MATH 103 - Trigonometry & CG.pdf

MATH103: Trigonometry and Coordinate Geometry DISTANCE LEARNING CENTRE Ahmadu Bello University Zaria, Nigeria MATH103: Trigonometry and Coordinate Geometry...

MATH103: Trigonometry and Coordinate Geometry DISTANCE LEARNING CENTRE Ahmadu Bello University Zaria, Nigeria MATH103: Trigonometry and Coordinate Geometry Course Material Programme Title: BSc. Computer Science Distance Learning Centre ABU, Course Material i  MATH103: Trigonometry and Coordinate Geometry © 2018 Distance Learning Centre, ABU Zaria, Nigeria All rights reserved. No part of this publication may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the Director, Distance Learning Centre, Ahmadu Bello University, Zaria, Nigeria. Published and Printed by Ahmadu Bello University Press Limited, Zaria, Kaduna State, Nigeria. Tel: 08065949711 E-mail: [email protected], [email protected] Website: www.abupress.com.ng ii Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Course Writers/ Development Team Editor Prof. M.I Sule Course Materials Development Overseer Dr. Usman Abubakar Zaria Subject Matter Expert Yusuf Samuel Taiwo Subject Matter Reviewer Dr. Musa Balarabe Language Reviewer Enegoloinu Ojokojo Instructional Designers/Graphics Emmanuel Ekoja / Ibrahim Otukoya Course Coordinator Emmanuel Ekoja ODL Expert Dr. Abdulkarim Muhammad Distance Learning Centre ABU, Course Material iii  MATH103: Trigonometry and Coordinate Geometry ACKNOWLEDGEMENT We acknowledge the use of the Courseware of the National Open University of Nigeria (NOUN) as the primary resource. Internal reviewers in the Ahmadu Bello University have also been duly listed. iv Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Contents Copyright Page ii Course Writers/Development Team iii Acknowledgements iv Contents v COURSE STUDY GUIDE 1 i. Course Information 1 ii. Course Introduction and Description 2 iii. Course Prerequisites 2 iv. Course Learning Resources 3 v. Course objectives and outcomes 3 vi. Activities to meet Course Objectives 4 vii. Time (To complete Syllabus/Course) 4 viii. Grading criteria and Scale 4 ix. ABU Calendar 8 x. Course Structure and Outline 9 STUDY MODULES 15 Module 1: 15 Study Session 1: Trigonometric Ratios I 15 Study Session 2: Trigonometric Ratios II 32 Study Session 3: Inverse of Trigonometric Functions 44 Module 2: 61 Study Session 1: Trigonometric Identities and Trigonometric Equations 61 Study Session 2: Solution of Triangle (Sine and Cosine Values) and Angles Of Elevation and Depression 73 Distance Learning Centre ABU, Course Material v  MATH103: Trigonometry and Coordinate Geometry Study Session 3: Bearings 96 Study Session 4: Cartesian Coordinate System 109 Study Session 5: Coordinate Geometry (Circle) 125 vi Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Course Study Guide Course Information Course Code: MATH103 Course Title: Trigonometry and Coordinate Geometry Credit Units: 2CU Year of Study: First year Semester: First Semester Distance Learning Centre ABU, Course Material 1  MATH103: Trigonometry and Coordinate Geometry Course Introduction and Description Introduction: Trigonometry and Coordinate Geometry is a two credit, two modules with Eight study sessions course for physical science students. This course is taken at the first year of their B.Sc degree programme. Trigonometry as the name implies, involves the study or measurement of triangles in relation to their sides and angles. It is interesting to note that trigonometry has a very significant relevance in real life hunting, travelling and is well applied in the field of sciences, engineering and navigation of ships, aero planes and astronomy. Coordinate geometry will be introduced at the second part of this course. Description: This course entails Trigonometric ratios (sine, cosine and tangent) Trigonometric ratios of any angle (General angle) Inverse trigonometric ratios. Trigonometric identities (sum, and difference formulae, product formula). Applications of trigonometric ratios - solution of triangles (sine and cosine rules angles of elevation and depression. Bearings. Circle, general equation of a circle etc. i. COURSE PREREQUISITES You should note that although this course has no subject pre-requisite, you are expected to have: 1. Satisfactory level of English proficiency 2. Basic Computer Operations proficiency 3. Online interaction proficiency 4. Web 2.0 and Social media interactive skills 2 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry 5. O level Mathematics ii. COURSE LEARNING RESOURCES i. Course Textbooks Amazigo, J.C. (ed) (1991): Introductory University Mathematics I: Algebra, Trigonometry and Complex Numbers. Onitsha: Africana - feb Publishers Ltd David - Osuagwu, M; Anemelu C and Onyeozilu I. (2000): New School Mathematics for Senior Secondary Schools. Onitsha: Africana - Feb Publishers Ltd Egbe, E. Odili, G.A and Ugbebor, O. O. (2000): Further Mathematics. Onitsha: Africana - feb Publishers ltd Vygodsky, M. (1972): Mathematical Handbook: elementary Mathematics. Mosco: M/R Publishers. iii. COURSE OUTCOMES By the time you have successfully completed this course, you should be able to: - Define the trigonometric ratios and their reciprocals. - Compute trigonometric ratios of any given angle. - Identify with the use of tables the trigonometric ratios of given angles. - Draw the graphs of trigonometric functions. Determine the trigonometric ratios of angles from their graphs. State and derive the sine and cosine rules. - Determine the direction of your movement accurately. - Discuss intelligently the bearing in a given problem. - Define the angles of elevation and depression Distance Learning Centre ABU, Course Material 3  MATH103: Trigonometry and