75 2cis30 o = o 4cis45 64

28 De Moivre s Theorem It states that: n n n z = [ rcisθ] = r cisnθ ; it is useful in finding the p th root of a complex number. 1 p p p θ z = [ rcisθ ] = r cis p Without De Moivre s Theorem problems like; find the cube roots of 8; will not yield all possible roots. A p th root problem should yield p roots. To find all possible roots we can use the fact that: sin θ = sin( θ + 2π ) = sin( θ + 4π ) =... = sin( θ + 2kπ ) cos θ = cos( θ + 2π ) = cos( θ + 4π ) =... = cos( θ + 2kπ ) Therefore z p p = [ rcisθ ] = [ rcis( θ + 2 kπ)] 1 p p p θ + 2kπ θ + 2kπ θ + 2kπ = r cis( ) or r (cos + isin ) p p p Evaluating this formula for k = 0,1,2,...,p!1 will yield the p roots. Ex. Solve for all roots and represent them on the Argand plane. Solution: 8 means 8 + 0i where a = 8 and b = 0. Thus z = r = 8 and θ = θ = tan 1 0 b a Using r = 8, 2 = 0, p = 3, k = 0,1,2 p p θ + 2kπ θ + 2kπ z = r (cos + isin p p 3 3 when k = 0 8 = 8 (cos0+ isin0) = 2 when k = 1 2π 2π = 8 (cos + isin ) = i when k = 2 4π 4π = 8 (cos + isin ) = i 65

29 Verify: 2 3 = 8 ( 1 + 3i) = 8 ( 1 3i) = T o get the previous screens, use r = 8, 2 = 0, p = 3, k = 0,1,2 X 1T = r(to the desired root) cos T Y 1T = r(to the desired root) sin T T min = 2/p = 0/3 = 0 T max = T min + 2B = 2B T step = 2B/p = 2B/3 Press Trace and < 66

30 Ex 3 p.601 AMC Find 3 8i or the cube roots of 8i. Check your work by graphing on the TI-83. Solution π π Convert 8i to polar form. r = 8, 2 = B/2 8i = 8[cos + isin ] 2 2 Using DeMoivre s Theorem r = 8, 2 = B/2, p = 3, k = 0,1,2 π π + 2kπ 2kπ cos 2 + i sin 2 = θ z p r p cos k π θ + isin k π = + p p when k = π π 8 = 8 cos + isin = 3+ i 6 6 when k = π π 8 = 8 cos + isin = 3+ i 6 6 when k = π π 8 = 8 cos + isin = 2i 2 2 Using the TI-83 (use the steps on p.602). 67

31 To get the previous screens, use r = 8, 2 = B/2, p = 3, k = 0,1,2 X 1T = r(to the desired root) cos T Y 1T = r(to the desired root) sin T T min = 2/p = B/2/3 = B/6 T max = T min + 2B = 13B/6 T step = 2B/p = 2B/3 68

32 Ex 4 p.601 AMC Find the three cube roots of!2!2i and check your work by graphing on the TI-83. Solution Convert!2!2i to polar form. r = 2 2, 2 = 5B/4 5π 5π 2 2i = 8 cos + isin 4 4 p p θ + 2kπ θ + 2kπ z = r (cos + isin p p 5π 5π + 2kπ 2kπ 3 3 ( 2 2 i) (2 2) (cos 4 + isin 4 = when k = 0 1 5π 5π 3 ( 2 2 i) = 2(cos + isin ) = i when k = π 13π 3 ( 2 2 i) = 2(cos + isin ) = i when k = ( 2 2 i) 2(cos π = + i sin ) = 1 i Using the TI-83( and steps on p.602) 69

33 To get the previous screens, use 5π r = 2 2, θ =, p = 3, k = 0,1,2 4 X 1T = r(to the desired root) cos T Y 1T = r(to the desired root) sin T T min = 2/p = 5B/4/3 = 5B/12 T max = T min + 2B = 29B/12 T step = 2B/p = 2B/3 70

34 Ex 5 p.603 AMC Solve for all roots x 5! 32 = 0. Solution: x 5 = 32 This reads as: find the fifth roots of = i = 32(cos 0 + isin 0) r = 32, 2 = 0, p = 5, k = 0,1,2,3,4 p p θ + 2kπ θ + 2kπ z = r (cos + isin p p p 32 = 32 (cos 0 + isin 0) = 2 k = 0 2π 2π = 32 (cos + isin ) = i 5 5 k = 1 4π 4π = 32 (cos + isin ) = i 5 5 6π 6π = 32 (cos + isin ) = i 5 5 k = 2 k = 3 8π 8π = 32 (cos + isin ) = i 5 5 k = 4 cursor through the roots Pre ss Trace and < The points are vertices of a regular pentagon and are concyclic in nature. That is they are equally spaced on a circle. 71

35 X 1T = r(to the desired root) cos T Y 1T = r(to the desired root) sin T T min = 2/p = 0/3 = 0 T max = T min + 2B = 2B T step = 2B/p = 2B/5 Fundamental Theorem of Algebra(extension) One of the most important uses of complex numbers is in solving equations in engineering and the n n 1 sciences of the type: ax 0 + ax an 1x + an = 0 where a 0 0 and a 1,...,a n are complex numbers. The FTA states that the above equation has at least one complex root. 72