1

NAME: ___________________ PERIOD: ___ DATE: ______________ MATH ANALYSIS 2

MR. MELLINA

CHAPTER 11: POLAR COORDINATES & COMPLEX NUMBERS

Sections:v 11.1 Polar Coordinates and Graphs

v 11.2 Geometric Representation of Complex Numbers v 11.3 Powers of Complex Numbers v 11.4 Roots of Complex Numbers

HWSets

SetA(Section11.1)Pages400&201,#’s2-24even.

SetB(Section11.2)Page406,#’s2-24even.

SetC(Section11.3)Page410,#’s2-8even,12,14.

SetD(Section11.4)Page413,#’s2-14even.

2

11.1 POLAR COORDINATES AND GRAPHS (PAGE 394) Objectives:Tographpolarequations 11.1WarmUp!a. Usingthepoint(3,4),findthehypotenuseandtheangleofelevationoftheright

triangleformed.

b. Graphinyourcalculator:± 𝑥# + 𝑦# = '()(*+)*

PolarCoordinatesActivityWeuseorderedpairstoidentifyanddistinguishpointsfromoneanotherintheCartesianplane.Whendoingso,weneedtwocomponents,onetogivethehorizontaldistancefromtheoriginandasecondtogiveaverticaldistancefromtheorigin.Isthereanotherwaytorepresentpointsinaplane?RectangularPlanePlotthepointsA(1,4),B(-3,5),andC(2,-5)onthegraphprovided.WhyarethepointsAandBonoppositeSidesofthey-axis?WhyarethepointsBandCondifferentsidesOfthex-axis?

3

PolarPlaneBelow,youwillfindagraphoftheunitcircle(hasaradiusof___)withgridlinesandtheCartesianplane.LocatethepointinQuadrant1thatismarkedonthecircle.Whatistheapproximateorderedpairthatisassociatedwiththepointmarked?a. Theorderedpairusestwodistances,ahorizontaldistancefromthey-axisandavertical

distancefromthex-axis.Insteadoftwodistances,couldyougettothesamepointbyusinganangleandadistance?Ifso,whatwouldtheybeforthepointmarkedonthegraph?

b. Findthedistanceassociatedwiththepoint(3,2);thenfindtheangle.

4

Example1SketchthePolarCoordinatesa. P(3,20°) b. P(3,380°)c. P(2,50°) d. P(-2,50°)

Polar Coordinates The position of a point P in the plane can be described by giving its ______________ distance r from a fixed point O, called the _______, and the measure of an angle formed by OP and a reference ray, called the __________ __________.

The number r and the angle 𝜃 are called the polar coordinates of P = (r,𝜃) The angle theta can be measured either in degrees or radians.

To find a point when r is negative, find the ray that forms the angle theta with the polar axis, then go r units in the opposite direction from the ray. Another way to find a point when r<0, plot (|𝑟|, 𝜃 + 𝜋) or the point (|𝑟|, 𝜃 + 180°).

5

Example2Ploteachpointwhosepolarcoordinatesaregiven.Thengivetwootherpairsofpolarcoordinatesforthesamepoint.a. (6,50°)b. (-7,120°)c. 3, 9

'

d. −4, <9

=

Example3Ploteachpointwhosepolarcoordinatesaregiven.Thengivetwootherpairsofpolarcoordinatesforthesamepoint.a. (5,-45°)b. (-2,-60°)c. 1,− 9

#

d. −>

#, −3𝜋

6

Example4Fillinthefollowingtablesandsketchthepolargraphof𝑟 = 2 cos 𝜃.

0° 30° 45° 60° 90° 120° 135° 150° 180°r 210° 225° 240° 270° 300° 315° 330° 360°

r Example5Sketchthepolargraphof𝑟 = 2 sin 2𝜃inyourcalculator.Usethefollowingspecifications.Mode=Polar𝜃min=0𝜃max=2𝜋𝜃step= 9

>E

xmin=-2xmax=2xscl=1ymin=-2ymax=2yscl=1

An equation in r and 𝜃 is called a __________ ______________. The graph is called a polar graph. This consists of several points (r, 𝜃) that satisfy the equation.