Coordinate Geometry - Solve problems on trigonometric equations correctly, - Apply trigonometric ratios to problems on height, distances and bearing correctly, - Obtain general equation of a circle -Derive the parametric equation of a circle - Obtain equation of tangents to a circle iv. ACTIVITIES TO MEET COURSE OBJECTIVES Specifically, this course shall comprise of the following activities: 1. Studying courseware 2. Listening to course audios 3. Watching relevant course videos 4. Field activities, industrial attachment or internship, laboratory or studio work (whichever is applicable) 5. Course assignments (individual and group) 6. Forum discussion participation 7. Tutorials (optional) 8. Semester examinations (CBT and essay based). v. TIME (TO COMPLETE SYLABUS/COURSE) This course requires three hours weekly and span over a period of eleven (11) weeks. viii. GRADING CRITERIA AND SCALE Grading Criteria A. Formative assessment Grades will be based on the following: Individual assignments/test (CA 1,2 etc). 20 Group assignments (GCA 1, 2 etc). 10 Discussions/Quizzes/Out of class engagements etc. 10 4 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry B. Summative assessment (Semester examination) CBT based 30 Essay based 30 TOTAL 100% C. Grading Scale: A = 70-100 B = 60 – 69 C = 50 - 59 D = 45-49 F = 0-44 D. Feedback Courseware based: 1. In-text questions and answers (answers preceding references) 2. SELF-ASSESSMENT Questions and answers (answers preceding references) Tutor based: 1. Discussion Forum tutor input 2. Graded Continuous assessments Student based: 1. Online programme assessment (administration, learning resource, deployment, and assessment). IX. LINKS TO OPEN EDUCATION RESOURCES OSS Watch provides tips for selecting open source, or for procuring free or open software. School Forge and Source Forge are good places to find, create, and publish open software. Source Forge, for one, has millions of downloads each day. Distance Learning Centre ABU, Course Material 5  MATH103: Trigonometry and Coordinate Geometry Open Source Education Foundation and Open Source Initiative, and other organisation like these, help disseminate knowledge. Creative Commons has a number of open projects from Khan Academy to Curriki where teachers and parents can find educational materials for children or learn about Creative Commons licenses. Also, they recently launched the School of Open that offers courses on the meaning, application, and impact of "openness." Numerous open or open educational resource databases and search engines exist. Some examples include:  OEDb: over 10,000 free courses from universities as well as reviews of colleges and rankings of college degree programmes  Open Tapestry: over 100,000 open licensed online learning resources for an academic and general audience  OER Commons: over 40,000 open educational resources from elementary school through to higher education; many of the elementary, middle, and high school resources are aligned to the Common Core State Standards  Open Content: a blog, definition, and game of open source as well as a friendly search engine for open educational resources from MIT, Stanford, and other universities with subject and description listings  Academic Earth: over 1,500 video lectures from MIT, Stanford, Berkeley, Harvard, Princeton, and Yale  JISC: Joint Information Systems Committee works on behalf of UK higher education and is involved in many open resources and open projects including digitising British newspapers from 1620-1900! Other sources for open education resources Universities  The University of Cambridge's guide on Open Educational Resources for Teacher Education (ORBIT)  Open Learn from Open University in the UK Global 6 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry  Unesco's searchable open database is a portal to worldwide courses and research initiatives  African Virtual University (http://oer.avu.org/) has numerous modules on subjects in English, French, and Portuguese  https://code.google.com/p/course-builder/ is Google's open source software that is designed to let anyone create online education courses  Global Voices (http://globalvoicesonline.org/) is an international community of bloggers who report on blogs and citizen media from around the world, including on open source and open educational resources Individuals (which include OERs)  Librarian Chick: everything from books to quizzes and videos here, includes directories on open source and open educational resources  K-12 Tech Tools: OERs, from art to special education  Web 2.0: Cool Tools for Schools: audio and video tools  Web 2.0 Guru: animation and various collections of free open source software  Livebinders: search, create, or organise digital information binders by age, grade, or subject (why re-invent the wheel?) Distance Learning Centre ABU, Course Material 7  MATH103: Trigonometry and Coordinate Geometry X. ABU DLC ACADEMIC CALENDAR/PLANNER PERIOD Semester Semester 1 Semester 2 Semester 3 Activity JAN FEB MAR APR MAY JUN JUL AUG SEPT OCT NOV DEC Registration Resumption Late Registn. Facilitation Revision/ Consolidation Semester Examination N.B: - Semester Examinations 1st/2nd Week January - All Sessions commence Mid-February - 1 Week break between Semesters and 4 Weeks vocation at end of session. - Semester 3 is OPTIONAL (Fast-tracking, making up carry-overs & deferments) 8 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry x. COURSE STRUCTURE AND OUTLINE Course Structure WEEK MODULE STUDY SESSION ACTIVITY 1. Read Courseware for the corresponding Study Session. 