Technology Supported Learning Course
Mathematics C30
Module 1 – Lessons 1–5
2009 Printing
Mathematics C30
Copyright © Saskatchewan Ministry of Education
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Mathematics C30
Course Introduction
i
Course Introduction
Mathematics C30 is the third of three Grade 12 level mathematics courses.
The topics covered are



Trigonometry
The Conics
Mathematical Proof
There are ten lessons to this course, each followed by an assignment to be submitted for
evaluation. Each of the ten assignments must be completed satisfactorily for the students
to be eligible to write the departmental exam and for a term mark to be submitted.
Module 1 contains Lessons 1 to 5 and Module 2 contains Lessons 6 to 10. Lessons 9 and
10 are review lessons.
Although the course is self-contained, further examples and practice exercises may be
obtained from any one of the references listed below.
Text Book References
1.
Algebra and Trigonometry (sixth edition), Keedy, Bittinger, and
2.
Algebra and Trigonometry (fifth edition), Sobel and Lerner, (1995),
3.
Algebra and Trigonometry Structure and method, Book 2, Dolciani,
Beecher, (1993), Addison Wesley
Prentice-Hall Inc.
Sorgenfrey, Brown, and Kane, (1996 or 1998), Houghton Mifflin
Company.
Your school or public library may have other texts that contain topics in trigonometry and
the conics.
iii
Calculators
For this course, you are required to use, a scientific calculator. This may be a
graphing calculator, but this is not a definite requirement.
It is assumed that you are familiar with the use of the calculator and no instructions are
given in the course.
Internet Sites
The internet has several good mathematics sites dealing with trigonometry, the
conics, and mathematical proofs. These may be updated from time to time and
you may contact your teacher for any update, or to suggest a new interesting
site.
You will find internet sites by searching for the following topics.
•
•
•
•
Trigonometry
Function Plotter
Conics
Mathematics Proofs
www.centralischool.ca/web_resources.html/
choose Math 30
each chapter has examples and on-line quizzes - great for extra practice.
Also included is a glossary and formulae.
Need to review some basic algebra or geometry before beginning your course?
A Preparation Package is now available for Math C30. Contact your teacher
for further information about this package.
iv
Index
Addition identities
Adjacent side
Ambiguous case
Amplitude
Analogical argument
Angle ( standard
position)
Angle (common)
Angle (coterminal)
Angle (negative)
Angle (positive)
Angle (principal)
Angle (quadrantal)
Angle of depression
Angle of elevation
Angle
Angular speed
Arc length
Area of a triangle
Argument
Arms
Asymptotes
Axis of parabola
Bearing
Circle (general
equation)
Circle (standard
equation)
Circular function
Circular measure
Cofunction identities
Completing the
square
Conclusion
Conditional
Conic section
Conjugate axis
Contradiction
Contrapositive
Converse
Cosecant function
Cosine function
Cosine law
Cotangent function
Counter example
Cycle
Deductive argument
Direct deductive
proof
4.2
1.1
3.3
2.5, 3.1
7.3
1.1
1.1
1.1
1.1
1.1
1.1
1.2
3.2
3.2
1.1
2.3
2.3
3.5
7.1
1.1
6.2
5.5
3.2
5.4
5.4
2.2
2.1
4.1
5.4
7.1
7.2
5.4
6.2
7.4
7.2
7.2
2.4
2.4
3.4
2.4
7.2
2.4
7.4
Direction
Directrix
Domain
Double-angle
identities
Ellipse
Focal length
Focal radii (ellipse)
Focal radii
(hyperbola)
Foci of ellipse
Foci of hyperbola
Focus
General solution
Half-angle identities
Heading
Heron’s formula
Horizontal ellipse
Horizontal
hyperbola
Hyperbola
Hypotenuse
Identity
Inadmissible values
Indeterminate
Indirect proof
Inductive argument
Intervals
Inverse
Linear speed
Lithotriper
Major axis
Mathematical
induction
Maximum value
Minimum value
Minor axis
Opposite side
Parabola
Particular solution
Perfect square
Period
Periodic function
Phase shift
Premise
Principal cycle
Pythagorean
identities
3.2
5.5
2.4
4.3
6.1
6.2
6.1
6.2
6.1
6.2
5.5
5.1
4.3
3.2
3.5
6.1
6.2
6.2
1.1
4.1
4.1
4.1
7.4
7.3
2.4
7.2
2.3
6.1
6.1
Pythagorean
theorem
Radian measure
Range
Reciprocal identities
Reference angle
Reference triangle
Secant function
Sine law
Statement
Subtraction
identities
Table of identities
Tangent function
Transverse axis
Trigonometric ratios
Undefined
Unit circle
Venn diagram
Vertex of parabola
Vertex
Vertical ellipse
Vertical hyperbola
Vertical translation
Vertices of ellipse
Vertices of hyperbola
3.2
2.1
2.4
4.1
1.3
1.3
2.4
3.3
7.1
4.2
4.4
2.4
6.2
1.1
4.1
2.1
7.2
5.5
1.1
6.1
6.2
3.1
6.1
6.2
8.1
2.4
2.4
6.1
1.1
5.5
5.1
5.4
2.4
2.4
2.5, 3.1
7.1
2.4
4.1
7.4
v
Acknowledgement
Every reasonable effort has been made to trace the ownership of all copyright material
and to secure the necessary permissions to reprint it.
