Mathematics Class 9 Textbook PDF

Developed by the National Curriculum and Textbook Board as a textbook according to the National Curriculum 2022 for Class Nine from the academic year 2024 Mathematics Class Nine (Experimental Edition)...

Developed by the National Curriculum and Textbook Board as a textbook according to the National Curriculum 2022 for Class Nine from the academic year 2024 Mathematics Class Nine (Experimental Edition) Writers & Editors Dr. Md. Rashed Talukder Dr. Md. Abdul Hakim Khan Dr. Md. Abdul Halim Dr. Chandra Nath Podder Nowreen Yasmin Md. Ahasanul Arefin Chowdhury Ratan Kanti Mondal Md. Munjil Hossain Asif Bayezid Md. Kamaruddin Akan Md. Mokhlesur Rahman Mst. Nurunnesa Sultana Translated By Mian Md. Naushaad Kabir D. M. Zunayed Kamal Nibir Abdullah Al-Mamun National Curriculum and Textbook Board, Bangladesh  Published by National Curriculum and Textbook Board 69-70, Motijheel commercial Area, Dhaka-1000 [All rights reserved by National Curriculum and Textbook Board, Bangladesh] 1st Published : ---------------- 2023 Art Direction Monjur Ahmed Illustration Kamrun Nahar Mahmudul Hasan Siam Cover Mahmudul Hasan Siam Graphic Design Noor-E-Elahi K. M. Yusuf Ali For free distribution by the Government of the People’s Republic of Bangladesh Printed by:  PREFACE In this ever-changing world, the concept of life and livelihood is changing every moment. This process of change has been accelerated due to the advancement of technology. There is no alternative to adapting to this fast changing world as technology is changing rapidly ever than before. In the era of fourth industrial revolution, the advancement of artificial intelligence has brought about drastic changes in our employment and lifestyles that will make the relationship among people more and more intimate. Various employment opportunities will be created in near future which we cannot even predict at this moment. We need to take preparation right now so that we can adapt ourselves to that coming future. Although a huge economic development has taken place throughout the world, problems like climate change, air pollution, migrations and ethnic violence have become much more intense nowadays. The breakouts of pandemics like COVID 19 have crippled the normal lifestyle and economic growth of the world. Thus, different challenges as well as opportunities, have been added to our daily life. Standing amid the array of challenges and potentials, sustainable and effective solutions are required to transform our large population into a resource. It entails global citizens with knowledge, skill, values, vision, positive attitude, sensitivity, adaptability, humanism and patriotism. Amidst all these, Bangladesh has graduated into a developing nation from the underdeveloped periphery and is continuously trying to achieve the desired goals in order to become a developed country by 2041. Education is one of the most crucial instruments to attain the goals. Hence, there is no alternative to the transformation of our education system. This transformation calls for developing an effective and updated curriculum. Developing and updating the curriculum is a routine and important activity of National Curriculum and Textbook Board. The curriculum was last revised in 2012. Since then, more than a decade has elapsed. Therefore, there was a need for curriculum revision and development. With this view, various research and technical studies were conducted under NCTB from 2017 to 2019 to analyze the current state of education and identify the learning needs. Based on the researches and technical studies, a competency- based and seamless curriculum from K−12 has been developed to create a competent generation capable of surviving in the new world situation. Under the framework of this competency based curriculum, the textbooks have been prepared for all streams (General, Madrasah and Vocational) of learners for Class Nine. The authentic experience-driven contents of this textbook were developed with a view to making learning comprehensible and enjoyable. This will connect the textbooks with various life related phenomenon and events that are constantly taking place around us. It is expected that, through this, learning will be much more insightful and lifelong. In developing the textbooks, due importance has been given to all − irrespective of gender, ethnicity, religion and caste while the needs of the disadvantaged and special children are taken into special considerations. I would like to thank all who have put their best efforts in writing, editing, revising, illustrating and publishing the textbook. If any errors or inconsistencies in this experimental version are found or if there is any suggestions for further improvement of this textbook, you are requested to let us know. Professor Md. Farhadul Islam Chairman National Curriculum and Textbook Board, Bangladesh  Dear Students Welcome to the ninth grade students of secondary level. National Curriculum and Textbook Board, Bangladesh has planned to develop new textbooks for all secondary level students. In continuation of this, the new mathematics book of class 9 has been written. Fundamental changes have been made in the book’s presentation, decoration, topics and teaching-learning methodology of mathematics. Surely you have various curiosity about these changes and innovations in class 9 books. In experiential learning method it is very im-portant to link the subject matter with real life ex-perience. In this context two aspects have been giv-en utmost importance while preparing the book suitable for students class IX. First, they will have the opportunity to solve mathematical problems through hands-on work by observing the objects and events of the familiar environment around them. Second, they can acquire the techniques of using mathematical skills in various tasks of daily life. A total of nine learning experiences are planned in this book of class 9. You will participate in these experiences by mathematically analyzing and solv-ing real-life problems. Each learning experience is presented step by step so that you can master mathematical concepts and skills through active participation and use of real materials. This jour-ney of learning mathematics through mathematical inquiry will be as enjoyable for you as you will dis-cover for yourself the relation of mathematical concepts to real life. The teacher will give you full support in all activi-ties inside and outside the classroom. We also hope that you will be supportive of each other as you participate in the various activities of this learning program and explore different topics of mathemat-ics with your classmates. You will always remem-ber that when all of you have a cooperative spirit you can do any task successfully. We hope this book will play an important role in ensuring an effective and enjoyable learning journey for you in the world of mathematics. The textbook will serve as a helpful resource for you. Good luck to you all.  