2. View the Video(s) on this Study Session Study Session 1: 3. Listen to the Audio on this Study Session Week 1 Title: Trigonometric 4. View any other Video/U-tube Ratios I (address/sitehttps://goo.gl/8E22BB ,https://goo.gl/3tFMwg , https://goo.gl/S6NYs7 , https://goo.gl/J7mCLn , https://goo.gl/LNY5Ts, https://goo.gl/L6h1xy) 5. View referred OER (address/site) 6. View referred Animation (Address/Site) 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity STUDY 1. Read Courseware for the corresponding Study Session. MODULE 1 2. View the Video(s) on this Study Session Week 2 Study Session 2 3. Listen to the Audio on this Study Session Title: Trigonometric 4. View any other Video/U-tube Ratios II (address/sitehttps://goo.gl/VW4vZE , https://goo.gl/eF18Gg , https://goo.gl/jhnRpK , https://goo.gl/w6Qf2M , https://goo.gl/L6h1xy) 5. View referred OER (address/site) Distance Learning Centre ABU, Course Material 9  MATH103: Trigonometry and Coordinate Geometry 6. View referred Animation (Address/Site) 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity 1. Read Courseware for the corresponding Study Session. 2. View the Video(s) on this Study Session Week 3 Study Session 3 3. Listen to the Audio on this Study Session Title: Inverse of 4. View any other Video/U-tube (address/sitehttps://goo.gl/v4edht Trigonometric ,https://goo.gl/hZiLRz , https://goo.gl/qNhzKF , Functions https://goo.gl/U85jHu, https://goo.gl/BzCytE) 5. View referred OER (address/site) 6. View referred Animation (Address/Site) 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity 1. Read Courseware for the corresponding Study Session. 2. View the Video(s) on this Study Session Week 5 3. Listen to the Audio on this Study Session Study Session 1 4. View any other Video/U-tube Title: Trigonometric (address/sitehttps://goo.gl/zSzEk4 , https://goo.gl/AvdNLJ , Identities and https://goo.gl/FBR2kG , https://goo.gl/wbwsHx , Trigonometric https://goo.gl/XtLwoj , https://goo.gl/h5L8mA) STUDY Equations 5. View referred OER (address/site) MODULE 2 6. View referred Animation (Address/Site) 10 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity 1. Read Courseware for the corresponding Study Session. 2. View the Video(s) on this Study Session Week 6 Study Session 2 3. Listen to the Audio on this Study Session Title: Solution of 4. View any other Video/U-tube Triangle (Sine and (address/sitehttps://goo.gl/or8vAr , https://goo.gl/5AquLe Cosine Values) and ,https://goo.gl/4BNND7 , https://goo.gl/6SGsqe , Angles Of Elevation https://goo.gl/aRbKtm) and Depression. 5. View referred OER (address/site) 6. View referred Animation (Address/Site) 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity 1. Read Courseware for the corresponding Study Session. 2. View the Video(s) on this Study Session Study Session 3 3. Listen to the Audio on this Study Session Week 7& 8 Title: Bearings 4. View any other Video/U-tube (address/sitehttps://goo.gl/aZsWx3, https://goo.gl/of6JPA , https://goo.gl/7jgWVp , https://goo.gl/SekQFE , https://goo.gl/5NkLWw) 5. View referred OER (address/site) Distance Learning Centre ABU, Course Material 11  MATH103: Trigonometry and Coordinate Geometry 6. View referred Animation (Address/Site) 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity 1. Read Courseware for the corresponding Study Session. 2. View the Video(s) on this Study Session Week 9 & 10 Study Session 4 3. Listen to the Audio on this Study Session Title: Cartesian 4. View any other Video/U-tube Coordinate System (address/sitehttps://goo.gl/w3yCj9 , https://goo.gl/RKtXmY , https://goo.gl/xUNY9E, https://goo.gl/8rqyvN) 5. View referred OER (address/site) 6. View referred Animation (Address/Sitehttps://goo.gl/DSXMiT) 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity 1. Read Courseware for the corresponding Study Session. 2. View the Video(s) on this Study Session Week 11& 12 3. Listen to the Audio on this Study Session Study Session 5 4. View any other Video/U-tube Title: Coordinate (address/sitehttps://goo.gl/WciRVx, https://goo.gl/g8xKEp , Geometry (Circle) https://goo.gl/KKJrgR , https://goo.gl/ZSYvWk , https://goo.gl/djSpDn) 5. View referred OER (address/site) 6. View referred Animation (Address/Site) 12 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry 7. Read Chapter/page of Standard/relevant text. 8. Any additional study material 9. Any out of Class Activity Week 13 REVISION/TUTORIALS (On Campus or Online) & CONSOLIDATION WEEK Week 14 & 15 SEMESTER EXAMINATION Distance Learning Centre ABU, Course Material 13  MATH103: Trigonometry and Coordinate Geometry Course Outline MODULE 1: Study Session 1: Trigonometric Ratios i Study Session 2: Trigonometric Ratios ii Study Session 3: Inverse of Trigonometric Functions MODULE 2: Study Session 1: Trigonometric Identities and Trigonometric Equations Study Session 2: Solution of Triangle (Sine and Cosine Values) and Angles Of Elevation and Depression. Study Session 3: Bearings. Study Session 4: Cartesian Coordinate System Study Session 5: Coordinate Geometry (Circle) 14 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Study Modules MODULE 1: Circular Measures Contents: Study Session 1: Trigonometric Ratios I Study Session 2: Trigonometric Ratios II Study Session 3: Inverse of Trigonometric Functions Study Session 1 Trigonometric Ratios I Section and Subsection Headings: Introduction 1.0 Learning outcomes 2.0 Trigonometric ratios 2.1 Trigonometric ratios of acute angles 2.2 Relationships between trigonometric ratios 2.3 Trigonometric ratios of any angle. 3.0 Tutor - Marked Assignment. 