Lesson 4
“Suggestions for Improving Identities.” From “Strategies for Improving Identities” in
Mathematics C30, by Burt Thiessen. Copyright © 1999 The Board of Education of the
Saskatoon School Division Number 13 of Saskatchewan. Used by permission.
vi
Mathematics C30
Module 1
Lessons 1 - 5
vii
Mathematics C30
1
Lesson 1
Introduction to Module 1
Module 1 consists of five lessons with an assignment following each lesson. The content of
these lessons is trigonometry with an introduction to conics in the last section of Lesson 5.
Module 2 will continue with a further study of conics.
You should expect to complete each lesson in about ten hours since one hundred hours are
prescribed for the course and there are ten lessons in the course. Lesson 9, in Module 2,
allows you to review the content of module 1 before completing the course.
In the “real world” one of the first uses of trigonometry was to measure distances
indirectly. Some rivers are too difficult to cross, some mountains too dangerous to climb,
and some stars or planets impossible to reach in a lifetime. In these cases the direct
measurement approach, by using a measuring tape, is not practical. Trigonometry, a
mathematical technique, can be used to measure these distances indirectly, without
making the journey.
Mathematics C30
2
Lesson 1
Mathematics C30
3
Lesson 1
Mathematics C30
Module 1
Table of Contents
Lesson 1
Trigonometric Ratios and Functions
Introduction .....................................................................................
Objectives .........................................................................................
1.1 Trigonometric Ratios of Acute Angles Measured in Degrees .
1.2 Trigonometric Functions of General Angles ............................
1.3 Reference Angles .......................................................................
Answers to Exercises .......................................................................
Assignment 1 ...................................................................................
Lesson 2
3
5
7
20
26
43
55
Radian Measure and Trigonometric Functions
Introduction .....................................................................................
Objectives .........................................................................................
2.1 Radian Measure of an Angle ....................................................
2.2 The Trigonometric Functions ...................................................
2.3 Arc Length and Angular Speed ................................................
2.4 Graphs of Trigonometric Functions .........................................
2.5 Amplitudes, Period, and Phase Shift (Part 1) .........................
Answers to Exercises .......................................................................
Assignment 2 ...................................................................................
Lesson 3
Page
69
71
73
78
85
91
105
119
131
Applications of Trigonometric Functions
Introduction .....................................................................................
Objectives .........................................................................................
3.1 Amplitude, Period, and Phase Shift (Part 2) ...........................
3.2 Problems Involving Right Triangles ........................................
3.3 The Law of Sines .......................................................................
3.4 The Law of Cosines ...................................................................
3.5 The Area of a Triangle ..............................................................
Answers to Exercises .......................................................................
Assignment 3 ...................................................................................
Mathematics C30
4
147
149
151
168
176
186
193
199
213
Lesson 1
Lesson 4
Trigonometric Identities
Introduction .....................................................................................
Objectives .........................................................................................
4.1 The Fundamental Trigonometric Identities ............................
4.2 The Addition and Subtraction Identities .................................
4.3 The Double-angle and Half-angle Identities ...........................
4.4 Application of the Fundamental Identities .............................
Answers to Exercises .......................................................................
Assignment 4 ...................................................................................
Lesson 5
229
231
233
242
256
265
275
289
Trigonometric Equations
Introduction to the Conics
Introduction .....................................................................................
Objectives .........................................................................................
5.1 Solving Equations – General and Particular Solutions ..........
5.2 Solving Equations of the Form sin ax  b  c .........................
5.3 Using Algebraic Techniques and
Trigonometric Identities to Solve Equations .....................
5.4 Conic Sections – The Circle ......................................................
5.5 The Parabola .............................................................................
Answers to Exercises .......................................................................
Assignment 5 ...................................................................................
Mathematics C30
5
301
303
305
310
314
318
333
349
359
Lesson 1