Download Trigonometry I

MTH-4103-1 C1-C4 Trigonometry I_Layout 1 10-10-18 13:57 Page 1
T
MTH-4103-1
rigonometry
I
MTH-4103-1
TRIGONOMETRY I
This course was produced in collaboration with the Service de l'éducation
des adultes de la Commission scolaire catholique de Sherbrooke and the
Department of the Secretary of State of Canada.
Author: Monique Pagé
Content revision: Jean-Paul Groleau
Daniel Gélineau
Mireille Moisan-Sanscartier
Adult education consultant: Serge Vallières
Coordinator for the DGFD: Jean-Paul Groleau
Coordinator for the DFGA: Ronald Côté
Photocomposition and layout: Multitexte Plus
Translation: Consultation en éducation Zegray
Linguistic revision: Kay Flanagan and Leslie Macdonald
Translation of updated sections: Claudia de Fulviis
First edition: 1991
Reprint: 2004
© Société de formation à distance des commissions scolaires du Québec
All rights for translation and adaptation, in whole or in part, reserved for all countries.
Any reproduction, by mechanical or electronic means, including micro-reproduction, is
forbidden without the written permission of a duly authorized representative of the
Société de formation à distance des commissions scolaires du Québec (SOFAD).
Legal Deposit
–
2004
Bibliothèque et Archives nationales du Québec
Bibliothèque et Archives Canada
ISBN 2-89493-281-0
MTH-4103-1
Answer Key
Trigonometry I
TABLE OF CONTENTS
Introduction to the Program Flowchart ................................................... 0.4
The Program Flowchart ............................................................................ 0.5
How to Use this Guide .............................................................................. 0.6
General Introduction ................................................................................. 0.9
Intermediate and Terminal Objectives of the Module ............................ 0.11
Diagnostic Test on the Prerequisites ....................................................... 0.15
Answer Key for the Diagnostic Test on the Prerequisites ...................... 0.21
Analysis of the Diagnostic Test Results ................................................... 0.23
Information for Distance Education Students ......................................... 0.25
UNITS
1. Right Triangles .......................................................................................... 1.1
2. Trigonometry and Trigonometric Ratios.................................................. 2.1
3. Using Trigonometric Ratios to Determine Angles .................................. 3.1
4. Solving Right Triangles ............................................................................ 4.1
5. Everyday Problems ................................................................................... 5.1
6. Solving Any Given Triangle ...................................................................... 6.1
Final Summary.......................................................................................... 7.1
Terminal Objectives .................................................................................. 7.9
Self-evaluation Test .................................................................................. 7.11
Answer Key for the Self-evaluation Test ................................................. 7.19
Analysis of the Self-evaluation Test Results ........................................... 7.25
Final Evaluation........................................................................................ 7.26
Answer Key for the Exercises ................................................................... 7.27
Glossary ..................................................................................................... 7.93
List of Symbols .......................................................................................... 7.96
Bibliography .............................................................................................. 7.97
Review Activities ....................................................................................... 8.1
© SOFAD
0.3
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
INTRODUCTION TO THE PROGRAM FLOWCHART
Welcome to the World of Mathematics!
This mathematics program has been developed for the adult students of the
Adult Education Services of school boards and distance education. The learning
activities have been designed for individualized learning. If you encounter
difficulties, do not hesitate to consult your teacher or to telephone the resource
person assigned to you. The following flowchart shows where this module fits
into the overall program. It allows you to see how far you have progressed and
how much you still have to do to achieve your vocational goal. There are several
possible paths you can take, depending on your chosen goal.
The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2
(MTH-416), and leads to a Diploma of Vocational Studies (DVS).
The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2
(MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School
Diploma (SSD), which allows you to enroll in certain Gegep-level programs that
do not call for a knowledge of advanced mathematics.
The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2
(MTH-536), and leads to Cegep programs that call for a solid knowledge of
mathematics in addition to other abiliies.
If this is your first contact with this mathematics program, consult the flowchart
on the next page and then read the section “How to Use This Guide.” Otherwise,
go directly to the section entitled “General Introduction.” Enjoy your work!
0.4
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
THE PROGRAM FLOWCHART
CEGEP
MTH-5112-1
MTH-5111-2
MTH-536
MTH-5104-1
MTH-5103-1
Introduction to Vectors
MTH-5109-1
Geometry IV
MTH-5108-1
Trigonometric Functions and Equations
MTH-5107-1
Exponential and Logarithmic Functions
and Equations
Optimization II
MTH-5106-1
Real Functions and Equations
Probability II
MTH-5105-1
Conics
MTH-5102-1
Statistics III
MTH-5101-1
MTH-436
MTH-426
MTH-4110-1
MTH-216
MTH-116
© SOFAD
The Four Operations on
Algebraic Fractions
MTH-4109-1
Sets, Relations and Functions
Quadratic Functions
MTH-4107-1
Straight Lines II
MTH-4106-1
Factoring and Algebraic Functions
MTH-4105-1
Exponents and Radicals
MTH-4103-1
MTH-4102-1
MTH-4101-2
Complement and Synthesis I
MTH-4108-1
MTH-4104-2
MTH-314
Optimization I
MTH-4111-2
Trades
DVS
MTH-416
Complement and Synthesis II
MTH-5110-1
MTH-526
MTH-514
Logic
Statistics II
Trigonometry I
You ar e h er e
Geometry III
Equations and Inequalities II
MTH-3003-2
Straight Lines I
MTH-3002-2
Geometry II
MTH-3001-2
The Four Operations on Polynomials
MAT-2008-2
Statistics and Probabilities I
MTH-2007-2
Geometry I
MTH-2006-2
Equations and Inequalities I
MTH-1007-2
Decimals and Percent
MTH-1006-2
The Four Operations on Fractions
MTH-1005-2
The Four Operations on Integers
0.5
25 hours
= 1 credit
50 hours
= 2 credits
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
HOW TO USE THIS GUIDE
Hi! My name is Monica and I have been
asked to tell you about this math module.