Index Name of experiences Page No. Sets in daily life 1 - 28 Sequence and series 29 - 58 Concept and Application of Logarithm 59 - 80 Polynomial Expression in Nature and Technology 81 - 112 System of Equations in Real World Problems 113 - 140 Trigonometry in Measurement 141 - 156 Trigonometry for Angular Distance 157 - 178 Measuring Regular and composite solids 179 - 210 Measures of Dispersion 211 - 235  Sets in Daily Life You can learn from this experience- Concept of sets Types of sets Operations of sets Venn diagram Cartesian product Application of sets Ant Bee Can Fly 6 legs moves by walking collects nectar lives in a colony hunters in nature makes hives in controlled by makes anthills open space a queen inside soil Academic year 2024 1  Sets in daily life Sets in daily life You are given a set of books on being promoted to a new class. When you started class VIII, what books were included in the set of books you were given? Write in the blank space below Set of books of class VIII: Think about the last time you used coloured pencils, what colours were in your set of coloured pencils? Colours in a set of coloured pencils: Many of you must love to play or watch cricket. Below is a picture of a set of cricket playing equipments. Look at what are in the set and write in the blank next to it: Set of cricket playing equipments: By now you can understand that we are discussing sets of different things. You have seen the set of textbooks, color pencils, cricket equipment etc. A set can be of as many students as there are in your class. The colors of the national flag of Bangladesh can be a Academic year 2024 set. A set can also be made of whatever is on your reading table. These are real objects. Abstract objects can also be in sets. For example, the set of names of the players in your school’s football team. There can also be different sets of numbers, for example, the set of integers. 2  Mathematics Let’s know German mathematician Georg Cantor (Georg Ferdinand Ludwig Philipp Cantor) is known as the father of set theory. He was born in Russia. Cantor and his lifelong friend Richard Dedekind agreed after exchanging letters that a set is a collection of finite or infinite objects that have a certain property and each object retains its uniqueness. Georg Cantor Then we can say, A specific collection of different objects, real or abstract, is called a set. 1.1 Importance of sets in mathematics By now you may have started wondering what is the importance of sets in mathematics? A careful consideration of the following example will make this clear to you. Example 1 Sixteen students of Mitu’s school participated in a local Mathematical Olympiad, where students were given various quizzes to test their intelligence with a total score of 100. Based on the results, it was decided who among them would go to the National Mathematical Olympiad. Students who secured more than 60% marks would represent the school at the national level. Name Score Name Score Name Score Name Score Sagor 58 Dipti 45 Koli 77 Maruf 50 Kamal 72 Avijit 63 Tribijoy 74 Andrew 76 Mitu 79 Jeba 90 Choity 81 Akash 59 Selim 33 Rowson 35 Nahar 78 Tasnim 80 Academic year 2024 Now, if we express the set of scores more than 60% by A and less than or equal to 60% by B, we get, 3  Sets in daily life A = {72, 79, 63, 90, 77, 74, 81, 78, 76, 80} and B = {58, 33, 45, 35, 50, 59} What do we understand from this? Notice that we can understand the following matters clearly. More than half of the students got more than 60%. About one-third of the students got less than 60%. Scores less than 60% are between 33 and 59. What other matters can be understood from these sets? Write in the blank space below. Note, in the above example we created two different sets based on a condition of some mathematical data. Now tell me what other decisions can be made to improve the skills of the students participating in Math Olympiad? Suppose we can draw the following conclusion. The school and the mathematics teachers need to take urgent measures to improve the mathematical understanding of students whose scores are in the B set. One such decision was made because we divided the students into two sets. Did you understand the need for sets through this example? By set we can denote a collection of similar mathematical or abstract data. By separating similar information or data, it is possible to gain a clear understanding of data processing and relevant issues. So let us try to know about one such important matter. 1.2 Expressing the sets Academic year 2024 By now you know the definition and importance of sets. There is also a nice way to express the set. The objects that will be expressed are separated by commas inside the second bracket. For example, 4  Mathematics The set of colours in the national flag of Bangladesh = {Green, Red} Work in pairs Express as a set: 1. The set of books of all subjects in class eight = 2. The set of colours in your colouring pencils = 3. The set of cricket equipments shown in the figure = 1.3 Method of writing sets Sets are usually named using capital letters A, B, C,..., X, Y, Z of English alphabet. In a set, the objects are called elements. Elements are usually expressed using small letters a, b, c,..., x, y, z of English alphabet. If B = {a, b}, elements of set B are a and b. The symbol to express elements is ∈ , That is, a ∈ B means that a is an element of set B or a belongs to B. If c is not an element of set B then we write c ∉ B and it is read as c is not an element of B or c does not belong to B. Individual task: 1. Construct the set of prime factors of 210 and write in the empty box below. Academic year 2024 5  Sets in daily life Individual task 2. If X = {5, 7, 9, 11, 13} write ∈ or ∉ in appropriate boxes below. ∈ ∉ 9 X 10 X 3 X 13 X 1.4 Methods of expressing sets As you can see, we can specifically express a collection of objects or numbers by means of sets. That is, it can be said precisely whether an object is an element of a set or not. For example- A = {1, 3, 5, 7, 9} is the set of all odd positive numbers less than 10. Here we can specifically say which are the elements of A. For example- 3 ∈ A but 4 ∉ A. B = {a, e, i, o, u} is the set of vowels in English alphabet. Here i ∈ B but b ∉ B. Sets are expressed in two ways. Roster Method or Tabular Method and Set Builder Method. 1.4.1 Roster Method or Tabular Method In this method, all elements of the set are separated by commas and written in second brackets. For example, The set formed by numbers 1, 2, 3 : A = {1, 2, 3} The set of prime numbers: Ρ = {2, 3, 5, 7, 11, ⋯} The set of even numbers: E = {⋯, –8, –6, –4, –2, 0, 2, 4, 6, 8,⋯} 1.4.2 Set builder method In this method all the elements of the set are expressed by specifice properties or condi- tions. For example, A = {x : x is an odd natural number} Academic year 2024 Notice that, there is a ‘:’ (colon) after x. The sign ‘:’is meant ‘such that’. Since in this method rules are given for determining the elements of the set, this method is also called Rule Method. Example 1. Express the set A = {0, 3, 6, 9, 12, 15} in set builder method. 