4.0 Conclusion and Summary 5.0 Self-Assessment Questions 6.0 Additional Activities 7.0 References/Further Reading INTRODUCTION Before starting any discussion in trigonometric ratios, you should be able to: (i) Identify the sides of a right-angled triangle in relation to a marked angle in the triangle. If this is not the case do not worry. You can quickly go through Distance Learning Centre ABU, Course Material 15  MATH103: Trigonometry and Coordinate Geometry This now: In the diagram (See Fig. 1.1.1) showing a Right-angled triangle ABC, right angled at B, with angle at C marked and the sides marked Is called the hypotenuse I.e. the side facing the marked angle at C is called the opposite side of the angle at C adjacent side to the angle at C. A b c Fig. 1.1.1: right-angled B triangle in relation to aCmarked angle a (ii) Again, you should recall that the ratios of two numbers " and " can either be expressed as or. If you have forgotten this, please, refresh your memories for this is important in the unit you are about to study. 1.0 STUDY SESSION LEARNING OUTCOMES After studying this session, I expect you to be able to: 1. Define trigonometric ratios of a given angle. 2. State the relationship between the trigonometric ratios 3. Locate the quadrant of the trigonometric ratios of given angles 4. Find the basic angles of given angles. 2.0 MAIN CONTENTS 2.1.1. TRIGONOMETRIC RATIOS Having refreshed your minds on the sides of a right-angled triangle and the concept of ratios you are now ready to study the trigonometric ratios (sine, cosine and tangent). This has to do with the ratio of the sides of a right-angled triangle. Here is an 16 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry EXAMPLE. A b c C B a Fig. 1.1.2: ratio of a right-angled triangle With and and the sides of Marked a, b, c, respectively, Then is called or simply is called Cosine or simply Cos is called or You can see that Using the notation of the sides of ΔABC * + * + In the above, at an acute angle and with the knowledge that the sum of the interior angles of a triangle is. What do you think will happen to the trigonometric rations? This takes us to the relationships between trigonometric ratios. Distance Learning Centre ABU, Course Material 17  MATH103: Trigonometry and Coordinate Geometry 2.1.2. RELATIONSHIP BETWEEN TRIGONOMETRIC RATIOS. A b c C B a Fig.1.1.3: relationship between trigonometric ratios In in Fig. 1.1.3 with the usual notations and , therefore,. Once more finding the trigonometric ratios in relation to the angle at A. = You might wonder what happens to this will be discussed later. In summary, given ΔABC as shown The conclusion from the summary of these trigonometric ratios is that the sine of an acute angle equals the cosine of its complement and vice versa. Thus , etc.(these angles are called complementary angles 0 because their sum is 90 i.e. Having known what trigonometric ratios are, you will now proceed to finding trigonometric ratios of any angle. 18 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry IN-TEXT QUESTION 1 If the sum of the measures of two angles is 180 degrees, the angles are called? IN-TEXT ANSWER 1. Supplementary angles 2.2 TRIGONOMETRIC RATIOS OF ANY ANGLE. It is possible to determine to some extent the trigonometric ratios of all angles using the acute angles in relation to the right-angled triangle. But since all problems concerning triangles are not only meant for right angle triangles, ~it is then good to extend the concept of the trigonometric ratios to angles of any size (i.e. between and any angle). To achieve the above, you take a unit circle i.e. a circle of radius I unit, Drawn Y 2 1 n st X d 0 3 4 Fig. r 1.1.4: t Quadrants of the circle In the Cartesian plane (x and y plane) d the h circle is divided into four equal parts each of which is called a quadrant (1st, 2nd, 3rd. 4th respectively). Angles are either measured positively in an anticlockwise direction (See Fig. 1.1.4) Distance Learning Centre ABU, Course Material 19  MATH103: Trigonometry and Coordinate Geometry Y A Positive X 0 Fig. 1.1.5: Positive measurements in an anticlockwise Or negatively in a clockwise direction. Fig. 1.1.6: negative measurements in a clockwise direction Example. In the diagrams below 20 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Fig. 1.1.7: Positive to Negative measurements Fig. 1.1.8: positive to negative measurements Note: Since this concerns angles at a point their sum is. But angles of sizes greater than 360° will always lie in any of the four quadrants. This is determined by first trying to find out how many revolutions (one completed revolution = 360°) there are contained in that angle. For example, (b) contains plus i.e. is called the basic angle of and since is in the first quadrant, is also in the first quadrant. (a) , since is in the third quadrant, is also in the third quadrant. Distance Learning Centre ABU, Course Material 21  MATH103: Trigonometry and Coordinate Geometry To find the basic angle of any given angle, you should subtract 360° (1 complete revolution) from the given angle until the remainder is an angle less than 360° , then locate the quadrant in which the remainder falls that becomes the quadrant of the angle. For you to determine the signs whether positive or negative of the angles and their trigonometric ratios in the four quadrants; First, choose any point on the circle and is the centre of the circle. Fig. 1.1.9: negative or positive signs of angles in the four quadrants P = r, is the radius and makes an angle of α with the positive - axis. Since P is any point, isrotated about 0 in the anticlockwise direction, Hence in the 1st quadrant , using your knowledge of trigonometric ratios. = ⁄ = ⁄ = ⁄ Therefore in first quadrant (acute angles) all the trigonometric ratios are positive. 