What’s your name?
Whether you are
registered at an
adult education
center or at
Formation à
distance, ...
Now, the module you have in your
hand is divided into three
sections. The first section is...
I’m Andy.
... you have probably taken a
placement test which tells you
exactly which module you
should start with.
... the entry activity, which
contains the test on the
prerequisites.
0.6
You’ll see that with this method, math is
a real breeze!
My results on the test
indicate that I should begin
with this module.
By carefully correcting this test using the
corresponding answer key, and recording your results on the analysis sheet ...
© SOFAD
1
Answer Key
2
MTH-4103-1
3
... you can tell if you’re well enough
prepared to do all the activities in the
module.
And if I’m not, if I need a little
review before moving on, what
happens then?
Trigonometry I
In that case, before you start the
activities in the module, the results
analysis chart refers you to a review
activity near the end of the module.
I see!
In this way, I can be sure I
have all the prerequisites
for starting.
START
The starting line
shows where the
learning activities
begin.
Exactly! The second section
contains the learning activities. It’s
the main part of the module.
?
The little white question mark indicates the questions
for which answers are given in the text.
The target precedes the
objective to be met.
The memo pad signals a brief reminder of
concepts which you have already studied.
?
Look closely at the box to
the right. It explains the
symbols used to identify the
various activities.
The boldface question mark
indicates practice exercices
which allow you to try out what
you have just learned.
The calculator symbol reminds you that
you will need to use your calculator.
?
The sheaf of wheat indicates a review designed to
reinforce what you have just learned. A row of
sheaves near the end of the module indicates the
final review, which helps you to interrelate all the
learning activities in the module.
FINISH
Lastly, the finish line indicates
that it is time to go on to the self-evaluation
test to verify how well you have understood
the learning activities.
© SOFAD
0.7
1
2
MTH-4103-1
3
There are also many fun things
in this module. For example,
when you see the drawing of a
sage, it introduces a “Did you
know that...”
It’s the same for the “math whiz”
pages, which are designed especially for those who love math.
For example. words in boldface italics appear in the
glossary at the end of the
module...
Answer Key
A “Did you know that...”?
Yes, for example, short tidbits
on the history of mathematics
and fun puzzles. They are interesting and relieve tension at
the same time.
Trigonometry I
Must I memorize what the sage says?
No, it’s not part of the learning activity. It’s just there to
give you a breather.
They are so stimulating that
even if you don’t have to do
them, you’ll still want to.
And the whole module has
been arranged to make
learning easier.
... statements in boxes are important
points to remember, like definitions, formulas and rules. I’m telling you, the format makes everything much easier.
The third section contains the final review, which interrelates the different
parts of the module.
Great!
There is also a self-evaluation
test and answer key. They tell
you if you’re ready for the final
evaluation.
Thanks, Monica, you’ve been a big
help.
I’m glad! Now,
I’ve got to run.
See you!
0.8
Later ...
This is great! I never thought that I would
like mathematics as much as this!
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
GENERAL INTRODUCTION
DISCOVERING THE PROPERTIES OF TRIANGLES
As everyone knows, surveyors measure the dimensions of lots. Most complex
geometric figures are formed by the juxtaposition of simple figures. Surveyors
therefore split up the area to be surveyed into triangles and then measure the
sides or angles in the triangles in order to determine the exact boundaries of the
lot in question.
Problems in subdividing land into lots or of measuring area led people in ancient
times to develop techniques which allowed them to state the relationships
between the lengths of the sides in a triangle and the size of its angles.
Astronomy was also developed in order to allow navigators to determine their
bearings and their location by taking the position of the stars as reference points:
a star, its projection on the earth's surface, and the position of an observer form
a triangle.
Our predecessors thus developed a great deal of knowledge based on the ratios
between the measures of the sides in a right triangle. In so doing, they gave
birth to trigonometry, often abbreviated to "trig" (triangle measurement),
which, when developed further, allows you to discover the relationships between
the sides or angles in any triangle.
To reach the objective of this module, you should be able to solve triangulation
problems, that is, problems that involve measuring angles and sides in
triangles. You should also be able to evaluate trigonometric ratios and solve
problems involving carpentry or navigation by using trigonometric ratios.
© SOFAD
0.9
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
Lastly, you will extend this knowledge to surveying by trying to solve problems
related to any triangle. To do so, you will apply the sine and cosine laws which
are the key tools for solving these problems. As you will see, this knowledge is
very useful in many sectors of human activity. In addition to its role in surveying,
carpentry and astronomy, it is useful in sewing, metalwork, architecture,
engineering and so on.
0.10
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
INTERMEDIATE AND TERMINAL OBJECTIVES OF
THE MODULE
Module MTH-4103-1 consists of nine units and requires fifty hours of study
distributed as shown below. Each unit covers either an intermediate or a
terminal objective. The terminal objectives appear in boldface.
Objectives
Number of Hours*
% (evaluation)
1 to 4
7
30%
5
9
40%
6
8
30%
* One hour is allotted for the final evaluation.
1. Right Triangles
To solve a right triangle, that is, to determine the measure of the three angles
and the three sides in this triangle, using the Pythagorean Theorem, and to
establish the relationships between these angles. Two cases are possible:
• a right triangle, given the measures of one acute angle and two sides
• a right triangle, given an angle which measures either 30° or 45°, and the
length of one side
The situations are presented in the form of word problems borrowed from
everyday life. The steps in the solution must be described.
© SOFAD
0.11
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
2. Trigonometry and Trigonometric Ratios
To calculate the numerical value of any one of the three trigonometric ratios
of an angle A:
• sine A (sin A)
• cosine A (cos A)
• tangent A (tan A)
given the lengths of the three sides in a right triangle ABC, which is rightangled in C.