6  Mathematics Solution: Here each element of the set is an integer, not less than 0, not greater than 15, and a multiple of 3. So in set builder method we can write, A = {x : x is an integer and multiple of 3, 0 ≤ x ≤ 15} Example 2. Express the set A = {x : x is an integer, x2 ≤ 25} in tabular method. Solution: Here each element of the set is an integer whose square is less than or equal to 25. The numbers are 0, ±1, ±2, ±3, ±4, ±5. Therefore, in tabular method we can write, A = {0, ±1, ±2, ±3, ±4, ±5} = {–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5} Individual task: 1. Express the following sets using set builder method. a) A = {–28, –21, –14, –7, 7, 14, 21, 28} b) B = {0, 1, 2, 3, 5, 8,…} 2. Express the following sets using tabular method. a) D = {x:x,is a multiple of 5 and less than 30} b) F = {x:x is a factor of 30} c) G = {x:x is a positive integer and x2 < 17} d) H = {x : x2 + 3x + 2 = 0} Examples of some special sets N : Set of all natural numbers Z : Set of all integers Q : Set of all rational numbers R : Set of all real numbers Z+ : Set of all positive integers Academic year 2024 Q+ : Set of all positive rational numbers R+ : Set of all positive real numbers 7  Sets in daily life 1.5 Types of sets 1.5.1 Universal Set If elements of any set are collected from a particular set, then the particular set from which the elements are collected is called the universal set. The universal set is usually denoted by U. But the universal set can also be expressed with other symbols. For example, If we consider the set of all even natural numbers E = {2, 4, 6,...} and the set of all natural numbers N = {1, 2, 3, 4, 5, 6,...} then N is the universal set of E. Example: The set A = {x, y} is from the set of English lowercase alphabet. So the set of English lowercase letters is the universal set of A = {x, y}. 1.5.2 Finite Set A set containing a finite number of elements is called a finite set. For example- 1. A = {2, 4, 6, 8} 2. B ={a, e, i, o, u} 3. F = {x : x is a prime number and 30 < x < 70} The number of elements of set A and set B are respectively 4 and 5. Brain storm What is the number of elements in set F ? Write how did you determine in the blank space below. 1.5.3 Infinite Set A set whose numbers of elements is not finite is called an infinite set. We cannot finish counting the number of elements in an infinite set. For example, a) A ={x : x is an odd natural number} b) Set of natural numbers N = {1, 2, 3, 4,...} c) Set of integers Z = {...,–3, –2, –1, 0, 1, 2, 3,...} Academic year 2024 {a d) Set of rational numbers Q = b : a and b are integers and b ≠ 0 } e) Set of real numbers R 8  Mathematics Brain storm Why are the above sets infinite? Team task Some sets are described in the left-hand column of Table 1.1 below. Your task is to decide whether each set is finite or infinite by placing a tick (∙) in the correct blank. Also write your logic in the blank column on the right side. Table 1.1 Serial Set Finite Infinite Your logic 1 All odd numbers less than 10 2 Rivers of Bangladesh 3 Vowels of English alphabet 4 Prime factors of 210 5 B = {x : x is a factor of 30} 6 D = {x : x2 + 3x + 2 = 0} 7 Ρ = {2, 3, 5, 7, 11, ⋯} 8 A = {x ∈ N : 0 < x 0 and a perfect square then the roots are real, unequal and rational. 2 Academic year 2024 If b2 - 4ac > 0 and not a perfect square then the roots are real, unequal and irrational. If b2 - 4ac < 0 there is no real root of the equation. 132  Mathematics Pair Work Some equations are given below. Complete the following list by determining the nature of the roots of the equations. Discriminant b2 Nature of the Sl. Equation Nature of roots - 4ac discriminant 2x2 - 10x + 9 = 0 = (-10)2 - 4.2.9 b2 - 4ac > 0 and Real, unequal 1 = 100 - 72 not a perfect and irrational. = 28 square. 2 7x2 - x + 2 = 0 3 -5 + 7x + 6x2 = 0 4 -2x + 5 - 3x2 = 0 5 -14x + x2 + 49 = 0 = (-5)2 - 4.3.4 6 = As we mentioned earlier, solving the equation x2 - 7x - 10 = 0 by expanding the middle-term is not easy. Let us now solve this equation using the general method. Problem: Solve the equation x2 - 7x - 10 = 0. Solution: Comparing the equation x2 - 7x - 10 = 0 with ax2 + bx + c = 0 we get, a = 1, b = -7, c = -10. then, -b ± b2 - 4ac -(-7) ± (-7)2 - 4.1(-10) 7 ± 49 + 40 x= 2a = 2.1 = 2 7 ± 89 ∴x= 2 7 + 89 7 - 89 Therefore, the roots are: x1 = Academic year 2024 2 and x2 = 2 133  System of Equations in Real World Problems Individual task Solve the following equations using the method you have learned. Write the roots in the blank spaces. Sl. Equation Roots of the equation 1 3x2 - 5x + 1 = 0 2 12x2 - 11x + 2 = 0 3 5x2 - 8x + 4 = 0 Solving quadratic equations with graphs Graphing the quadratic equation ax2 + bx + c = 0 requires the value of x along with the value of y. Let y = ax2 + bx + c. Then for values of x such that y = 0 i.e., the points at which the graph intersects the x-axis, all those values of x are solutions of the equation ax2 + bx + c = 0. Example: Solve the equation 2x2 - 3x - 2 = 0 using graphs. Solution: Suppose, y = 2x2 - 3x - 2 We calculate some values of y for some values of x. x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 y 12 7 3 0 -2 -3 -3 -2 0 3 7 12 18 On graph paper, take the length of the smallest square as one unit and plot the points to draw the graph. 20 Notice that the graph intersects the x-axis at the (4, 18) 1 points (- , 0) and (2, 0). The values of x at these 2 15 two points are the solution of the given equation. (-2, 12) So, the required solution: x1 = - 1 and x2 = 2 2 10 Individual task (3, 7) Apply the formula to solve 2x2 - 5 Academic year 2024 3x - 2 = 0 and also solve it using (-1, 3) graph. Check if both solutions O are the same -10 -5 (2, 0) 5 -5 134  Mathematics A real-life problem and solution Problem: Setu’s uncle Mr. Hasan is a businessman. He bought a few packets of pens from a wholesale shop for Tk 50000. In another shop he bought pens of the same amount as he got the price Tk 2 less per packet of pens than the first shop and he got 25 more packets of pens. How many packets of pens did Mr. Hasan buy first and what was the price per packet of pens? What should be his selling price so that his total profit would be Tk 12000? Solution: Can you represent the problem with equations? Let me help you. Suppose Mr. Hasan first bought x packets of pens. Now write the answers to the following questions in terms of x. In first shop the price of each packet of pens = Tk In second shop the price of each packet of pens = Tk In second shop he bought = packets Price of pen bought in second shop = Tk Total price of pens = Tk According to the conditions, ( 50000 x - 2) (x + 25) = 50000 or, (50000 - 2x) (x + 25) = 50000x or, 50000x - 2x2 + 50000 × 25 - 50x = 50000x or, - 2x2 + 50000 × 25 - 50x = 0 or, 2x2 + 50x - 50000 × 25 = 0 or, x2 + 25x - 25000 × 25 = 0 This is a quadratic equation of one variable. Solving it using the general formula we get, -25 ± (25)2 - 4 × (-25000) × 25 -25 ± (25)2 + 4 × 25000 × 25 Academic year 2024 x= = 2 2 -25 ± 25 1 + 4000 = 2 135  System of Equations in Real World Problems -25 + 25 1 + 4000 = , [taking positive value as the number of packets can’t be negative] 2 25 × ( 4001 - 1) = ≈ 778 2 Solving a system of simultaneous linear equation of two variables and quadratic equation of one variable In practice, there are many problems that are easier to solve by converting them to linear equations in two variables and quadratic equations in one variable. An example of how to solve a mathematical problem is given first. Example: Solve the following system of linear equation of two variables and quadratic equation of one variable. y = 2x2 - x - 3 x - 5y + 13 = 0 Solution: Suppose, y = 2x2 - x - 3 …… (1) Again given that, x - 5y + 13 = 0 or, x + 13 = 5y or, 5y = x + 13 or, y = x + 13 ………..