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Senior 4 Pre-Calculus Mathematics Solutions for \ 1,." Cumulative ExercisesI, .:

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A Supplement to A 'Foundation for Implementation,"

Manitoba Education and Training

Education et Formation professionnelle Manitoba

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SENIOR 4 ' . PRE-CALCULUS MATHEMATICS SOLUTIONS TO CUMULATI'VE EXERCISES, '

A Supplement to, A Foundation for Implementation'I

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2000Manitoba Education and Training

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Manitoba Education and Training Cata logu ingi n Publication,J

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'Senior 4 pre-calculus mathematics. Solutions to cumulative exercises: a supplement to a foundation for implementation ISBN 0-7711-2787-1 1. Mathematics-Problems, exercises, etc. 2. Calculus-Problems, exercises, etc. 3. Mathematics-Study and teaching (Secondary). , 4. Calculus-Study and teaching (Secondary). 1. Manitoba. Dept. of Education and Training.\ J

Copyright 2000, the Crown in Right of Manitoba 'as represented by the Minister of Education and Training. Manitoba Education and Training, School Programs Division, 197.0 Ness Avenue, Winnipeg, Manitoba R3J OY9.,

Every effort has ,been made to acknowledge original sources and to comply with copyright law. If cases are identified where this has not been done, please notify Manitoba Edu~ation and Training. Errors or omissions will be corrected in a future edition.Bincere thanks to the authors and publishers who allowed their original material to be adapted or reproduced. ,

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Solutions to Cumulative ,"

Exercises

Senior 4 Pre-Calculus

Mathematics

ACKNOWLEDGEMENTSManitoba Education and Training gratefully acknowledges the contributions of the following individuals in the development of Senior 4 Pre-Calculus Mathematics Solutions to Cumulative Exercises: A Supplement to A Foundation for Implementation. Senior 4 Pre-Calculus John Barsby Lynn Batten (until December 1996) Carole 'Bilyk (sinceFebruary 1999) Joseph Combiadakis '(until June 1998) Abdou Daoudi. \ I

Mathematics Revision Committee St. John's Ravenscourt Department of Mathematics School Programs Division Bureau de l'education francaiseI

Independent Schools University of Manitoba Manitoba Education and TrainingI

Manitoba Education and TrainingI

Dave Klasse'n William Korytowski

Vivian'e Leonard Jayesh Maniar Doli Nichol Hilliard Sawchuck,

Connie Shaver Kathy Tetlock Don Trim (since January 1997) Barb Vukets (until June 1997) Wayne Watt Alan Wells

Notre Dame Collegiate (until June 1998) Bureau de l'education francaise (since September 1998) R.D. Parker Collegiate Sisler High (until January 1998) Balmoral Hall (since January 1998) College Beliveau Oak Park High Whitemouth School .(until June 1998) Rivers Collegiate (until June 1996) Silver Heights Collegiate School Programs Division Department of Mathematics River East Collegiate School Programs Division West Kildonan Collegiate (until June 1995)

Mountain S.D. No. 28I'f

Mystery Lake S.I? No. 2355 Winnipeg S.D. No.1

St. Boniface S.D. No.4 Assiniboine South S.D. No.'3 Agassiz S.D. No. 13 Rolling River S.D. No. 39 St. James-Assiniboia S.D. No.2 Manitoba Education and Training University of Manitoba River East S.D. No.9 .Manitoba Education and Traiing Seven Oaks S.D. No_ 10