3. Using Trigonometric Ratios to Determine Angles
To calculate the measures of the acute angles in right triangles, given the
lengths of two of the sides, by applying the properties of trigonometric ratios
and by using a trigonometric table or a scientific calculator.
4. Solving Right Triangles
To determine the missing dimensions in right triangles by applying
the properties of trigonometric ratios, the Pythagorean Theorem
and the relationships between the angles in this triangle, given:
• the length of two sides, or
• the length of one side and the measure of one acute angle.
The problems are solved either by using a trigonometric table or a
scientific calculator. The steps in the solution must be described.
0.12
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
5. Everyday Problems
To solve word problems dealing with right triangles which require
the application of the properties of the six trigonometric ratios and
the use of a trigonometric table or a scientific calculator.
The
measures to be determined may be those of angles, sides, or angles
and sides in a right triangle that is given or is to be drawn. The
situations are borrowed from everyday life and the steps in the
solution must be described.
6. Solving any Given Triangle
To solve word problems dealing with any given triangle, by applying:
• either the sine law: a = b = c ,
sin A sin B sin C
• or the cosine law: a2 = b2 + c2 – 2bc cos A.
The measures to be determined may be those of angles, sides, or
angles and sides in any type of triangle that is given or is to be drawn.
Use of trigonometric tables or a scientific calculator is required. The
situations are borrowed from everyday life and the steps in the
solution must be described.
© SOFAD
0.13
1
2
3
Answer Key
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
DIAGNOSTIC TEST ON THE PREREQUISITES
Instructions
1. Answer as many questions as you can.
2. To answer these questions, you should have a protractor and a
calculator.
3. Write your answers on the test paper.
4. Do not waste any time. If you cannot answer a question, go on
to the next one immediately.
5. When you have answered as many questions as you can, correct
your answers, using the answer key which follows the diagnostic test.
6. To be considered correct, your answers must be identical to
those in the key. For example, if you are asked to describe the
steps involved in solving a problem, your answer must contain
all the steps.
7. Transcribe your results onto the chart which follows the answer
key. It gives an analysis of the diagnostic test results.
8. Do only the review activities listed for each of your incorrect
answers.
9. If all your answers are correct, you have the prerequisites to
begin working on this module.
© SOFAD
0.15
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
1. Match the names of the different triangles with the corresponding geometric
figures below.
a) An isosceles right triangle: ....................
b) An isosceles triangle: ....................
c) A scalene triangle: ....................
d) An equilateral triangle: ....................
2. a) Measure the three angles
in the adjacent triangle,
A
using a protractor.
N.B. Your measurements
must be accurate to the
B
C
nearest 2°.
m∠A = ......................, m∠B = ......................, m∠C = ......................
b) Which angle in the triangle is obtuse? ......................................................
0.16
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
3. In the adjacent figure, identify the following:
a) the hypotenuse: .................................
B
b) an acute angle: ..................................
c
a
c) the sides adjacent to angle A:
...........................................................
C
A
b
d) the side opposite angle A: .................
e) the right angle: ..................................
4. Given the adjacent triangle ABC,
A
determine the measure of angle A
without using a protractor. A com20°
plete solution is required.
B
5. a) Referring to the adjacent figure,
B
calculate the length of the hypotenuse in this right triangle by
applying the Pythagorean Theorem. Round your answers to the
nearest centimetre. A complete
solution is required.
© SOFAD
0.17
C
c
4 cm
C
10 cm
A
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
V
b) Determine the length of side u in
adjacent triangle TUV, using the
Pythagorean Theorem. A comp-
u=?
t = 5 cm
lete solution is required.
T
v = 4 cm
U
6. Convert each of the following fractions to a decimal. Your answers must
contain 4 decimal places.
a)
7 = ..............
16
b) 12 =...............
17
c) 4 = ................
9
7. Calculate the value of x in the following expressions by applying the fundamental property of proportions. A complete solution is required. If necessary,
round your answers to the nearest hundredth.
125
a) 725
x = 150
b)
0.18
x = 240
210 715
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
8. a) Garfield deposits $75.00 each week in a savings account. The remainder
of his salary, namely $250.00, is used for various expenses such as rent,
food, travel and so on. Garfield wants to save enough money to buy a new
television and a video cassette recorder worth $925.00. If he already has
$150.00 in his account, how many more weeks will it take him to
accumulate this amount? A complete solution is required.
© SOFAD
0.19
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
b) Julianna is having fun flying a kite. This kite is shaped like a quadrilateral with two 40 cm sides and two 55 cm sides. Julianna is standing
50 m from her house and she has let out all the string. If she sights the
kite at a 30° angle from where she stands and if the string attached to the
kite is 45 m long, calculate how high above Julianna's head the kite is
flying as well as the horizontal distance which separates Julianna and the
kite. The steps in the solution as well as the answers are required.
0.20
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
ANSWER KEY FOR THE DIAGNOSTIC
TEST ON THE PREREQUISITES
1. a) ➃
b) ➁ or ➃ or ➅
c) ➄
d) ➅
2. a) m∠A = 32°; m∠B = 130°; m∠C = 18°
N.B. The sum of the measures of the three angles of a triangle must equal
180°.
b) ∠B
3. a) c or AB
d) a or BC
b) ∠A or ∠B
c) b and c or AC and AB
e) ∠C
4. m∠A = 90° – 20° = 70°
5. a) c2 = a2 + b2
b) u2 = t2 – v2
c2 = 42 + 102
u2 = 52 – 42
c2 = 16 + 100
u2 = 25 – 16
c2 = 116
u2 = 9
c = 116
u= 9
c = 10.77 cm
u = 3 cm
c = 11 cm
6. a) 0.437 5
7. a)
725 = 125
x
150
125 × x = 725 × 150
b) 0.705 9
c) 0.444 4
b)
x = 725 × 150
125
x = 870
© SOFAD
x = 240
715
210
x × 715 = 240 × 210
x = 240 × 210
715
x = 70.49
0.21
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
8. a) • We want to determine how many more weeks it will take Garfield to
save this amount.