(2) 5 We can write from (1) and (2), 2x2 - x - 3 = x + 13 5 or, 10x2 - 5x - 15 = x + 13 or, 10x2 - 5x - 15 - x - 13 = 0 or, 10x2 - 6x - 28 = 0 or, 2(5x2 - 3x - 14) = 0 Academic year 2024 or, 5x2 - 3x - 14 = 0 or, 5x2 - 10x + 7x - 14 = 0 or, 5x(x - 2) + 7(x - 2) = 0 136  Mathematics or, (x - 2) (5x + 7) = 0 ∴ x = 2 or, x = - 5 7 Therefore, x - 2 = 0 or 5x + 7 = 0 If x = 2, from equation (2) we obtain, y = 2 + 13 = 15 5 =3 5 7 Again, if x = - , from equation (2) we obtain, 5 - 75 + 13 -7 + 65 5 58 5 58 y= 5 = 5 = 5 = 25 7 58 Required solution: (x, y) = (2, 3), (- , ) 5 25 Solving using graphs Given equations are 6 y = 2x2 - x -3 5 x - 5y + 13 = 0 4 Here, x - 5y + 13 = 0 is a linear equation 3 and y = 2x2 - x -3 is a quadratic equation. 2 You have learned to graph linear equations and quadratic equations. Using your 1 experience, graph the two equations on the same plane. The graph is given in the -4 -3 -2 -1 O 1 2 3 4 figure on the side. Match it with your graph. It is observed from the figure that the two -1 equations intersect at the points (2, 3) -2 7 58 and (- , ). Both methods got us the 5 25 -3 same solution. Thus, the correctness of the solution is verified. Academic year 2024 137  System of Equations in Real World Problems Team Project: Determinig supply according to demand To make a factory profitable, it must produce goods equal to the demand of the consumer. This condition is called market equilibrium. Given below is the equation of supply in relation to the demand for the product produced in a factory. q = p2 - 2p + 44 … … … (i) p - q + 2 = 0 … … … (ii) Here, p is the price of the product and q is the quantity. Find the values of p and q for market equilibrium. Work instruction: 1. Collect a poster paper, a graph paper and other necessary materials. 2. Solve algebraically. Describe the steps in the solution. 3. Plot the graphs of both equations (i) and (ii) on the same two dimensional coordinate axes on graph paper. Determine the point of intersection of the two graphs obtained. 4. Present your group’s work methods and results on a poster paper or on the back of an old calendar. Consult with teacher if necessary. Writes arguments in favour of your group findings on poster paper. Match the solution [ p = 1, q = 3 or, p = 2, q = 4 ] Pair work Following the instructions of class teacher, divide into several groups and solve the following system of equations algebraically. Then solve the equations graphically to prove that the solutions obtained both methods are the same. Write your group activities on posters and present them to the class. Academic year 2024 y = x2 - x - 2 x - 2y + 5 = 0 138  Mathematics Exercise 1. Compare the given systems of equations with a1x + b1y = c1, a2x + b2y = c2 and fill in the blanks. Sl. System of a1 b1 c1 Comparison Position Consistent/ Algebraic equations a2 b2 c2 of ratios on graph Inconsistent decision (i) x + 3y = 1 2x + 6y = 2 (ii) 2x - 5y = 3 x + 3y = 1 (iii) 2x - 4y = 7 x - 3y = -2 (iv) 1 - x-y=0 2 x - 2y = 1 2. Sketch the graph and solve the following pairs of equations that are solvable. Write at least three solutions if they have infinite solutions: x y i) 2x + y = 8 ii) 2x + 5y = -14 iii) 2 + 3 = 8 iv) -7x + 8y = 9 5x 2x - 2y = 5 4x - 5y = 17 4 - 3y = -3 5x - 4y = -3 3. Solve using substitution method: x y i) 7x - 3y = 31 ii) (x + 2)(y - 3) = y(x - 1) iii) a + b = 2 9x - 5y = 41 5x - 11y -8=0 ax + by = a2 + b2 x y iv) 14 + 18 = 1 v) p (x + y) = q(x - y) = 2pq x+y 3x + 5y =2 Academic year 2024 2 + 2 139  System of Equations in Real World Problems 4. Solve using elimination method: x+1 4 3 i) 3x - 5y = -9 ii) y + 1 = 5 iii) 2x + y = 5 iv) ax + by = 1 x-5 1 2 2ab 5x - 3y = 1 = 2 5x - y = 3 bx+ ay = a2 + b2 y-5 5. Solve using cross multiplication method: x y i) 3x - 2y = 2 ii) + = 8 iii) px + qy = p2 + q2 iv) ax - by = ab 2 3 5x 7x + 3y = 43 - 3y = -3 2qx - py = pq bx - ay = ab 4 6. Apu has a rectangular vegetable garden. The perimeter of the garden is 120 meters. Doubling the width and subtracting 3 meters from the length the perimeter will 150 meters. a) The garden is enclosed on 3 sides and is open on one side along its length. How much money will it cost to enclose the empty side with a fence of tk. 10 Per meter? b) If organic fertilizer costs tk.7 per square meter, how much will Apu have to spend on fertilizer in total? 7. Find the nature of the roots of x2 - 3=0 and solve it. 8. Solve 3x2 - 2x - 1 = 0 using formula. Again solve it with the help of graph and show that there is the same solution in both the method 9. Setu’s mother keeps ducks and chickens at home.She bought 25 ducklings and 30 chickens for tk. 5000. If she had bought 20 ducklings and 40 chickens at the same rate, he would have spent tk. 500 less. a) What is the price of a duckling and a chicken? b) How much total profit will she make if she sells each duck at tk. 250 and each chicken at tk.1 60 after rearing for a few days? 10. Solve the following system of equations: y = x2 - 2x - 3 Academic year 2024 x - 3y + 1 = 0 11. Construct 3 sets system of equations of two variables( one is linear and another is quadratic) as you like and solve it. 140  Trigonometry in Measurement You can learn from this experience- Concept of trigonometry Tigonometric ratios. Values of different trigonometric ratios. Elevation and Depression Angle. Solution of real life problem related to distance and height. P h 2nd friend 1st friend 30° 60° B 500 A x Q θ C Academic year 2024 θ θ A B 141  Trigonometry in Measurement Trigonometry in Measurement Suppose, one day, Abhi, Mita and Rina were discussing their studies sitting under a tree. Mita asked Avi if he could tell her the height of the tree. Abhi said, “Yes. I am going to measure the height of the tree right now.” Rina gave a condition that he could not climb up the tree and she proposed to learn how to measure the height of the tree without climbing up. All of a sudden, Mita took their attention. She showed the shadow of the tree and they decided to find out a way of measuring the height of the tree by measuring the shadow. Abhi said, “Actually, the shadow is at the right angle to the tree. If we imagine some lines from the end point of the shadow to the top point of the tree, we will get a right angled triangle. Can it be of any use?” Rina said, “Yes. We can use the Pythagoras theorem.” Mita said, “If we have the length of two sides of a right angled triangle, we can find out the length of the third side through Pythagoras theorem. Here, we can only measure the length of the shadow of the tree. That would be the length of the base. But without measuring the length of the hypotenuse, we cannot find out the height. Maybe, we need a formula for that. Tomorrow, we will discuss this matter with our mathematics teacher.” The next day, Abhi asked their teacher how they could measure the height of a tree without climbing up. The teacher told them to attend some classes on trigonometry. He also assured them that they would be able to solve the problem on their own after attending the classes. 