III

,~Senior 4 Pre-Calculus Mathematics Solutions to Cumulative Exercises

Kindergarten to Senior 4 Mathematics Steering Committee, Lynda Baxter Carole Bilyk (since February 1999) Cheryl Collins Joseph Combiadakis (until June 1998) Brent Corrigan Marcel Druwe (since September 1998) Emmalou Evanson Jack Fraser Lenna Glade Lars Jansson Gene Karlik Gretha Pallen Joanne, Peters Gail Scheer Martin Simmons Richard Skarban Wayne Watt Grant Woods Senior 4 Pre-Calculus John Barsby Barry Boothe Judy Bowes Ruth Dyck John Enns Lyn Jannuska Dennis Kasian David Klassen Bill Korytow.ski Kelly Livesley Joyce McCallum Don Nichol Christine Ottawa Larry Ruppel Gordon Smith Ketri Wilkes St. Vital S.D. No.6' School Programs Division Arthur Day School Bureau de l'education francaise St. -Iames-Aeainiboia S.D. No.2I

Manitoba Association of School Superintendents " Manitoba Education and Training Transcona-Springfield S.D. No. 12 'II I Manitoba Education and Training Manitoba Associati~n of Mathematics Teachers' .Manitoba Education and TrainingI'

Bureau de l'education francaise Fort la Bosse S.D. No. 41 H.L. Wyatt School Bernie Wolfe Community School Faculty of Education Red River Com:munity College School Programs Division Winnipeg S. D. No.1

River East S.D. No.9 I'J ackhead School Mystery Lake S.D. No. 2355 School Programs Division Department of Mathematics

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Manitoba Teachers" Society , St. Vital S. D. No.6 Transcona-Springfield S.D. No. 12 , University of Manitoba Community Colleges I I Manitoba Education and Training Manitoba School Library , Association I Manitoba Association of School Trustees , ,, Ma'~itoba Association of 'Parent Councils Manitoba Association of Principals Matutoba Education and Training University of Manitoba 'I I

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Mathematics Cumulative Exercises Committee St. John's-Ravenscourt Fort Richmond Collegiate St. -Iohn's-Ravenscourt Westgate Mennonite Collegiate Westgate Mennonite Collegiate Crocus Plains Regional Secondary School River East Collegiate R.D. Parker Collegiate Balmoral Hall John Taylor Collegiate Morden Collegiate Whitemouth School St. John's High School River East Collegiate MacGregor Collegiate John Taylor Collegiate Independent Schools Fort Garry S.D. No.5, Independend Schools Independent Schools Independent Schools Brandon S.D. No. 40I

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River East S.D. No.9 Mystery Lake S.D. No. 2355 Independent Schools St. James Assiniboia S.D. No.2 Western S.D. No. 47 Agassiz S.D. No. 13 Winnipeg S.D. No.1 River East S.D. No.9 Pine Creek S.D. No. 30 St. James-Assiniboia S.D. No.2

iv

Solutions to Cumulative

Exercises

Senior 4 Pre-Oalcutus Mathematics"

Senior 4 Pre-Calculus

Mathematics Field Validation (Pilot) Project (1997-0( Brandon S.D. No. 40 River East S.D. No.9 Fort Garry S.D. No.5 Portage la Prl:!-irie,$.D.No. 24 Assiniboine South S.D. No.3 Western S.D. f-Jo.47 Winnipeg S.D. No.1 Midland S.D. No. 25 ' Swan Valley S.D. No. 3~,j

Dave Barnes (1998-99) Neelin High School River East Collegiate Darryl Bohart , Barry Boothe (1997-99) Fort Richmond Collegiate Portage Collegiate Raf Borelli (1998-99) Chris Boumford (1998-00) Shaftesbury High School Paul Carpenter 'Morden Collegiate Pat Clark Gordon Bell School Carman Collegiate , ', Kim Deamal (1999-00) Darren Deserranno (1997-98) Swan Valley Regional , Secondary School Ruth Dyck Westgate Mennonite Collegiate John Enns John Fast (1997-99) Dean Favoni Arlyn Filewich Harland Garinger -Ian Gibson (1998-00) Ewa Goldfeld (1998-00) Chuck Hatfield (1999-00) Jerry Hildebrand (1998-00) Erwin Hildebrandt (1999-00) Greg Jakubec (1998-00) Lyn Jannuska Brian Johnston (1998-99) Dennis Kasian Doug Kinney (1998-00) Dave Klassen (1997-99) Peter Kula (1999-00) Doug Kutcher (1999-00) Philippe LeClercq (1998-00) Don Lepper Kelly Livesley (1997-98) Tanis MacDonald (1998-00) Bill Marshall Joyce McCallum Melanie McCasin (1998-00) Pat McCrea (1998-00) Ralph McKay (1998-00) Patty Miller Les Mitchell Mike Nantais Don Nichol (1997-98) Joe Oczerklewicz (1998-99) Rob Pehura (1998-00)