7

Example6Givepolarcoordinates 𝑟, 𝜃 ,where𝜃isindegrees,foreachpoint.a. (-2,2) b. 2,− 2 c. 3, 1 d. (5,0) Example7Givepolarcoordinates 𝑟, 𝜃 ,where𝜃isinradians,foreachpoint.

a. F#, >

# b. (0,12)

c. (-1,-1) d. (-2,0)

Polar vs. Rectangular Coordinates Converting Polar to Rectangular: 𝑥 = 𝑟______𝜃, 𝑦 = 𝑟______𝜃 Converting Rectangular to Polar: 𝑟 = ±H𝑥#𝑦#, tan 𝜃 =

8

Example8Givetherectangularcoordinatesforeachpoint.a. (4,120°) b. (-3,90°)c. 1, <9

= d. 2, >9

'

Example9Findtherectangularcoordinatesforeachpointtothreedecimalplaces.a. (1,20°) b. (2,20°)c. (1,2) d. (1,-2)

9

Example10:ConvertingPolarEquationstoRectangularUseagraphingcalculatortosketchthepolargraphofeachequation.Also,givearectangularequationofeachgraph.Leaveequationsintermsofxandywithnoplusorminusandnoradicals.a. 𝑟 = sin 𝜃 b. 𝑟 = −3cos 𝜃c. 𝑟 = 1 − sin 𝜃

10

Example11:ConvertingRectangularEquationstoPolarUsewhatyouknowtoconvertfromrectangulartopolarform.Thenuseyourcalculatortosketchthegraph.

a. ± 𝑥# + 𝑦# = )± (*+)*

b. ± (*+)*

#(= F

± (*+)*

c. 𝑥# + 𝑦# = 5𝑥 − ± 𝑥# + 𝑦# d. ± 𝑥# + 𝑦#>= 4𝑥𝑦

11

11.2 GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS (PAGE 403) Objectives:Towritecomplexnumbersinpolarformandtofindproductsinpolarform. 11.2WarmUp!Expanda. 1 − 𝑖 4 + 𝑖 b. 1 + 𝑖 3 1 − 𝑖 3 c. 2 − 𝑖 MExample1:GraphingandFindingtheAbsoluteValueofComplexnumbersGrapheachnumberinthecomplexplaneandfinditsabsolutevalue.a. z=3+2i b. z=4i

Complex Plane Complex numbers can be graphed in the complex plane. The complex plane has a ______ axis and an _______________ axis. The real axis is horizontal, and the imaginary axis is vertical. The complex number a + bi is graphed as the ordered pair ( , ) in the complex plane. The complex plane is sometimes called the ______________ plane.

Recall that the absolute value of a real number is its distance from ________ on the number line. Similarly, the absolute value of a complex number is its distance from zero in the complex plane. When a + bi is graphed in the complex plane, the distance from zero can be calculated using the ______________ theorem.

So if 𝑧 = 𝑎 + 𝑏𝒊, then |𝑧| = √

12

Example1Expressthefollowinginrectangularform:a. 2𝑐𝑖𝑠50° b. 25𝑐𝑖𝑠90° c. 3𝑐𝑖𝑠𝜋d. 4𝑐𝑖𝑠45° e. 6𝑐𝑖𝑠30° f. 9𝑐𝑖𝑠 '9

>

Complex Plane The point representing the complex number z= a + bi can be given in rectangular coordinates (a, b) or polar coordinates (r, 𝜃) Rectangular form: z = a +bi Polar form: z = ______ + ( )i

z = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) is abbreviated as _______

Ex: 2𝑐𝑖𝑠30° = 2(cos 30° + 𝑖 sin 30°)

13

Example2Expressthefollowinginpolarform:a. -1–2i b. 1+i c. id. -3 e. 3 −i f. 2 3 + 2𝑖Example3Giveeachproductinpolarandrectangularform.a. 4𝑐𝑖𝑠25° 6𝑐𝑖𝑠35° b. 5𝑐𝑖𝑠 9