1. Concept of Trigonometry In the case of measuring, the right angled triangle plays an important role. You are aware of an interrelationship among the three sides of a right angled triangle. You learnt it in your previous class. Square on the hypotenuse is equal to the sum of the area of the square drawn on the other two sides. This relationship has been made based on the three sides of the triangle. But a right angled triangle has three angles along with the three sides. We can find out different relationships considering the angles and sides of the triangle. In ancient times, people used to solve different problems through the use of the ratio between the angles and sides of a triangle, for example how to measure the height of a tree without climbing up; how to measure the width of a river standing on one side of the river etc. These mathematical strategies have created a Academic year 2024 special branch of mathematics called trigonometry. The word ‘trigonometry’ originated from the Greek words ‘tri’ (three), ‘gon’ (side) and ‘metron’ (measure). The Egyptians used trigonometry for land measurement and engineering. Trigonometry deals with the problems related to triangles. 142  Mathematics 2. Introduction to Different Sides and Angles of Right Angled Triangle In a right angled triangle, the opposite side of the C right angle is called hypotenuse. There are two Opposite side (height) acute angles along with one right angle. The two Hy acute angles are adjacent to the hypotenuse. One po ten of the sides adjacent to the hypotenuse is called us e base and the other one is called height. The horizontal side parallel to the ground is the base and the vertical side located on the base is the θ height. If we turn the triangle and set the height A Adjacent side (base) B parallel to the ground, the height will become the base and the base will become the height. So, the name of the sides depends on the position of the triangle. It is not a constant thing. Sometimes, it creates confusion. So, in order to avoid the confusion, we have to name the sides in reference to the angles. Suppose, we want to name the sides considering the angles adjacent to the base and hypotenuse. In this case, the base is regarded as the adjacent side and the height is regarded as the opposite side. C In geometric figures, for indicating the top points, capital letters (A, B, C etc.) are used and for indicating the sides, small letters (a, b, c etc.) are used. For b a indicating the angles, usually, Greek alphabets are used. Mathematicians from ancient Greece have been using these alphabets in geometry and trigonometry. θ Some of the alphabets are as follows: A c B Angle α β γ θ δ Name alpha beta gamma theta delta In the above figure of right angled triangle, the angle ∠ABC has been referred as θ. Individual task: Identify the hypotenuse, adjacent side and opposite side of the following triangles considering the angles θ and α. Academic year 2024 143  Trigonometry in Measurement (a) (b) (c) G r 17 8 p q θ F θ θ α 15 E Name of right Acute angle Hypotenuse Opposite side Adjacent angled triangle side a θ 17 b θ c θ d α EF 3. The Sides of Different Sides of a Right Angled Triangle Considering the Interim Angle between the Hypotenuse and the adjacent side Pair work: Draw a right angled triangle in your notebook. You can draw the length of the sides according to your wish but the acute angle adjacent to the base has to be 30°. After drawing the triangle, measure the length of every side by using a scale and complete the following table. (1) (2) (3) (4) (5) (6) (7) (8) (9) Adjacent Opposite Hypote- Adjacent Opposite Opposite Hypote- Adjacent side side nuse side side side nuse Hypotenuse side Hypote- Hypote- Opposite Adjacent Adjacent Opposite nuse nuse side side side side Academic year 2024 Match the six ratios from number 4 to number 9 with your classmates. What have you noticed from the task? You have found the ratio of the sides in reference to a 30° acute angle. Though the sides have different lengths, the ratios are the same. If you find out the ratio of the sides in reference to any acute angle of a right angled triangle, we will see that the ratios are the same though the length of the slides are different. From this experiment, we can say that- 144  Mathematics If the interim angle of base and hypotenuse of right angled triangles are equal, the ratio of the sides of those right angled triangles will be equal. If the measure of the of the interim angles are different, the ratio of their sides will be different too. 4. Naming Different Ratios in Reference to a Particular Angle In a right angled triangle, the ratio of the sides in reference to a particular acute angle is equal. We have three sides: adjacent side, opposite side and hypotenuse. We can create ratios by using any two of the three sides. Do you have any idea about this? We can create six ratios in total. The ratios are as follows: Opposite side Hypotenuse Adjacent side Hypotenuse Opposite side Adjacent side Hypotenuse Opposite side Hypotenuse Adjacent side Adjacent side Opposite side C Opposite side (height) Mathematicians have set six different names for these six Hy po ratios. If the acute angle adjacent to hypotenuse and base ten us is pointed by θ, the ratios will be: sinθ , cosθ , tanθ , cscθ, e secθ and cotθ. The relationships these six ratios create with the sides are as follows: θ A Adjacent side (base) B Opposite side AC AB AC Adjacent side Opposite side sinθ = Hypotenuse = cosθ = Hypotenuse = tanθ = = BC BC Adjacent side AB Hypotenuse BC Hypotenuse BC Adjacent side AB cscθ = Opposite side = secθ = Adjacent side = cotθ = Opposite side = AC AB AC These ratios are Trigonometric Ratios. Usually, the names of trigonometric names are written in short forms. Their full name are as follows: Academic year 2024 145  Trigonometry in Measurement Full name sine cosine tangent cotangent secant cosecant Short form sin cos tan cot sec csc Pair work: After analyzing the trigonometric ratio, try to find out whether all the other ratios can be expressed through sinθ and cosθ. Two examples have been given below. Now, think and complete the table. Hypotenuse 1 1 cscθ, = Opposite side = Opposite side = sinθ Hypotenuse Opposite side Hypotenuse Opposite side sinθ tanθ, = = = Adjacent side Adjacent side cosθ Hypotenuse secθ = ? cotθ = ? 