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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