'2𝑐𝑖𝑠 >9

' c. 3 2𝑐𝑖𝑠45° 2𝑐𝑖𝑠60°

How to Multiply Complex Numbers in Polar Form

1. _________ their absolute values 2. _______ their polar angles

Ex: 𝑧F = 𝑟𝑐𝑖𝑠𝛼, 𝑧# = 𝑠𝑐𝑖𝑠𝛽 𝑧F𝑧# = (𝑟𝑐𝑖𝑠𝛼)(𝑠𝑐𝑖𝑠𝛽) =

14

Example4:IndependentWork1.Find𝑧F𝑧#inrectangularformbymultiplying𝑧Fand𝑧#.2.Find𝑧F, 𝑧#,and𝑧F𝑧#inpolarform.Showthat𝑧F𝑧#inpolarformagreeswith𝑧F𝑧#inrectangularform.a. 𝑧F = 2 + 2𝑖 3, 𝑧# = 3 − 𝑖 b. 𝑧F = 2 + 2𝑖, 𝑧# = 2 − 2𝑖

15

11.3 POWERS OF COMPLEX NUMBERS (PAGE 407) Objectives:TouseDeMoivre’sTheoremtofindpowersofcomplexnumbers. 11.3WarmUp!Fora:Expressthecomplexnumberinpolarform.Forb:Writetheexpressioninrectangularform.Forc:Expand

a. 3 − 𝑖 b. 2<𝑐𝑖𝑠 5 ∙ 27°

e. 2 − 𝑖 MExample1Expandthefollowingexpressiona. 𝑟𝑐𝑖𝑠𝛼 # b. 𝑟𝑐𝑖𝑠𝛼 >

Powers of Complex Numbers in Polar Form De Moivre’s Theorem (𝑟𝑐𝑖𝑠𝛼)\ = If 𝑧 = 𝑟𝑐𝑖𝑠𝜃, then 𝑧\ = 𝑟\𝑐𝑖𝑠𝑛𝜃

16

Example2Find𝑧#, 𝑧>, 𝑧', 𝑧<, 𝑧=forthegivencomplexnumberinpolarform.

a. 𝑧 = F#+ >

#𝑖

Example3Givethepolarformofeachofthefollowing

a. 2𝑐𝑖𝑠45° # b. 2𝑐𝑖𝑠 −18°' c. 4𝑐𝑖𝑠 9

=

>

Example4Expresszinpolarform.Calculate𝑧#and𝑧>byusingDeMoivre’sTheorem.Show𝑧, 𝑧#and𝑧>inanArganddiagram.

a. 𝑧 = >#+ F

#𝑖

17

Example5:IndependentWorkExpresszinpolarform.Find𝑧^F, 𝑧E, 𝑧, 𝑧#, 𝑧>, 𝑧', 𝑧<, 𝑧=, 𝑧_,and𝑧M.a. 𝑧 = 1 − 𝑖

18

11.4 ROOTS OF COMPLEX NUMBERS (PAGE 412) Objectives:Tofindrootsofcomplexnumbers 11.4WarmUp!Fora,Evaluatetheexpressionfork=0andk=1.Forb,Solve.a. 𝑐𝑖𝑠 aE°

#+ b∙>=E°

# b. 𝑧' = −16

Example1Findthenthrootofthefollowingcomplexnumbera. Findthecuberootsof8i(indegrees)

Roots of Complex Numbers A nonzero complex number will have:

- 2 square roots. - 3 cube roots. - k kth roots

19

b. Findthefourthrootsof-16(inradians)Example2Findthenthrootofthefollowingcomplexnumbera. cuberootsofi

The nth Roots of a Complex Number The n nth roots of 𝑧 = 𝑟𝑐𝑖𝑠𝜃 are: √𝑧c = 𝑧

dc = 𝑟

d𝑐𝑖𝑠 ef

\+ b∙

\g

20

b. cuberootsof8c. fourthrootsof16

21

Example3:IndependentPracticeUse𝑧F = 3 + 𝑖, 𝑧# = − 3 + 𝑖, 𝑧> = −2𝑖.a. Showthat𝑧F + 𝑧# + 𝑧> = 0.b. Showthat𝑧F𝑧#𝑧> = 8𝑖c. Anycuberootof8imustsatisfytheequation𝑧> = 8𝑖.Showthatthisistrue.Example4:IndependentPracticeThethreecuberootsof8mustsatisfytheequation𝑧> − 8 = 0.Solvethisequation.YouranswersshouldagreewiththoseofExample2b.