• ($925 – $150) ÷ $75/week = 10.3 weeks
• Garfield must save for 11 weeks.
b)
A
45 m
22.5 m
30°
B
39m
C
• We want to determine how high above Julianna’s head the kite is
flying.
• In a right triangle, the length of the side opposite the 30° angle is equal
to half that of the hypotenuse.
b = 45 m = 22.5 m.
2
The kite is flying 22.5 m above Julianna’s head.
• We want to determine the horizontal distance which separates
Julianna from the kite.
• The horizontal distance required is calculated by applying the
Pythagorean Theorem.
c 2 = a 2 + b2
a 2 = c 2 – b2
a2 = 452 – 22.52
a2 = 2 025 – 506.25
a2 = 1 518.75
a = 1 518.75
a = 38.97
The horizontal distance between Julianna and the kite is 39 m.
0.22
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
ANALYSIS OF THE DIAGNOSTIC
TEST RESULTS
Question
1. a)
b)
c)
d)
2. a)
b)
3. a)
b)
c)
d)
e)
4.
5. a)
b)
6. a)
b)
7. a)
b)
8. a)
b)
Answer
Correct
Incorrect
Section
Review
Page
Before Going
to Unit(s)
8.2
8.2
8.2
8.2
8.2
8.2
8.2
8.2
8.2
8.2
8.2
8.3
8.4
8.4
8.5
8.5
8.6
8.6
8.1
8.1
8.20
8.20
8.20
8.20
8.22
8.22
8.26
8.26
8.26
8.26
8.26
8.29
8.32
8.32
8.39
8.39
8.41
8.41
8.4
8.4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3
3
4
4
1, 5 and 6
1, 5 and 6
• If all your answers are correct, you may begin working on this module.
• For each incorrect answer, find the related section listed in the Review
column. Do the review activities for that section before beginning the units
listed in the right-hand column under the heading Before Going to Unit(s).
© SOFAD
0.23
1
2
3
Answer Key
1
2
MTH-4103-1
3
Answer Key
INFORMATION
FOR
EDUCATION STUDENTS
Trigonometry I
DISTANCE
You now have the learning material for MTH-4103-1 together with the homework assignments. Enclosed with this material is a letter of introduction from
your tutor indicating the various ways in which you can communicate with him
or her (e.g. by letter, telephone) as well as the times when he or she is available.
Your tutor will correct your work and help you with your studies. Do not hesitate
to make use of his or her services if you have any questions.
DEVELOPING EFFECTIVE STUDY HABITS
Distance education is a process which offers considerable flexibility, but which
also requires active involvement on your part. It demands regular study and
sustained effort. Efficient study habits will simplify your task. To ensure
effective and continuous progress in your studies, it is strongly recommended
that you:
• draw up a study timetable that takes your working habits into account and
is compatible with your leisure time and other activities;
• develop a habit of regular and concentrated study.
© SOFAD
0.25
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
The following guidelines concerning the theory, examples, exercises and assignments are designed to help you succeed in this mathematics course.
Theory
To make sure you thoroughly grasp the theoretical concepts:
1. Read the lesson carefully and underline the important points.
2. Memorize the definitions, formulas and procedures used to solve a given
problem, since this will make the lesson much easier to understand.
3. At the end of an assignment, make a note of any points that you do not
understand. Your tutor will then be able to give you pertinent explanations.
4. Try to continue studying even if you run into a particular problem. However,
if a major difficulty hinders your learning, ask for explanations before
sending in your assignment.
Contact your tutor, using the procedure
outlined in his or her letter of introduction.
Examples
The examples given throughout the course are an application of the theory you
are studying. They illustrate the steps involved in doing the exercises. Carefully
study the solutions given in the examples and redo them yourself before starting
the exercises.
0.26
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
Exercises
The exercises in each unit are generally modelled on the examples provided.
Here are a few suggestions to help you complete these exercises.
1. Write up your solutions, using the examples in the unit as models. It is
important not to refer to the answer key found on the coloured pages at the
end of the module until you have completed the exercises.
2. Compare your solutions with those in the answer key only after having done
all the exercises. Careful! Examine the steps in your solution carefully even
if your answers are correct.
3. If you find a mistake in your answer or your solution, review the concepts that
you did not understand, as well as the pertinent examples. Then, redo the
exercise.
4. Make sure you have successfully completed all the exercises in a unit before
moving on to the next one.
Homework Assignments
Module MTH-4103-1 contains three assignments.
The first page of each
assignment indicates the units to which the questions refer. The assignments
are designed to evaluate how well you have understood the material studied.
They also provide a means of communicating with your tutor.
When you have understood the material and have successfully done the pertinent exercises, do the corresponding assignment immediately. Here are a few
suggestions.
1. Do a rough draft first and then, if necessary, revise your solutions before
submitting a clean copy of your answer.
© SOFAD
0.27
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
2. Copy out your final answers or solutions in the blank spaces of the document
to be sent to your tutor. It is preferable to use a pencil.
3. Include a clear and detailed solution with the answer if the problem involves
several steps.
4. Mail only one homework assignment at a time. After correcting the assignment, your tutor will return it to you.
In the section “Student’s Questions,” write any questions which you may wish
to have answered by your tutor. He or she will give you advice and guide you in
your studies, if necessary.
In this course
Homework Assignment 1 is based on units 1 to 4.
Homework Assignment 2 is based on units 5 and 6.
Homework Assignment 3 is based on units 1 to 6.
CERTIFICATION
When you have completed all the work, and provided you have maintained an
average of at least 60%, you will be eligible to write the examination for this
course.
0.28
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
DÉPART
UNIT 1
RIGHT TRIANGLES
1.1
SETTING THE CONTEXT
Triangles Everywhere
Young Trigon really likes kites. His father, Mr Trigon, capitalizes on this one day
when they are out flying a kite in order to teach his son some physics concepts.