" Manitoba Federa tion ,of, Independent , Schools Westgate Mennonite Collegiate Manitoba Federation of Independent Schools River East Collegiate River East S.:O. Nb. 9 St. Vital S.D. No.6 Dakota Collegiate ,Fort Richmond Collegiate Fort Garry S.D. Nb. 5 Hapnot Collegiate Flin Flon S.D. No. fo6 ,,' Kelvin High School Winnipeg S.D. No.1 Daniel McIntyre Collegiate Winnipeg S.D. No.1 Assiniboine South S.D. 'No. 3 Shafteshury High School Vincent Massey Collegiate " Fort Garry S.D. No.5, River East Collegiate River East S.D. No.9, ': Assiniboine South'S.D. No.3 Oak Park High Crocus Plains Regional ~ Brandon S.D. No. 40, ' Secondary School Agassiz S.D. No. 13 White mouth School, River East S.D. No'. 9 River East Collegiate Brandon S.D. No. 40 Neelin High School R.D. Parker Collegiate Mystery Lake S.D~No.;2355 Transcona-Springfield S.'D,.No. 12 Springfield Collegiate Agassiz D.D. ~o.' 13., \, Whitemouth School Fort Garry S.D. Nb. 5 " Vincent Massey Collegiate Sanford Collegiate Morris-Macdonald S.D. No. 19 St. James-Assinib~ia S.D. No, 2 John Taylor Collegiate Vincent Massey High School Brandon S.D. No. 40 Swan Valley Regional Swan Valley S.D. No. 35 Secondary School Morden Collegiate Western S.D. No. 47 Kelvin High School Winnipeg S.D. No.1 Portage la Prairie S.D. No. 24 Portage Collegiate Windsor Park Collegiate St. Boniface S.D. No.4 R.D, Parker Collegiate Mystery Lake S.D. No. 2355 Fort Garry S'.D. No.5 Fort Richmond Collegiate Turtle Mountain S.D. No. 44 Boissevain School Agassiz S.D. No. 13 Whitemouth School John Taylor Collegiate St. James-Assiniboia S.D. No.2 Winnipegosis Collegiate Duck Mountain S.D. No. 34

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Senior 4 Pre~CalculusMathematics '"

Solutions to Cumulative

Exercises

Senior 4 Pre-Calculus

Mathematics Field Validation (Pilot) Project (1997-00) (continued) Kelvin High School Grant Park High School River East Collegiate R.D. Parker Collegiate Dakota Collegiate Vincent Massey' High School ' Macflregor Collegiate Grant Park High School Miami Collegiate Kelvin High School Oak Park, High Springfield Collegiate John Taylor Collegiate Arthur Meighen High Vincent Massey Collegiate Winnipeg S.D. No.1 Winnipeg S.D. No.1 River East S.D: No.9 Mystery Lake S.D. No. 2355 St. Vital S.D. No.6 Brandon S.D. No. 40 Pine Creek S.D. No. 30 Winnipeg S.D. No.1 Midland S.D. No. 25 Winnipeg S.D. No.1 Assiniboine South S.D.' No.3 , Transcona-Springfield S.D. No. 12 St. James-Assiniboia S.D. No.2 Portage la Prairie S.D. No. 24 Fort Garry S.D. No.5