2nd quadrant (obtuse angles) 22 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Fig. 1.1.10: Fig. 1.1.9: negative or positive signs of angles in the four quadrants In at O is , here is - (it lies on the negative axis) but and are positive. The trigonometric ratios are = ⁄ is positive = ⁄ is negative = ⁄ is negative So, only the sine of the obtuse angle is positive, the other trigonometric ratios are negative. Guess what happens in the 3rd quadrant (reflex angles). 3rd quadrant (reflex angles) Note: P = r (i.e.) the radius is always positive. Reference is made to 180°, so the angle is (180 + α)° or α - 180° IN-TEXT QUESTION 2 Cosine in the second quadrant is negative TRUE/FALSE IN-TEXT ANSWER Distance 2. Learning TRUECentre ABU, Course Material 23  MATH103: Trigonometry and Coordinate Geometry Fig. 1.1.11: negative or positive signs of angles in the four quadrants – is negative is negative is positive So if the angle lies between and the sine, cosine of that angle are negative while the tangent is positive. 4th quadrant (Double Reflex angles) y 24 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Fig. 1.1.12: negative or positive signs of angles in the four quadrants Here PA is negative but OA and OP are positive. – is negative – is positive is negative. Here again, we have sine and tangent of any angle that lies between 270° and 360° are negative the cosine of that angle is positive. Looking at the figures above, it is seems that, the sign of a cosine is similar to the sign of the – axis (and coordinate) while the sign of a sine is similar to the sign of - coordinate (i.e. - axis). The signs can then be written in the four quadrants as shown below Distance Learning Centre ABU, Course Material 25  MATH103: Trigonometry and Coordinate Geometry Fig. 1.1.13: Summary of the signs in their quadrants Fig. 1.1.14: Summary of the signs in their quadrants A summary of the signs in their respective quadrants, thus going in the anticlockwise direction, the acronym is; (i) CAST(from the 4th to 1st to 2nd and then 3rd) (ii) ACTS (from 1st 4th 3rd then 2nd ) Clockwise 26 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry (iii) All Science Teachers Cooperate (ASTC) (from the 1st 2nd 3rd 4th ). The letters in Fig.1.1.14 (marked quadrants) show the trigonometric ratios that are positive (iv) SACT (2nd 1st 4th 3rd) (v) TASC (3rd 2nd 1st 4th) 3.0 TUTOR MARKED ASSIGNMENT Find the values of in the following equations (1) = (2) (3) (4) Find the basic angles of the following and hence indicate the quadrants in which they fell. (1) (2) (3) (4) (5) (6) 4.0 SUMMARY In this study session, you have seen that the trigonometric ratios with respect to a right angled triangle is = i.e. SOH =.e. CAH = TOA Hence the acronym SOH CAH TOA which is a combination of the above meaning can be used to remember the trigonometric ratios Again, you saw the relationships between the trigonometric ratios. The sine or cosine of an acute angle equals the cosine or sine of its complementary angle. That i.e. (1) Distance Learning Centre ABU, Course Material 27  MATH103: Trigonometry and Coordinate Geometry For obtuse angle, (2) (3) (4) (360 - ) = - sin (360 - ) = cos CONCLUSION In this study session, you have learnt the definition of the trigonometric ratios sine, cosine and tangent and how to find the trigonometric ratios of any given angle. You should have also learnt that the value of any angle depended on its basic angle and its sign depends on the quadrant in which it is found. Thou now understand that the most commonly used trigonometric ratios are the sine, cosine and tangent; and the basic angle lies between and 360° i.e. In the next session a detail study of the relationship between the trigonometric ratios and their reciprocals and many more shall be presented. 5.0 SELF-ASSESSMENT QUESTIONS 1. Find the value of in the following (i) (ii) (iii) (IV). In case you are finding it difficult, the following are the 2. Find the trigonometric ratios in their following triangle B 28 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry B 5 4 C A Fig. 1.1.15: Acute angle 3. In the following, angle is acute and angle α is acute. Find the following trigonometric ratios. A 17 B C 15 Fig. 1.1.16: Acute angle α and (a) (b) (c) (d) (e) (f) 4. Indicate the quadrants of the following angles and state whether their trigonometric ratios of each is positive or negative. (1) Distance Learning Centre ABU, Course Material 29  MATH103: Trigonometry and Coordinate Geometry 5. Show in which of the quadrant each of the following angles occurs and state whether the trigonometric ratio of the angle is positive or negative. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 6. Find the values of in the following equations (1) (2) (3) (4) (5) find the value of and if Answer 1. i. ii. iii. iv. 2. i. ii. iii. ta 3. i. b. c. d. e. 4. i. 2nd ii. 2nd iii. 1st iv.4th v. 3rd 5. i. 2nd ii. 2nd iii. 2nd iv. 1st v. 1st vi. 3rd vii. 3rd viii. 4th ix. 3rd.x. 3rd 6. i. 420 ii. -79067’ iii. 720 iv. 560 v. 6.0 ADDITIONAL ACTIVITIES Visit U-tube https://www.youtube.com/watch?v=JmLN3QxshlE Watch the video & summarise in 1 paragraph 30 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry 7.0 REFERENCES Amazigo, J.C. (ed) (1991): Introductory University Mathematics I: Algebra, Trigonometry and Complex Numbers. Onitsha: Africana - feb Publishers Ltd David - Osuagwu, M; Anemelu C and Onyeozilu I. (2000): New School Mathematics for Senior Secondary Schools. Onitsha: Africana – Feb Publishers Ltd Egbe, E. Odili, G.A and Ugbebor, O. O. (2000): Further Mathematics. Onitsha: Africana - feb Publishers ltd Vygodsky, M. (1972): Mathematical Handbook: elementary Mathematics. Mosco: M/R Publishers. Distance Learning Centre ABU, Course Material 31  MATH103: Trigonometry and Coordinate Geometry Study Session 2 Trigonometric Ratios II Section and subsection headings: Introduction 1.0 Learning Outcomes 2.0 Main Contents 2.1 Trigonometric Ratios - Reciprocals 2.1.1 Relationship between the Trigonometric Ratios and their Reciprocals 2.2 Use of Trigonometric Tables 2.2.1 Use of Natural Trigonometric Tables 2.2.2 Use of Logarithm of Trigonometric Tables 3.0 Tutor - Marked Assignment. 