"There are two forces," he says, "which work together to prevent your kite from
being carried off into the sky by the wind; one comes from the tension which you
apply with your arm on the string which is holding the kite and the other is
caused by the earth's gravitational attraction.* These two forces form a right
triangle with a line which is parallel to the ground (Fig. 1.1)."
* All objects are attracted towards the centre of the earth. This gravitational force is therefore
exerted downwards, perpendicular to the earth's surface.
© SOFAD
1.1
1
Answer Key
2
MTH-4103-1
3
T
Trigonometry I
A
37°
The forces acting
on the kite
37°
Ground
Fig. 1.1 The physics lesson
Thus we can see the shape of a right triangle each time we project a perpendicular and an oblique (slanted) line to the ground from the same point above the
ground. Knowing how to measure the sides and the angles in this type of
triangle is very useful in everyday situations for measuring distances or
determining the measures of various angles.
Have you noticed to what extent right angles, which measure 90°, are found
everywhere? A pole carrying electrical wires forms a right angle with the ground;
a staircase which connects one floor to another meets the floor at a right angle,
and the stairs-wall-floor together define a right triangle. Look around you
carefully: you will discover a great many right angles and right triangles.
1.2
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
A triangle is a polygon with 3 sides and with 3 angles, the sum
of whose measures is always equal to 180°. The point of
intersection of two sides is called a vertex.
There are 5 types of triangles:
1. equilateral triangles, which have 3 congruent angles of 60°
and 3 congruent sides;
2. isosceles triangles, which have 2 congruent sides and 2
congruent angles;
3. right triangles, which have 1 right angle;
4. isosceles right triangles, which have 1 right angle, 2 congruent angles of 45° and 2 congruent sides;
5. scalene triangles, which have sides and angles that are not
congruent.
You know that a right triangle is a triangle which has an angle of 90°. It is
important to remember that the side opposite the right angle is called the
hypotenuse.
Since a triangle is the simplest polygon possible, any other polygon can be
decomposed into triangles by drawing appropriate diagonals inside the figure
(Fig. 1.2). Also, by dropping a perpendicular from a vertex in each of these 3 types
of triangles (equilateral, isosceles or scalene) we obtain right triangles (Fig. 1.2).
G
G
F
A
A
B
C
G
G
H
H
F
E
A
D
Fig. 1.2
© SOFAD
H
Decomposition of polygons into right triangles
1.3
F
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
The study of various geometric shapes, for example in surveying or in landscaping, usually involves the study of right triangles.
To reach the objective of this unit, you should be able to solve problems
dealing with right triangles where the measures of one acute angle and
of two sides are given. In the case of a right triangle where one of the
acute angles measures 30° or 45°, you should be able to solve these
problems given only the length of one leg (side adjacent the right angle)
in the right triangle.
To solve problems concerning triangles, you must determine the measures of its
6 dimensions, that is, the measures of the 3 sides and the 3 angles.
The hypotenuse is the longest side in a right triangle. It is the side
opposite the triangle's right angle.
An acute angle is an angle whose measure is less than 90°.
The side opposite an acute angle is the side of the triangle located
across from this angle.
The side of the right angle adjacent to an angle is the side in the
triangle which is common to the right angle and to this angle.
c
B
a
Fig. 1.3
A
c: hypotenuse
b
a: side opposite angle A
or
side adjacent to angle B
C
b: side opposite angle B
or
side adjacent to angle A
Triangle, right-angled in C
1.4
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
Angles are measured in degrees. Hipparchus of Alexandria (2nd century BC) is
credited with having introduced this measurement unit to the Western world.
He divided a circle into 360 parts called degrees. Each of these degrees was
divided into 60 minutes of 60 seconds each. Up to that time, this method of
dividing by 60, called the sexagesimal system, had been used only by the
Babylonians.
This measurement unit can be compared to our method of
measuring time; a day is divided into 24 hours, which are in turn divided into
60 minutes, each of which contains 60 seconds. Of course, it is much more precise
to say 2 hours 4 minutes 30 seconds than it is to say about 2 hours. The
sexagesimal system is a very precise system of measurement.
1 degree =
1 of a circle and is written 1°.
360
1 minute = 1 of a degree and is written 1'.
60
1 second = 1 of a minute and is written 1".
60
Example 1
Angle A measures 22° or 22 of a circle.
360
A
Fig. 1.4 Measuring a portion of a circle
Angle measures can also be expressed in decimal form. An angle measure of
20° 15' can be converted to decimal form as follows:
15' ÷ 60' = 0.25; therefore 20° 15' = 20.25°.
You can also perform the inverse operation. For example, to convert an angle
measure of 65.32° to degrees, minutes and seconds:
© SOFAD
1.5
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
• Calculate how many minutes 0.32° represents.
32 × 60' = 19.2'
100
• Keep the 19' and calculate how many seconds 0.2' represents.
2 × 60" = 12"
10
• Therefore, 65.32° = 65° 19' 12".
Now see if you can measure dimensions, using the sexagesimal system, and if you
have learned the vocabulary relevant to right triangles.
Exercise 1.1
1. How many degrees are there in 1 of a circle? ...............................................
4
2. Convert the following angles, which are expressed in minutes or seconds, to
angles expressed in degrees and minutes.
a) 150' = ....................
b) 162 000" = .............
c) 1 830' = ......................
3. Convert the following angle measures to decimal form.
a) 32° 15' = ....................................... b) 53° 32' 24" = ...................................
4. Convert the following angle measures to degrees, minutes and seconds.
a) 72.2° = .......................................... b) 18.32° = ..........................................
1.6
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
B
5. Using a ruler, measure:
a) the hypotenuse in right triangle ABC:
C
...............................................................
A
b) the side opposite the 30° angle: ...................