5. The Value of Trigonometric Ratio in Reference to Different Angles 5.1. For 45° angle C Suppose, △ABC is a right angled triangle. ∠B = 1 is a right angle and A = 45°. So, ∠C = 45° [∵ The sum of three angles of a triangle is equal to the sum of two right angle] Then, AB = BC [∵ The opposite sides of equal angles of a θ B triangle are equal to each other] A Academic year 2024 Suppose, AB = BC = a. 146  Mathematics What we get by using the Pythagoras theorem is that AC2 = AB2 + BC2 = a2 + a2 = 2a2 ∴ AC = 2 a opposite side BC a 1 So, sin 45° = ‍sin A = == = = Hypotenuse AC 2a 2 adjacent side BC a 1 In the same way, cos 45° = cos A == = = = Hypotenuse AC 2a 2 Pair work: Find out the value of the following trigonometric ratios. One sample has been provided. 1 1 csc 45° = = = √2 sin 45° 1 √2 tan 45° = ? sec 45° = ? cot 45° = ? 5.2. For 30° and 60° acute angles In the adjacent figure, △ABC is an equilateral triangle. ∴A = B = C = 60° [Every angle of the equilateral triangle is 60°] Draw a perpendicular AD on BC from the point A. The point D divides BC into two equal parts. So, BD = CD. Again the line AD divides the angle ∠BAC into two equal angles. A So, ∠BAD = ∠CAD = 30° Suppose, AB = 2a. So, BD = 1.BC = 1.2a = a and Academic year 2024 2 2 300 AD = AB2 - BD2 = 4a2 - a2 = 3a2 = 3a So, we can write: 600 B D C 147  Trigonometry in Measurement AD 3a 3 BD a 1 cos30° = = = , cos60° = = = AB 2a 2 AB 2a 2 Pair work Find out the value of the following trigonometric ratios and show these to your teacher. sin 30°, sin 60°, tan 30°, tan 60°, sec 30°, sec 60°, csc 30°, csc 60°, cot 30°, cot 60° 5.3. For 0° angle We have learnt how to find out the trigonometric ratio of the angles 30°, 45° and 60°. Now, we will see what will be the shape of a triangle if its angle is 0° or 90°. We will also learn how to find out the value of the ratio of a triangle if its angle is 0° or 90°. Suppose, △ABC is a right angled triangle. If the value of the angle ∠A of the triangle becomes smaller, the length of BC will get smaller too. The closer the value of ∠A gets to 0, the closer the length of BC gets to zero. C C C C C C A B A B A B A B A B A B opposite side BC In the triangle △ABC, the value of sin A = = will be close to 0. In Hypotenuse AC this case, the length of AC will be nearly equal to AB. adjacent side AB Then, the value of cos A = = will be 1. Hypotenuse AC This idea helps us to define sinA and cosA when A = 0°. We can write: sin 0° = 0 and cos 0° = 1 Individual task: Find out the value of tan 0°, cot 0°, sec 0° and csc 0° by using the value of sin 0° and cos 0°. Academic year 2024 5.4. For 90° angle When the angle ∠A in △ABC become larger, the length of AB gets smaller. 148  Mathematics C C C C C B A B A B A B A B A The closer the value of ∠A gets to 90° , the closer the length of AB gets to 0. adjacent side AB In the triangle △ABC, the value of cos A = = will be close to 0. Hypotenuse AC opposite side BC Then, the value of sin A = =will be close to 1. Hypotenuse AC This idea helps us to define cos A and sin A when A = 90°. Then we can write: cos 90° =0 and sin 90° =1. Individual task Find out the value of tan 90°, cot 90°, sec 90° and csc 90° by using the value of sin 90° and cos 90°. We can write the value of the trigonometric ratios in the following manner. θ° 0° 30° 45° 60° 90° ratios 1 1 3 sin 0 1 2 2 2 3 1 1 cos 1 0 2 2 2 1 tan 0 1 3 Undefined 3 Academic year 2024 1 cot Undefined 3 1 0 3 149  Trigonometry in Measurement 2 sec 1 2 2 Undefined 3 2 csc Undefined 2 2 1 3 We can solve many problems by using the above table. Problem: In a right angled triangle △ABC, ∠A=30° and AB = 7cm. Find out the length of BC and AC. C Solution: From △ABC, we get: tan A = BC. AB 7 ∴ tan 30° = ⟹ ⟹ BC = BC 1 BC = = 4.04 cm (Approx.) 7 3 7 3 Again, cosA = AB 30° AC A B 7 3 7 14 ∴ cos30° = ⟹ 2 = ⟹ AC = = 8.08 cm (Approx.) AC AC 3 Pair work Make a problem like the above problem and give it to your pair to solve the problem. Check the solution by your teacher. 6. Finding out the Trigonometric Ratios in Reference to Different Angles through Calculator You have found the trigonometric ratios in reference to a particular angle by using the rules of the right angled triangle. It is very difficult to find out the trigonometric ratio in reference to an angle. Luckily, we have scientific calculators or computers that can be used to find out the trigonometric ratios. For your practice purpose, use your calculator and find out the ratios given to you as pair work. Academic year 2024 150  Mathematics Pair work: 1) With the help of teacher, find out the ratio of 40°, 55°, 62°, 83° by using a scientific calculator or computer. 2) Find out the value of the following ratios. sin 32°,cos 36°,tan 52°,cot 61.5°,sec 72.6°,csc 15° The mathematics teacher said to Rina, Abhi and Mita, “Now you have all the knowledge that you need to measure the height of a tree without climbing up. Let’s complete the following task with all the other students.” Group project All the students in the class will be divided into several groups. Each group will pick a stick or tree branch of different lengths according to their convenience. When the sun is in a reclining position, every group will go to a tree. After that they will set the stick or tree branch vertically on the ground and measure the length of the shadow. At the same time, they will measure the length of the shadow of the tree. θ C θ θ A B Now draw a triangle ∆ABC like the above figure where the ratio of AC and AB is equal to the ratio of the length of stick and the length of the shadow of stick. It means: AC Length of the stick Academic year 2024 = Length of the shadow of the stick AB AC Length of the stick Suppose, ∠ABC = θ. Then, tan θ = = Length of the shadow of the stick AB Find out the value of tan θ by using the measurement that you did earlier and write it down in your notebook. 151  Trigonometry in Measurement Suppose, the height of the tree is h. The line from the top point to the ended point of the shadow of the tree makes an acute angle θ. So, h tanθ = Length of the shadow of the tree That is, h = tanθ × Length of the shadow of the tree You have the value of tan θ and you know the length of the shadow of the tree. Now find out the value of h. The groups you measured the shadow of the same tree will get the same or close value of h. If any group's value of h. doesn’t match with others in case of the same shadow, there might be an error in the process. The group should try again. 