4.0 Conclusion and Summary 5.0 Self-Assessment Questions 6.0 Additional Activities 7.0 References/Further Reading INTRODUCTION You are welcome to study session 2, in the previous study session, you learnt about the basic trigonometric ratios sine, cosine and tangent. You also saw the relationship between the sine and cosine of any angle; nothing was mentioned about the relationship of the tangent except that it is the sine of an angle over its cosine. Also in our discussion, form our definition of ratios only one aspect is treated i.e. what happens when it is ⁄ An attempt to answer this question will take us to the study session on the reciprocals of trigonometric ratios - secant, cosecant and cotangent. 1.0 LEARNING OUTCOMES After you have finished studying this session, you should be able to: 32 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry 1. Define the reciprocals of trigonometric ratios in relation to the right angled triangle. 2. Establish the relationship between the six trigonometric ratios 3. Use trigonometric tables to find values of given angles. 2.0 MAIN CONTENTS 2.1 TRIGONOMETRIC RATIOS II From the previous study session, using , right-angled and and with the usual notations, the knowledge of the ratio of two numbers " and " expressed as was used to find the sine, cosine and tangent of. In this study session, the expression will be used. A b c C B a Fig. 1.2.1: right angled triangle = = = Now if this relationship is viewed in this order. it is called cosecants or cosecs is calculate cotangent opposite of or Distance Learning Centre ABU, Course Material 33  MATH103: Trigonometry and Coordinate Geometry Now study the above ratios carefully, what can you say of their relationship? This leads us to the following sub-heading 2.1.2 RELATIONSHIPS BETWEEN THE TRIGONOMETRIC RATIOS. As you can see and for example are related in the sense that Sin = and cosec = from which means that = = = * + This then means that is the reciprocal of and sin is the reciprocal of. From the above ratios also, you can see; Note from the sum of angles of a triangle giving , the following relations can be proved. A b c C B a Fig. 1.2.2 You should recall that in study session 1, And Now let us, see the tangent. (90° - θ) = BC/AC in Fig. 1.2.2 i.e. 34 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry IN-TEXT QUESTION 1 The reciprocal of and are respectively ________ and ________ ANSWER 1. and 2. Yes, they can be used to find their reciprocals Also –. This brings us to the conclusion that the tangent of an acute angle is equal to the cotangent of its complement. I.e. = and ; also Now move to the next step, the relationship between trigonometric ratios of other angles. It has been established that: (1) The secant of any angle is the reciprocal of the cosine of the angle i.e. (2) And (3) It then means that whatever applies to the trigonometric ratios their reciprocals, so the following are true in the first quadrant i.e.; (Acute) all the reciprocals trigonometric ratios are positive. and. In the second quadrant (obtuse) since only the sine is positive only its reciprocal the cosecant will also be positive in the third and fourth quadrants respectively only the tangent and cotangent for are positive and cosine and secant in are positive respectively. So the following relationships are established as, 1. 2. is negative, lies between 90° and 180° Distance Learning Centre ABU, Course Material 35  MATH103: Trigonometry and Coordinate Geometry is positive is negative. 3. is negative, lies between and is negative is positive 4. is positive, lies between and Is negative Is negative Having seen the relationships between the trigonometric ratios and their Reciprocals, let us move on to find angles using the trigonometric tables. 2.2 USE OF TRIGONOMETRIC TABLES In the trigonometric tables, the sine, cosine and tangent of angles can be use to find the values of their reciprocals. In the four figure tables available only the tables for sine, cosine and tangent are available So whatever is obtain in their case also applies to their reciprocals. The exact values of the trigonometric ratios can be obtained using the four figure tables or calculators. The tables to be used here are extracting of the Natural sine and cosine of Selected angles between and at the interval of or. The fullTrigonometric tables will be supplied at the end (are tables for log sine, log cos and log tan) 36 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Table 1.2.1: natural sine and cosine of selected angles IN-TEXT QUESTION 2 Can trigonometric table for sine of angles be used to find their reciprocals? 2. Yes, they can be used to find their reciprocals. Note that the difference column always at the extreme right - hand corner of the table is omitted Extracts from natural for ( WAEC, four figure table) Distance Learning Centre ABU, Course Material 37  MATH103: Trigonometry and Coordinate Geometry Table 1.2.2 Again the difference column is omitted. 2.2.2 USE OF LOGARITHMS OF TRIGONOMETRIC FUNCTIONS At times, you might be faced with problems which require multiplication and direction in solving triangles. Here the use of tables of trigonometric. A function becomes time consuming and energy sapping. It is best at this stage to use the tables of the logarithms of trigonometric functions directly. 3.0 Tutor Marked Assignment In the diagram below, find the trigonometric ratios indicated. 