People have long been interested in measuring the dimensions in a right
triangle. The Babylonians already understood the concept that later made
Pythagoras famous: "If c is the length of the hypotenuse in a right triangle and
if a and b are the lengths of its two legs, then c2 = a2 + b2." This is the famous
Pythagorean Theorem.
In a right triangle, the square of the
length of the hypotenuse is equal to the
B
sum of the squares of the lengths of the
other two sides. If a = 3, b = 4, c = 5, then:
3
5
4
A
C
2
2
c =a +b
2
52 = 32 + 42
25 = 9 + 16
25 = 25
Fig. 1.5 Illustration of the
Pythagorean Theorem
In any triangle ABC, the length of the segment joining vertex A to vertex B is
symbolized either by mAB or by the lower-case letter of the opposite angle,
B
namely c.
mAB = mBA = c
a
mBC = mCB = a
c
mAC = mCA = b
C
Fig. 1.6
© SOFAD
b
Lengths of the sides in a right triangle
1.7
A
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
The measure of an angle, for example, angle A, is symbolized by m∠A, which is
read "the measure of angle A." Protractors are essential in trigonometry; proceed
now with a few short exercises in order to brush up your skills in using this
instrument.
Exercise 1.2
1. Measure the 3 angles in each of the following triangles to the nearest degree,
using a protractor. Then add these measurements.
C
a) m∠A = ..............................................
m∠B = ..............................................
m∠C = ..............................................
B
m∠A + m∠B + m∠C = ....................
A
b) m∠A = ..............................................
D
m∠2 = ..............................................
3 4
m∠C = ..............................................
m∠3 = ..............................................
m∠4 = ..............................................
2
1
C
B
A
m∠A + m∠2 + m∠4 = .....................
m∠A + m∠C + (m∠3 + m∠4) = .................................
2. How many minutes are there in 0.6°? .............................................................
1.8
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
3. Measure each of the sides in these right triangles and verify whether they
satisfy the Pythagorean Theorem. Measurements and calculations should be
given to an accuracy of two decimal places.
a)
D
e
f
E
d
F
e2 = d2 + f2
d = ..................
e = ...................
f = ...................
b)
C
a
b
A
c
B
c2 = a2 + b2
a =...................
b = ...................
c = ...................
© SOFAD
1.9
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
4. Can the following groups of three numbers be lengths of the sides in a right
triangle? Explain your answer using the Pythagorean Theorem.
a) 17, 144, 145
b) 8, 10, 15
The Pythagorean Theorem allows you to calculate the length of a side in a right
triangle when the lengths of the other 2 sides are known.
Example 2
You travel 25 km east, then turn 90° south and travel a distance of 50 km;
calculate the distance from your departure point A to point B as the crow flies.
Figure 1.7 represents your trip, to a scale of 1 cm ^
= 10 km.
A
C
N
W
E
S
B
Fig. 1.7
Scale diagram of your trip
1.10
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
The straight-line distance between points A and B on this trip is the
hypotenuse of a right triangle formed by the dotted line joining the departure
point to the destination point. Thus you can apply the Pythagorean Theorem
to calculate the distance between these two points.
c 2 = a 2 + b2
c2 = 252 + 502
c2 = 625 + 2 500
c2 = 3 125
c = 3 125
c = 55.901 699...
The straight-line distance is 56 km, to the nearest kilometre.
Now find out whether you have learned the Pythagorean Theorem.
Exercise 1.3
1. A tilting tree is straightened using a guy wire fixed to its trunk 1.5 m above
the ground; the other end of the guy wire is held to the ground by a peg placed
3 m from the trunk.
a) Draw a scale diagram of this situation.
b) Using the Pythagorean Theorem, determine the length of the wire used
to hold the tree. Round your answer to the nearest tenth.
a)
© SOFAD
b)
1.11
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
2. A glass door is 3 m high and 1 m wide. Determine the length of its diagonal
to the nearest centimetre.
The hypotenuse is the longest side in a right triangle since it is
equal to the sum of the squares of the other two sides. Thus:
c2 = a2 + b2
c>a
c>b
Also, since the sum of the measures of the interior angles of any
triangle is always equal to 180°, the two non-right angles are
acute angles.
m∠A + m∠B + m∠C = 180°
B
m∠A + m∠B + 90° = 180°
thus m∠A = 90° – m∠B
m∠B = 90° – m∠A
Fig. 1.8
90°
C
180°
A
Calculating angle measurements in a right triangle
When you know the measure of one acute angle in a right triangle, you can derive
the measure of the other acute angle. These two acute angles are said to be
complementary, in other words, the sum of their measures is equal to 90°.
1.12
© SOFAD
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
Example 3
A
m∠A = 180° – m∠C – m∠B
m∠A = 90° – m∠B
m∠A = 90° – 39°
39°
m∠A = 51°
B
C
Fig. 1.9 Right triangle ABC
Are you able to draw right triangles given the measures of certain sides or certain
angles? The following exercise will show whether you have acquired this ability,
which is essential for further study.
Exercise 1.4
1. a) Sides BC, AC and AB in a right triangle measure 11 cm, 60 cm and 61 cm
respectively. Draw this triangle, using a set-square and a ruler.
Suggested scale: 1 cm ^
= 10 cm.
b) What are the measures of the three angles in the preceding triangle?
m∠A = ............... , m∠B = ............... , m∠C = ...............
© SOFAD
1.13
1
2
MTH-4103-1
3
Answer Key
Trigonometry I
2. a) A 6.4 m ladder AB is placed against a wall BC. The foot A of the ladder
is placed 3.2 m from the wall. At what height does the ladder touch the
wall? Using a scale of 1 cm ^
= 1 m, represent the position of the ladder and
measure the angles formed by this right triangle.
m∠A = ............... , m∠B = ............... , m∠C = ...............
3. A kite B is flying 14 m above the ground. The string AB which is attached to
it is 20 m long. By drawing a right triangle representing this situation and
dropping a perpendicular from the kite to the ground, determine the distance
between the point C (where this perpendicular meets the ground) and point
A, the position of the person who is holding the string. Draw the right triangle
that represents this situation.
