Post on 03-Dec-2019
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NAME: ___________________ PERIOD: ___ DATE: ______________ PRE-CALCULUS
MR. MELLINA
UNIT 7: 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.
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LESSON 1: POLAR COORDINATES AND GRAPHS Objectives:Tographpolarequations Lesson1WarmUp!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?
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PolarPlaneBelow,youwillfindagraphoftheunitcircle(hasaradiusof___)withgridlinesandtheCartesianplane.LocatethepointinQuadrant1thatismarkedonthecircle.Whatistheapproximateorderedpairthatisassociatedwiththepointmarked?a. Theorderedpairusestwodistances,ahorizontaldistancefromthey-axisandavertical
distancefromthex-axis.Insteadoftwodistances,couldyougettothesamepointbyusinganangleandadistance?Ifso,whatwouldtheybeforthepointmarkedonthegraph?
b. Findthedistanceassociatedwiththepoint(3,2);thenfindtheangle.
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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°).
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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
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Example4Fillinthefollowingtablesandsketchthepolargraphof𝑟 = 2 sin 3𝜃.
𝜃 0° 30° 45° 60° 90° 120° 135° 150° 180°r 𝜃 210° 225° 240° 270° 300° 315° 330° 360°r
Example5Sketchthepolargraphof𝑟 = 2 sin 2𝜃inyourcalculator.Usethefollowingspecifications.Mode=Polar𝜃min=0𝜃max=2𝜋𝜃step= 9
>C
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.
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Example6Givepolarcoordinates 𝑟, 𝜃 ,where𝜃isindegrees,foreachpoint.a. (-2,2) b. 2,− 2 c. 3, 1 d. (5,0) Example7Givepolarcoordinates 𝑟, 𝜃 ,where𝜃isinradians,foreachpoint.
a. D#, >
# 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 = ±F𝑥#𝑦#, tan 𝜃 =
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Example8Givetherectangularcoordinatesforeachpoint.a. (4,120°) b. (-3,90°)c. 1, <9
= d. 2, >9
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Example9Findtherectangularcoordinatesforeachpointtothreedecimalplaces.a. (1,20°) b. (2,20°)c. (1,2) d. (1,-2)
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Example10:ConvertingPolarEquationstoRectangularUseagraphingcalculatortosketchthepolargraphofeachequation.Also,givearectangularequationofeachgraph.Leaveequationsintermsofxandywithnoplusorminusandnoradicals.a. 𝑟 = sin 𝜃 b. 𝑟 = −3cos 𝜃c. 𝑟 = 1 − sin 𝜃
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Example11:ConvertingRectangularEquationstoPolarUsewhatyouknowtoconvertfromrectangulartopolarform.Thenuseyourcalculatortosketchthegraph.
a. 𝑥# + 𝑦# = )*
(*+)* b. (*+)*
'(*= D
(*+)*
c. 𝑥# + 𝑦# = 5𝑥 − 𝑥# + 𝑦# d. 𝑥# + 𝑦#>= 4𝑥𝑦
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LESSON 2: GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS Objectives:Towritecomplexnumbersinpolarformandtofindproductsinpolarform. Lesson2WarmUp!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| = √
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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°)
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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: 𝑧D = 𝑟𝑐𝑖𝑠𝛼, 𝑧# = 𝑠𝑐𝑖𝑠𝛽 𝑧D𝑧# = (𝑟𝑐𝑖𝑠𝛼)(𝑠𝑐𝑖𝑠𝛽) =
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Example4:IndependentWork1.Find𝑧D𝑧#inrectangularformbymultiplying𝑧Dand𝑧#.2.Find𝑧D, 𝑧#,and𝑧D𝑧#inpolarform.Showthat𝑧D𝑧#inpolarformagreeswith𝑧D𝑧#inrectangularform.a. 𝑧D = 2 + 2𝑖 3, 𝑧# = 3 − 𝑖 b. 𝑧D = 2 + 2𝑖, 𝑧# = 2 − 2𝑖
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LESSON 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
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Example2Find𝑧#, 𝑧>, 𝑧', 𝑧<, 𝑧=forthegivencomplexnumberinpolarform.
a. 𝑧 = D#+ >
#𝑖
Example3Givethepolarformofeachofthefollowing
a. 2𝑐𝑖𝑠45° # b. 2𝑐𝑖𝑠 −18°' c. 4𝑐𝑖𝑠 9
=
>
Example4Expresszinpolarform.Calculate𝑧#and𝑧>byusingDeMoivre’sTheorem.Show𝑧, 𝑧#and𝑧>inanArganddiagram.
a. 𝑧 = >#+ D
#𝑖
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Example5:IndependentWorkExpresszinpolarform.Find𝑧^D, 𝑧C, 𝑧, 𝑧#, 𝑧>, 𝑧', 𝑧<, 𝑧=, 𝑧_,and𝑧M.a. 𝑧 = 1 − 𝑖
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LESSON 4: ROOTS OF COMPLEX NUMBERS Objectives:Tofindrootsofcomplexnumbers Lesson4WarmUp!Fora,Evaluatetheexpressionfork=0andk=1.Forb,Solve.a. 𝑐𝑖𝑠 aC°
#+ b∙>=C°
# 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
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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
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b. cuberootsof8c. fourthrootsof16
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Example3:IndependentPracticeUse𝑧D = 3 + 𝑖, 𝑧# = − 3 + 𝑖, 𝑧> = −2𝑖.a. Showthat𝑧D + 𝑧# + 𝑧> = 0.b. Showthat𝑧D𝑧#𝑧> = 8𝑖c. Anycuberootof8imustsatisfytheequation𝑧> = 8𝑖.Showthatthisistrue.Example4:IndependentPracticeThethreecuberootsof8mustsatisfytheequation𝑧> − 8 = 0.Solvethisequation.YouranswersshouldagreewiththoseofExample2b.
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Example5:IndependentPracticea. Factor𝑧' + 4bywritingitas 𝑧' + 4𝑧# + 4 − 4𝑧#,whichisthedifferenceoftwo
squares.b. Usepart(a)tosolve𝑧' = −4c. UseDeMoivre’stheoremtoverifyyouranswertopart(b).
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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
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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
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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°
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Fornumbers26&27:Grapheachnumberinthecomplexplaneandfinditsabsolutevalue.26. -2–3i 27. 3+4i
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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
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3. Sketchthepolargraphofeachequation. a. 𝑟 = 5 b. 𝜃 = −1
c. 𝑟 = 1 + sin 𝜃
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4. Let𝑧D = − 3 + 𝑖and𝑧# = 4 + 4𝑖. a. Express𝑧D, 𝑧#,and𝑧D𝑧#inpolarform. b. Show𝑧D, 𝑧#,and𝑧D, 𝑧#inanArganddiagram.5. Let𝑧 = 3𝑐𝑖𝑠150°. a. Find𝑧#inpolarformandinrectangularform. b. Showthat𝑧#inpolarformagreeswith𝑧#inrectangularform
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6. a. Express𝑧 = 1 − 𝑖inpolarform b. Show𝑧, 𝑧#, 𝑧>, 𝑎𝑛𝑑𝑧'inanArganddiagram. c. Find𝑧DC.8. Showthat 2i 𝑐𝑖𝑠130°isacuberootof 3 + 𝑖,andfindtheothertwocuberoots.
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