22

Example5:IndependentPracticea. Factor𝑧' + 4bywritingitas 𝑧' + 4𝑧# + 4 − 4𝑧#,whichisthedifferenceoftwo

squares.b. Usepart(a)tosolve𝑧' = −4c. UseDeMoivre’stheoremtoverifyyouranswertopart(b).

23

11.1&11.2QuizReviewFornumbers1-3:Plotthefollowingpointswhosepolarcoordinatesaregiven.Converttorectangularcoordinates.Giveanswersinsimplestradicalform.1. 2, 𝜋 2. 3,− <9

= 3. −5, 135°

Fornumbers4-7:Converttherectangularcoordinatestopolar,graphthepolarcoordinate,andgivethetwootherrepresentationsofthepolarcoordinate,onewithapositiverandonewithanegativer.4. −1, 1 inradians 5. 0, 4 inradians6. 5,− 15 indegrees 7. −2 3,−2 indegrees

24

Fornumbers8&9:Converttherectangularequationtopolarform8. 𝑥# + 𝑦# = 2𝑥 9. 𝑦 = 9Fornumbers10&11:Convertthepolarequationtorectangularform.10. 𝑟 1 + cos 𝜃 = 2 11. 𝑟 = 3 sin 𝜃Fornumbers12&13:Graphthefollowingpolarequations.12. 𝑟 = −4 sin 𝜃 13. 𝑟 = 3 cos 2𝜃

25

Fornumbers14-19:Expresseachcomplexnumberinpolarform:14. 3+3i 15. 6–8i 16. -2i17. -3+4i 18. 2–2i 19. 3Fornumbers20-25:Expresseachcomplexnumberinrectangularform:20. 3𝑐𝑖𝑠 9

' 21. 2𝑐𝑖𝑠 '9

> 22. 2𝑐𝑖𝑠 <9

'

23. 4𝑐𝑖𝑠 9

> 24. 2𝑐𝑖𝑠3 25. >

#𝑐𝑖𝑠360°

26

Fornumbers26&27:Grapheachnumberinthecomplexplaneandfinditsabsolutevalue.26. -2–3i 27. 3+4i

27

Chapter11PracticeTest1. Givepolarcoordinatesorrectangularcoordinatesforeachpoint,asindicated. a. 3,−3 ,polar b. 6,−90° ,rect c. 0, 2 ,polar d. 8,−𝜋 ,rect e. 2, 60° ,rect f. 1,− 3 ,polar2. a. Sketchthepolargraphof𝑟 = 2 sin 2𝜃 b. Givearectangularequationofthisgraph

28

3. Sketchthepolargraphofeachequation. a. 𝑟 = 5 b. 𝜃 = −1

c. 𝑟 = 1 + sin 𝜃

29

4. Let𝑧F = − 3 + 𝑖and𝑧# = 4 + 4𝑖. a. Express𝑧F, 𝑧#,and𝑧F𝑧#inpolarform. b. Show𝑧F, 𝑧#,and𝑧F, 𝑧#inanArganddiagram.5. Let𝑧 = 3𝑐𝑖𝑠150°. a. Find𝑧#inpolarformandinrectangularform. b. Showthat𝑧#inpolarformagreeswith𝑧#inrectangularform

30

6. a. Express𝑧 = 1 − 𝑖inpolarform b. Show𝑧, 𝑧#, 𝑧>, 𝑎𝑛𝑑𝑧'inanArganddiagram. c. Find𝑧FE.8. Showthat 2i 𝑐𝑖𝑠130°isacuberootof 3 + 𝑖,andfindtheothertwocuberoots.

31