1.14
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
4. Two cyclists leave the same point simultaneously. One goes directly south at
10 km/h while the other moves directly east at 14 km/h. What is the distance
between them after 2 1 hours?
2
N
W
E
S
At the beginning of this module you were told that right triangles are found just
about everywhere. Here are two right triangles frequently encountered in
construction.
C
30°
B
A
Fig. 1.10 Right triangle with an acute angle of 30°
© SOFAD
1.15
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
F
45°
D
E
Fig. 1.11 Right triangle with an acute angle of 45°
These right triangles are used so frequently that you will find them in your
geometry sets in the form of set-squares.
Measure the sides and the angles in these triangles carefully.
?
m∠A = ............. ,
?
a = ............. ,
?
?
?
m∠D = 45°,
d = ............. ,
m∠B = 30°,
m∠C =...................... .
b = .......... ,
c =..................... .
m∠E = .......... ,
m∠F =..................... .
e = .......... ,
f =..................... .
}
b = ..........
c
}
d = ..........
e
What can you deduce from looking at these measures?
...........................................................................................................................
...........................................................................................................................
You can verify whether your deductions are correct by doing the following
exercises.
1.16
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
Exercise 1.5
1. a) Measure the angles in the right triangle in the figure below.
A
m∠A =
............. , a =
m∠B = 30°
m∠C =
..............
, b=
..............
............. , c =
..............
B
30°
C
b) What is the ratio of the length of the side opposite the 30° angle to that of
the hypotenuse?
b =
c
2. a) Measure the angles and sides in this triangle.
E
m∠E = ...............
e = ...............
m∠F = ...............
f = ...............
m∠G = ...............
g = ...............
F
G
g
b) Calculate: e =
3. What connection can you see between the measure of the angle opposite the
30° angle in a right triangle and the measure of the hypotenuse?
...........................................................................................................................
...........................................................................................................................
© SOFAD
1.17
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
4. a) Measure the angles and sides in this triangle.
A
m∠A = ...............
a = ...............
m∠B = ...............
b = ...............
m∠C = ...............
c = ...............
B
C
b) What connection can you see between the acute angles in this triangle and
the sides opposite these angles?
.......................................................................................................................
.......................................................................................................................
You will now finish this unit by studying two other right triangles whose
properties you will find very useful.
Consider the following equilateral triangle whose sides are 1 unit long.
An equilateral triangle has 3 congruent sides. The 3 congruent
sides are opposite congruent angles each measuring 60°.
A
c=1
a=b=c=1
b=1
m∠A = m∠B = m∠C = 60°
60°
B
a=1
C
Fig. 1.12 Equilateral triangle ABC
1.18
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
From any vertex in equilateral triangle ABC, for example A, drop a perpendicular to the opposite side. Let this perpendicular be AD. You thereby obtain the
2 right triangles ABD and ADC. These two triangles are congruent under the
A-S-A congruence property.
If two angles and the contained side of one triangle are equal to
two angles and the contained side of another triangle, then the
triangles are congruent.
A
1. b = c =1;
2. m∠B = 60°, by definition;
30° 30°
3. m∠BAD = 90° – 60° = 30° since acute
c=1
angles in a right triangle are complementary.
b=1
h
60°
60°
B
C
D
Fig. 1.13 Equilateral triangle
ABC divided into 2 right triangles
Now find the lengths of the sides of right triangle ABD.
1. AB is the hypotenuse of the right triangle and mAB = 1;
2. mBD = 1 , since mBD = mDC = 1 mBC and mBC = 1;
2
2
3. mAD2 = mAB2 – mBD2
(Pythagorean Theorem)
2
30°
mAD2 = 1 2 – 1
2
mAD2 = 1 – 1
4
mAD2 = 3
4
3 = 3
mAD =
4
4
3
mAD =
or 0.866
2
© SOFAD
A
d
h
60°
B
a
D
C
Fig. 1.14 Right triangle
ABD derived from equilateral
triangle ABC
1.19
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
The lengths of the sides in these two right triangles derived from the decompoA
sition of the equilateral triangle are 1, 1 and 3 .
2
2
• The side opposite the 30° angle measures 1 .
2
• The side adjacent to the 30° angle measures
30°
3
2
1
3 .
2
60°
B
D
1
2
Fig. 1.15 Measures of the sides
1
• The side adjacent to the 60° angle measures . of right triangle ABD derived
2
from equilateral triangle ABC
• The side opposite the 60° angle measures
3 .
2
Thus, after examining the results obtained by partitioning this equilateral
triangle, a first theorem can be stated about triangles.
Theorem 1
In any right triangle, the length of the side opposite the 30°
angle is equal to half the length of the hypotenuse and the
length of the side adjacent to it is equal to 3 × the length of
2
the hypotenuse.
Apply this theorem to solve the following problems.
Exercise 1.6
1. When Martin is standing 2.2 m from
the foot of the mast on his sailboat, he
measures a 60° angle from his position relative to the top of the mast.
1.20
60°
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
a) What is the measure of the third angle formed at the top of the mast
relative to Martin's position? .....................................................................
b) What is the length of the hypotenuse in this right triangle?
c) How long is the sailboat's mast?
2. Calculate the length of the third side BC in the triangle below.
Hint: You may decompose the figure into two right triangles.
A
25
19
30°
B
© SOFAD
C
1.21
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
3. According to safety standards, the slope of a slide must not exceed a 30° angle.
If these standards were applied, what would the height of the ladder E be if
the slide were 50 m long?
S
E
30°
Consider now the isosceles right triangle in Figure 1.16 whose hypotenuse
measures c.
All isosceles triangles have 2 congruent sides.
The two congruent sides are opposite congruent angles.