7. Elevation and Depression Angle Let’s look into the adjacent figure. A person is looking at the t top of the tree. If we imagine a line along with the person’s s igh e ey vertical line eyesight, a geosynchronous line from the eye and a vertical g n alo ion ang line from the bottom to top point of the tree, we will get a le e right angled triangle. In this case, the angle situated in between lin leva t e the line along with the eyesight and the geosynchronous line horizontal line from the eye is an elevation angle. If we know the measurement of this elevation angle and the length of one side of the imagined triangle, we can find out the lengths of other sides of the triangle by using trigonometric ratio. Now look into the adjacent figure. A child is looking down at an object from the balcony of the 1st floor. If we imagine a line along with the child’s eye sight, a vertical line from the child’s eye to the ground, a line from the object to the end point of the vertical line, we will get a right angled triangle. In this case, the angle situated in between the line along horizontal line with the eyesight and the line imagined on the dep ground from the object to the end point of the ress ion ang vertical line is a depression angle. If we know le the measurement of this elevation angle and the length of one side of the imagined triangle, we can find out the lengths of other sides of the triangle by using trigonometric ratio. P Academic year 2024 7.1. Elevation and Depression Angle in Reference to a Particular Point of a Particular Side elevation Suppose, AB is a line parallel to the ground. O is a point on angle AB. Two angles ∠POB and ∠BOQ has been drawn so that A O depression P A, O, B, P and Q can be located on the same vertical plane. angle Q 152  Mathematics Here the point P is located above the line AB. So, the elevation angle P in reference to O is ∠POB. Again, the point Q is located below the line AB. So, the depression angle of Q in reference to O is ∠QOB. 8.The Importance of Measuring Trigonometric Ratio You have already understood the importance of trigonometric ratio. We use trigonometric ratios for different purposes in our daily life. We can measure the distance of an object by measuring the angle. We do not need to go near the object. This discovery was a revolution. So, we will concentrate more on acquiring the knowledge of trigonometry. We can solve many difficult problems with this knowledge. 9. Distance and Height: Real Life Problems and Their Solutions Let’s solve some real life problems by using the knowledge that we have gathered so far. Problem 1: A ladder has been kept leaning against the edge of the roof of a house. The length of the ladder is 12 feet and the ladder has created a 45° angle with the land. What is the height of the roof from the ground? Solution: Suppose, the top point of AC is C and the point C is at the edge of the roof. So, the altitude drawn from the ground to the point C will be the height of the roof. According to the figure, BC = h (suppose), the height of the roof and the base AB have created a 45° angle. Therefore, ∠CAB = 45°. From the right angled triangle △ABC, we get: C sin45° = BC = h AC 12 ⟹ 1 = h 12 2 12 h ⟹ 2 h = 12 450 ⟹ h = 12 = 8.49 feet (Approx.) B 2 A So, the height of the wall is 8.49 feet (Approx.) P Problem 2: Two friends were Academic year 2024 standing 500 m away and they saw a plane flying over them. At a h certain instant, the elevation angle 2nd friend of the plane from the 1st friend was 1st friend 60° 60° and the elevation angle of the 30° plane from the 2nd friend was 30°. B 500 A x Q 153  Trigonometry in Measurement How high was the plane flying? If the plane passed over the 2nd friend 2 seconds later, what was the speed of the plane? Solution: Suppose, the position of the 1st friend is A, the 2nd friend’s position is B and the position of the plane is P. Suppose, the altitude imagined from P to ground is PQ = h and AQ = x. From the right angled triangle ∆APQ, we get: tan 60° = h x ⟹ 3 = h x ⟹x= h ……… (1) 3 From the right angled triangle ∆BPQ, we get: tan 30° = h x + 500 ⟹ 1 = h 3 x + 500 ⟹ x + 500 = h 3 ⟹ h + 500 = h 3 [Substituting the value of x from the equation (1)] 3 ⟹ h + 500 3 = 3h ⟹ 2h = 500 3 ⟹ h = 250 3 So, the plane is flying through an altitude of 250 3 m. We get from the equation (1): x = h = 250 3 = 250 3 3 The plane covers a distance of 500 + 250 = 750 m. in 2 seconds. So, the speed of the plane is 750 ÷ 2 = 375 m/s. Problem 3: A pole broke in such a way that its broken part touched the ground 10 m from the base of the pole. The depression angle is 30°. So, what is the length of the pole? Academic year 2024 154  Mathematics Solution: L Let’s assume, the length of the pole was BL= h m and the pole broke at the height of BC = x m. The broken part touched the ground AB = 10 m from the C base of the pole. So, AC = CL. D 300 x meter 300 Here the depression angle is ∠ACD = 30°. So, A 10 meter B ∠BAC = ∠ACD=30° [Alternate angle] According to the rule, AC = BL - BC = (h - x) m. So, we get from the triangle △ABC, tan 30° = BC = x. ⟹ x = 10 tan 30° = 10 × 1 = 10 m AB 10 3 3 AB 10 Aganin, cos 30° = = AC h - x 3 10 ⟹ = 2 h-x ⟹h - x = 20 3 ⟹h=x+ 20 = 10 + 20 = 30 = 17.32 m 3 3 3 3 So, the length of the pole is 17.32 m (Approx.) Pair work You are standing on the bank of a river and looking at a tree on the other side of the river. Move 50 m in such a way that the intersection of your current position with that tree makes an angle of 30°. What is the distance of your previous position from the tree across the river? 300 50 Group project According to the instruction of your teacher, divide into several groups and measure the height of the highest building of your school by Academic year 2024 using the knowledge of trigonometric ratio. After measuring the height, present the whole process on a poster paper. 155  Trigonometry in Measurement Exercise 3 1. If cosθ = , find out the other ratios of the angle θ. 4 2. If 12 cotθ = 7, what is the value of cosθ and cscθ ? 3. In a right angled triangle ∆ABC, ∠B = 90°, AC = 12 cm, BC = 13 cm and ∠BAC = θ. Find out the value of sinθ ,secθ and tanθ. 4. If θ = 30°, prove that (i) cos2θ = 1 - tan2θ , (ii) tan2θ = 2tanθ. 1 + tan2θ 1 - tan2θ 5. If the elevation angle of the top point of the tree is 60° at a point on the ground 15 m from its base, find the height of the tree. 6. A ladder with a length of 6 m creates an angle of 60° with the base. What is the height of the roof? 7. The elevation angle of the top point of a tower is 60° from a point on the ground. The elevation angle of the tower will be 45° if is 20 m behind from the previous place. What is the height of the tower? 8. A man standing on the bank of the river observes that on the other side of the river, the elevation angle of the top point of a 100 m tower is 45°. The man starts journey by boat to the tower. But due to water current, the boat reaches the river side with 10 m. distance from the tower. Determine the distance from starting point of the man to the ending point. 9. A man standing on a tower at the bank of the sea observed 300 that a ship was coming towards the port. At that time, the depression angle of the ship was 30°. A few moments 450 later, the depression angle became 45°. If the height of the tower was 50 m, how far did the ship cross during that time? 10. The elevation angle is 45° at a distance of 10 m from your office building. If the elevation angle is θ at a distance of 20 m, what is the value of Academic year 2024 sinθ and cosθ ? 