Peter Pura (1998-00) , Norm Roseman (1999-00) Larry Ruppel Gerry Sapinsky (1997-98) John Scott ' , Mark Sefton (1998-00) Gordon Smith Roy Spivak (1998-00) Wanda Stockford (1998-00) Jim Stuparyk (1998-00) Hans Tauchnitz (1998-00) Egon Ulr~ch (1998-99) Ketri' Wil~es (1998-99) Rick Williamson (1998-00) Fra,Qk WiJson (1998-00) ,

Mani~oba,Education and Training Staff School Programs Division Carole Bilyk Lee-Ila Bothe'I, ,',

I''''' 'Curriculum Unit , Program Development Branch Production Support Unit Program Development Branch Curriculum Unit ' Program Development Branch Learning Technologies Unit Program Development Branch Curriculum Unit , Program Development Branch

Consultant Coordina tor Project Manager Consultant Project Leader

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DianeCoo~ey ,,

Katharine Tetlock Wayne Wfltt '

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SolutiQns to Cumulative

Exercises

Senior 4 Pre-Calculus

Mathematics

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CONTENTSAcknowledgements m Introduction Exercise1

1 Degree and Radian Measure 3 The Unit Circle 5 Special Angles and the Trigonometric Functions 10 Solving Trigonometric Equations on a Specified Interval General Solution of Trigonometric Equations. 15 Graphing Circular Functions 18 Translations 21 \ I'~ , Horizontal and Vertical Stretches 25, Symmetry, 'Reflections, and Inverses 28 Graphing Reciprocals 32 Graphing Absolute Values 36 Practice with Transformations 40 I Transformations with Trig Functions 13 Trigonometric Identities I 46 Trigonometric Identities II 48 Sum and Difference Identities I 51 Sum and Difference Identities II 55 Double Angle Identities 58 Exponential Functions 61 Solve Exponential Equations 64 Logarithmic Functions 66 Logarithmic Theorems I 69 Logarithmic Theorems II 72 Exponential and Logarithmic Equations I 75 Exponential and Logarithmic Equations II 79 Natural Logarithms 82 Applications of the Exponential Function 85 Counting Principles 89 Permutation with Repetitions and Restrictions 92 Permutations 95 Circular Permutations 98 Permutations with Case Restrictions 102 Combinations 105

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78 9 10 11

12 13 14 15 16 1718 " 19

20 21 22 23 24 25 26 27 2829

30 31 3233

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Senior

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Mathematics

Solutions to Cumulative Exercises

Exercise

.'Binomial Theorem 108 Permutations, Combinations, and Binomial The'brem 111 Parabola 114 Circle and Ellipse 117 Hyperbola 121 '" Sample Spaces 126 Probability of Independent .and Dependent Events 129 Combining Probabilities lif2 Conditional Probability I 'J 35 Conditional Probability II 138 " Probability Using Permutations and Combinations .. ~41 Geometric Sequences 144 Geometric Series 147 Infinite Geometric Series 150 Review I 153 Review II 157 ,,' Review III 161I

34 3536

3738 39 40 41

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Solutions to Cumulative

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Senior 4 Pre-Calculus

Mathematics

INTRODUCTIONSenior 4 Pre-Calculus Mathematics Solutions to Cumulative Exercises: A Supplement 'to A Foundation for Implementation initially was prepared by members of the Senior 4 Pre-Calculus Mathematics Cumulative Exercises Committee. "I The initial solutions were provided to the field test teachers during the 1997-98 school year. As the cumulative exercises were revised, pther Manitoba educatora'participated in the' further , development of Senior 4 Pre-Calculus Mathematics Solutions ~ Cumulative Exercises.,

This document is provided to Manitoba teachers of Senior 4 Pre-Calculus Mathematics to support implementation of the new curriculum. It is intended to provide both teachers and students' with sample solutions and/or alternate solutions.

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Solutions to Cumulative Exercises

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Exercise 1: Degree and Radian Measure

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Solutions to Cumulative

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Senior 4 Pre-Calculus

Mathematics

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