C
a=b
a
b
m∠A = m∠B = 180° – 90° = 45°
2
A
c
B
Fig. 1.16 Isosceles right triangle ABC
1.22
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
Apply the Pythagorean Theorem to this triangle:
a2 + b2 = c2
a2 + a2 = c2
2a2 = c2
2
a2 = c
2
c2
2
a=
a= c
2
a= c ×
2
2
2
a = c 2 or 2 c
2
2
N.B. In order to make the denominator a rational number, the fraction was
multiplied by 2 = 1 since it is easier to divide by 2 than by 2 , which is equal
2
to 1.414... .
• The side opposite a 45° angle measures
2c.
2
The same procedure can be used to derive the length of the hypotenuse. Thus:
c 2 = a 2 + b2
c2 = a2 + a2
c2 = 2a2
c=
2a 2
c= 2 ×a
• The hypotenuse measures 2 × a or 2 × b .
Theorem 2
In any isosceles right triangle, the length of the side opposite
the 45° angle is equal to 2 × the length of the hypotenuse
2
and the length of the hypotenuse is equal to 2 × length of the
side opposite the 45° angle.
© SOFAD
1.23
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
Here now is a little problem which will allow you to apply this second theorem
about right triangles immediately!
Exercise 1.7
Your house has a gabled roof. The gable
has a 45° angle of inclination. If the
m
gable is 4 m high, how long is slope
4
m and base b of the roof?
45°
b
Thus we can conclude that when we are confronted with a right triangle and
know the length of one of its sides, it is possible to determine the length of its
other two sides if one of the acute angles measures 30° or 45°. Look at the next
example.
Example 4
C
Derive the missing dimensions in
a
the adjacent triangle.
b
30°
A
c=8
B
Fig. 1.17 Right triangle, one of
whose acute angles measures 30°
1.24
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
1. You are given that m∠B = 30° and m∠C = 90°; thus
m∠A = 90° – 30° = 60°.
2. b is equal to half the length of the hypotenuse, thus
b = 1 c = 1 × 8 = 4.
2
8
3. a is equal to
3 × the length of the hypotenuse, thus
2
3 × 8 = 6.93
2
a = 6.93
a=
N.B. You should always round off the result according to the context of the
problem. For example, if the length of a side is in metres, you generally round
off your result to the nearest centimetre (i.e., to the nearest hundredth). If the
length is in centimetres, you generally round off your result to the nearest
millimetre (i.e., to the nearest thousandth). However, when a great deal of
precision is not required, you can round off your result to the nearest unit.
The same procedure can be applied to a triangle containing an acute angle of 45°.
Now you have a general idea about triangles and are ready to tackle more meaty
problems.
If you are uncertain about anything in the following practice
exercises, do not hesitate to review the preceding concepts.
Did you know that...
Egyptians were the first accurate surveyors? The taxes
collected by the pharaoh were set according to the size of the
cultivated land, so lands had to be marked precisely and
areas had to be calculated. It was therefore through surveying that the
Egyptians learned the fundamentals of geometry.
© SOFAD
1.25
1
Answer Key
2
MTH-4103-1
?
1.2
3
Trigonometry I
PRACTICE EXERCISES
36
cm
A
1. Here is a kite whose shape is given by
45°
quadrilateral ABCD. If the length of
E
B
45°
D
segment AB in the kite is 36 cm,
determine the lengths of the other
30°
sides in this quadrilateral.
N.B. Round your answers to the
nearest cm.
C
mAE = ...............................................................................................................
mBE = ...............................................................................................................
mBC = ...............................................................................................................
mEC = ...............................................................................................................
A
2. The angle from the feet of an observer to the top of a tower is 45° and
the distance between the observer
45°
and the base of the tower is 140 m.
N.B. Round your answers to the
nearest tenth metre.
1,70 m
{
O
140 m B
C
a) How high is this tower? ..............................................................................
b) How far is the eye of the observer from the top of the tower?
.......................................................................................................................
1.26
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
3. A carpenter is sawing rafters for the
A
roof of a shed which has a 30° angle
2.8
of inclination.
N.B. Round your answers to the
nearest tenth of a metre.
m
E
30°
B
D
C
a) What is the height AC of this roof?
.......................................................................................................................
b) What is the width BD of the shed?
.......................................................................................................................
.......................................................................................................................
c) How much does CE measure?
.......................................................................................................................
4. A student in surveying wants to
C
know the width RA of a marsh. To
60°
37 m
find it, she goes to a point A opposite
a reference point R located on the
opposite shore of the marsh. She
then backs up 37 m to B, turns 90°
and continues for another 37 m. She R
is now at point C, where she measures angle RCB. It is 60°. Determine the width of the marsh to the
nearest metre.
mRA = ..............................................
© SOFAD
1.27
A
37 m
B
1
Answer Key
2
MTH-4103-1
1.3
3
Trigonometry I
SUMMARY ACTIVITY
1. What has to be done to solve problems concerning triangles?
...........................................................................................................................
...........................................................................................................................
2. A tree casts a 6 m shadow when the
angle of elevation of the sun is 30°.
How high is this tree?
30°
1.28
© SOFAD
1
Answer Key
2
MTH-4103-1
3
Trigonometry I
3. An antenna that is 2.3 m long is
C
placed 1 m from the edge of the roof
of a building. An observer A is stan-
2.3 m
ding 50 m from the building. Given
this information, can you:
45°
A
50 m
B
F
1m
a) Determine the height of this building?
b) Determine how far the observer is from the top of the antenna, that is, the
length of AC, to the nearest tenth of a metre?
© SOFAD
1.29
1
Answer Key
2
MTH-4103-1
1.4
3
Trigonometry I
THE MATH WHIZ PAGE
Ready for a Challenge?
Here is a little challenge for surveying enthusiasts.
Two semi-circles are drawn on the sides of the right angle in a right
triangle.
What is the area of the shaded region?
S2
S1
b
a
S
c
Fig. 1.18
Semi-circles drawn on the sides of the right angle in
a right triangle
1.30
© SOFAD