38 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Fig. 1.2.3 (a) (b) (c) (d) 2. Express the following in terms of the trigonometric ratios of α (a) i. – ii. (b) i. – ii. (c) i. – ii. Sec (180 – α) (d) i. – ii. – 3. Find the basic angles of the following and their respective quadrants. (a) 1290o (b) -340o (c) -220o (d) 19o (e) 125o (f) 214o 4. Use trigonometric tables to find the value of the following: (a) (b) (c) (d) (e) 5. Use the logarithm table for trig. Functions to find the value of the following. (a) (b) (c) Distance Learning Centre ABU, Course Material 39  MATH103: Trigonometry and Coordinate Geometry 4.0 SUMMARY In these two study sessions you have seen that the trigonometric ratios and their reciprocals with respect to a right angled triangle is = = = The acronym SOH CAH TOA meaning S = sine, O = opposite over, H = hypotenuse C = cosine, A = adjacent over, H = hypotenuse T = tangent, O = opposite over, A = adjacent Can be used to remember the trigonometric ratios their reciprocals are obtained from these. You have also learnt that: (i) The sine or cosine or tangent of an acute angle equals the cosine or sine or cotangent of its complementary angle. This means that you can use the sine table find the cosine of all angles from 90 to 0 at the same interval of 61 or 0°.1° (ii) The tables of trigonometric functions can also be used in finding the ratios of given angles by bearing in mind the following where is acute or obtuse. (iv) (v) 40 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry (vi) In using the table sometimes angles may be expressed in radians, first convert the angles in radians to degrees before finding the trigonometric ratios of the given angles or convert from degrees to radians before finding the trigonometric ratios, if it is in radians CONCLUSION In study session 1 and 2, you have learnt the definition of the trigonometric ratios and their reciprocals, and how to find the trigonometric ratios of any given angle and the use of trigonometric tables in finding angles. You should have also learnt that the value of any angle depends on the basic angle and its sign depends on the quadrant in which it is found. However, you need be aware that the most commonly used trigonometric ratios are the sine, cosine and tangent and the basic angle lies between and 360 i.e. In the next session we are going to consider inverse trigonometric functions, definition and notation of inverse trigonometric functions and lots more. 5.0 SELF-ASSESSMENT QUESTIONS 1. Find the value of in the following (a) (b) (c) (d) 2. In the diagram, on the right find the following: Distance Learning Centre ABU, Course Material 41  MATH103: Trigonometry and Coordinate Geometry A 17 c C 15 B Fig. 1.2.4 3. Find the value of the following angles: (i) (ii) (iii) (iv) 4. At times, there might have problems involving minutes or degrees other than the one given in the table. You have to use the difference table when such is the case. For example Find (1) (II) 5. Find (3) (4) ° 6. Find (i) 7. Find the quadrant of the following angles and determine whether the trigonometric ratios (reciprocals) are positive or negative. (a) 1000 (b) 1100 (c) 1230 (d) 420 (e) 200 (f) 2310 (g) 2680 (h) 3120 (i) 5910 (j) 19990 Answer 1. a. 70 b. 0.766 c. 2.747 d. 50 2. a. b. 3. i. 0.3515 ii. 0.8650 iii. 0.9425 v. 0.7581 42 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry 4. i. 0.3445 ii. 0.6405 5. i. 0.8660 ii. -0.5 iii. 0.9848 iv. -0.1736 6. i. -.0286 ii. -3.884 iii. -0.0736 7. a. 2nd b. 2nd c. 2nd d. 1st e. 1st f 3rd g. 3rd h. 4th i. 3rd j. 3rd 6.0 ADDITIONAL ACTIVITIES Visit U-tube https://www.youtube.com/watch?v=QQKn__bZAoU Watch the video & summarise in 1 paragraph 7.0 REFERENCES Amazigo, J.C. (ed) (1991): Introductory University Mathematics I: Algebra, Trigonometry and Complex Numbers. Onitsha: Africana - feb Publishers Ltd David - Osuagwu, M; Anemelu C and Onyeozilu I. (2000): New School Mathematics for Senior Secondary Schools. Onitsha: Africana – Feb Publishers Ltd Egbe, E. Odili, G.A and Ugbebor, O. O. (2000): Further Mathematics. Onitsha: Africana - feb Publishers ltd Vygodsky, M. (1972): Mathematical Handbook: elementary Mathematics. Mosco: M/R Publishers. Distance Learning Centre ABU, Course Material 43  MATH103: Trigonometry and Coordinate Geometry Study Session 3 Inverse Trigonometric Functions Section and Subsection Headings: Introduction 1.0 Learning Outcomes 2.0 Main Content 2.1 Inverse Trigonometric Functions 2.1.1 Definition and Notation of Inverse Trigonometric Functions 2.1.2. Inverse Trigonometric Functions of Any Angle 2.1.3. Principal Values of Inverse Trigonometric Functions. 2.2. Trigonometric Ratios of Common Angles 2.2.1. Trigonometric Ratios of 30° And 60° 2.2.2. Trigonometric Ratios of 45° 2.2.3. Trigonometric Ratios of 0°, 90° And Multiples Of 90° 3.0 Tutor - Marked Assignment. 4.0 Conclusion and Summary 5.0 Self-Assessment Questions 6.0 Additional Activities 7.0 References/Further Reading INTRODUCTION Once again you are welcome to study session 3, in the previous study session, you have learnt the definition of the trigonometric ratios and their reciprocals, and how to find the trigonometric ratios of any given angle and the use of trigonometric tables in finding angles. Very often you see relations like is possible to find the value of y, if is known. On the other hand the need might arise to find the value of when y is known. What do you think can be done in this case? In the example above, , sine is a function of an angle and also the angle is a function of sine. 44 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry In this study session, you shall learn the inverse trigonometric functions, sometimes called circular functions and the basic relation of the principal value and trigonometric ratios of special angles 0°, 30°, 45°, 60°, 900, 180°, 270° and 1.0 LEARNING OUTCOMES After you have finished studying this session, you should be able to: 1. Define inverse trigonometric functions 2. Find accurately the inverse trigonometric functions of given values. 3. Determine without tables or calculators the trigonometric ratios of 0°, 30°, 45°, 60°, 90°, 180°, 270 ° and 360°. 4. Solve problems involving inverse trigonometric functions and trigonometric ratios of special angles correctly. 2.1 INVERSE TRIGONOMETRIC FUNCTIONS (CIRCULAR FUNCTIONS) 2.1.1. DEFINITION AND NOTATION The trigonometric ratios of angle are usually expressed as y = sin (where y and represents any value and angle respectively). Or Or. Above are examples when the values of is known, but the value of is unknown and y is known, the above relations can be expressed as: =( ) written as arc read as ark Or =( ), written as arc read as ark Or = ( ) written as arc read as ark These relations , and are called the inverse trigonometric functions or circular functions. Distance Learning Centre ABU, Course Material 45  MATH103: Trigonometry and Coordinate Geometry EXAMPLE (a) If , then = meaning that is the angle Whose is or the of is (b) If , at then , which implies that is the angle whose is 0.8594 or of = 0.8954. (c) If , then shows that is the angle whose is or the of is. PROCEDURES FOR FINDING INVERSE TRIGONOMETREIC FUNCTIONS Having been conversant with the use of the trigonometric tables, the task here becomes easy. In other to find the inverse trigonometric ratio of any angle, first look for the given value on the body of the stated trigonometric table and, read off the angle and minute under which it appeared. If the exact value is not found, the method of interpolation can be adopted. 2.1.2 INVERSE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. The inverse trigonometric functions here are extended to include values of given angles between 0° and 360° and beyond EXAMPLE Find the value of between and in the following: (a) (b) (c) t SOLUTIONS (a) ; Since sin is positive then the angle must either be in the 1st or 2nd quadrant thus 46 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry Fig. 1.3.1: values of across quadrants In the first quadrant and in the 2nd quadrant – (b) From the cosine tables but is negative, therefore, lies either in the 2nd or 3rd quadrant. Fig. 1.3.2: values of across quadrant In the 2nd `quadrant In the 3rd quadrant Distance Learning Centre ABU, Course Material 47  MATH103: Trigonometry and Coordinate Geometry (c) , here but since tan is negative, lies either in the 2nd or 4th quadrants. Fig. 1.3.3: values of across quadrants In the 2nd quadrant In the 4th quadrant Note that, from the previous units, there are several values of with the same value but in different quadrants. For example sin 30° = sin 150° = sin 390° = sin 750° etc. Hence the inverse trigonometric functions have many valued expressions. This mean that one value of is related to an infinite number of values of the function. Hence to obtain all possible angles of a given trigonometric ratio, we either add or subtract , where is any integer positive, negative or zero 2.1.3 Principal Values of Inverse Trigonometric Functions. In this section, your attention should be found on the value which lies in a specified range for example: (i) For , the range of values are – This value is called the principal value of the inverse of sine denoted by (Smalls). 48 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry For example if √ radians then the principal value of the inverse of sin 1/√ is or /4 (since it is within the √ range). (ii) If , then is the inverse cosine of and the principal value of the inverse of cosine is the value of in the range 0° to (180°). This is the same for arc cot , and arc sec Example, if , then arc sec the principal value √ ( ) the principal value is ( ) √ √ (iii) The principal value of the inverse of tangent is the value of in the Range This is the same for arc cosec. EXAMPLE OF PRINCIPAL VALUES The principal value of; (a) (b) √ (c) The relationship between the values of an inverse function and its principal value is given by the formulae below ( 1972: 366). (i) (ii) (iii) (iv) Where is any integer positive, negative or zero. Hence denotes arbitrary values of inverse trigonometric functions and denotes principal values of given angles. Distance Learning Centre ABU, Course Material 49  MATH103: Trigonometry and Coordinate Geometry IN-TEXT QUESTION 1 The principal value of for , is the length of the arc of a unit circle centred at the origin which subtends an angle at the centre whose sine is. TRUE/FALSE ANSWER 1. TRUE EXAMPLE (a) 2.2 TRIGONOMETRIC RATIOS OF COMMON ANGLES The angles 0°, 30°, 45°, 60°, 90° are called common angles because they are frequently used in mathematics and mechanics in physics. Although, the trigonometric ratios of common angles 0°, 30°, 45°, 60°, 90°, (and multiples of 90° up to 360° ) can be found from the trigonometric tables, they can be easily determined and are widely used in trigonometric problems. 2.2.1 THE ANGLE OF 30° AND 60° Consider an equilateral triangle ABC of sides 2cm. An altitude AD (see Fig.1.3.4 below) 50 Distance Learning Centre ABU, Course Material MATH103: Trigonometry and Coordinate Geometry A √ B D C Fig. 1.3.4: equilateral triangle ABC An altitude AD bisects < BAC so that < BAD = < CAD = 30° < ABC = < ACB = 60° by Pythagoras theorem AD √ units. Hence, the value of the trigonometric ratios of 60° and 30° are √ And And √ √ And √ √ And √ And √ √ And 2.2.2. TRIGONOMETRIC RATIOS OF 45° Consider a right-angled isosceles triangle ABC with AB= BC = 1 unit,