156  Trigonometry for Angular Distance You can learn from this experience- Concepts, importance, and techniques of measuring trigonometric angles. Difference between geometric angles and trigonometric angles. Standard positions of trigonometric angles and measures of angles relative to these positions. Concept and measurement of coterminal angle, quadrant angle and quadrantal angle. Trigonometric ratios in standard position. Interrelation of trigonometric ratios of various angles. Interrelation of trigonometry and coordinate geometry. Radian measurement of angles and the relationship between degrees and radians. α θ- θ α Academic year 2024 157  Trigonometry for Angular Distance Trigonometry for Angular Distance In previous class, we learned to find the linear distance between two objects. Interestingly, apart from linear D distance there is another type of distance called angular C distance. For example, the adjacent image shows some P players on a cricket field. Distances PA and PB directly θ angular distance from batsman P to fielders A and B are called linear distances. But if we want to measure the distance A B between fielders A and B with respect to the batsman P considered to be at the centre, then that distance is called angular distance. The difference in position of PA to PB with respect to the point P in the adjacent figure is called the angular distance, which is denoted by θ. Whether we know it or not, we use angular distance in various activities every day. For example, in the game of cricket, a batsman hits the ball keeping in mind the angular distance of the fielders and then runs. Again, we always use different angular distances when working with our hands. The hands of 12 our wall clocks are 11 1 constantly traversing 2 θ 10 angular distances. 9 3 θ When we measure 8 4 the distance from one 7 6 5 star to another in the night sky, that too is (a) (b) (c) essentially an angular distance. You will find numerous examples where angular distance is used. By measuring angles, we can know the position of many distant objects and determine their size, rotation properties etc. In this chapter we will try to solve such problems by measuring trigonometric angles. Work in pairs Think in pairs and write three examples in the table below where angular distance is used. Academic year 2024 To measure angular distance, we use the knowledge of trigonometry. Below this topic is discussed continuously in detail. 158  Mathematics 1. Measurement of trigonometric angle An angle is formed when a ray rotates from an initial position to a P terminal position with respect to its starting point. Let OA be an initial ray which turns about the point O and reaches the position OP. So, an angle ∠AOP is formed. Let ∠AOP = θ. Here O is the θ vertex, OA is the initial line, and OP is the terminal line. The O A amount by which the ray OP rotates with respect to the point O, keeping the ray OA fixed, is called the angular distance. That is, θ is the angular distance. Angular distance plays a very important role in measurement. Angular distance is usually measured in degrees. Degrees are indicated by the ° symbol. By keeping the ray OA fixed and the ray OP rotated about the point O, angles of different measure are formed. Initially when the ray OP is parallel to the ray OA the angle will be 0°. If the ray OP turns once with respect to the point O and again coincides with the ray OA, then the angle will be 360°. That is, 1° is the angular distance we get dividing one full rotation by 360. 1’ (1 minute) is the angular distance we get dividing 1° by 60. That is 1 1’ = × 1°. Again dividing 1’ by 60 gives the angular distance as 1’’ (1 second). 60 1 1 That is 1’’ = × 1’. Hence, 1’’ = × 1°. 60 3600 Have you noticed that the hands of your home wall clock or table clock or your school’s wall clock are always rotating? The hands are repeatedly moving over 12 o’clock. If we imagine a ray starting at 12 o’clock from the 11 12 1 center of the clock, what angular distance can you say these hands are 10 2 traversing? When the hands turn once over 12 o’clock, they traverse 9 3 an angular distance of 360°. Completing one full rotation once more 8 4 will cover a distance of 360° + 360° = 720°. Thus, completing each 7 6 5 full rotation will add 360° to the angular distance. So, we can see, angle measure can be more than 360° in terms of angular distance. That is, trigonometric angles can be greater than 360°. Work in pairs If the angular distance of two places A and B with respect to the center of the earth in the adjacent figure is 15°, express the angular distance of the two places in seconds. Write the answer in the box below. Academic year 2024 159  Trigonometry for Angular Distance 1.1 Positive and Negative Angles P Just as number system has positive and negative numbers, angular distances have positive and negative angles. If the terminal ray OP rotates clockwise with respect to the initial ray OA, then the angle θ θ is negative, and if the terminal ray OP rotates counterclockwise O A with respect to the initial ray OA, then the angle θ is positive. Arrows are used to indicate direction of positive and negative O -θ A angles. Also, negative angles are indicated by a ‘-’ sign before the angle like a numerical expression. Positive and negative angles are P indicated in the adjacent figure. If the terminal ray OP rotates more than 360° counterclockwise or clockwise, then the angle is greater than 360° and we can represent it as in the adjacent figure. Angles A O greater than 360° are observed in various P -(360 + θ) 0 objects in nature; For example, spiral galaxy, 3600+θ vine arms etc. Can you name more examples O that form angles greater than 360°? Think A P and write in the box below. Galaxy creeping tree Work in pairs Draw 200° and -230° angles in the blank below using a geometric ruler and protractor. Academic year 2024 160  Mathematics 2. Geometric Angles and Trigonometric Angles From geometry we know that when two different rays meet at a point, B an angle is formed at that point. In the figure ∠AOB is a geometric angle. Here the rays OA and OB meet at two points O. Hence angle ∠AOB is formed at point O. ∠AOB is always considered positive in measuring angle. So, the discussion of angles in geometry is limited O A to 0° to 360° or four right angles. On the other hand, in case of trigonometric angles, keeping the P ray OA fixed, rotate OP about the point O to form angles of different measures. Trigonometric angles can be both positive and negative and can be greater than 360°. In the adjacent figure θ = ∠AOP is a trigonometric angle. This is a positive angle, θ O because the line OP makes an angle θ with the origin line OA A in a counterclockwise direction. Individual task Draw a geometric angle of 120° in the blank space below. Draw a positive and a negative trigonometric angle of the same measure. 3. Standard position of trigonometric angle We can represent any trigonometric angle in two-dimensional Y coordinates or the xy-plane. If a trigonometric angle θ is placed in the xy-plane in such a way that the vertex of the angle is at P O and the initial ray lies on the positive side of the x-axis, then Academic year 2024 θ x this position is called the standard position of the angle. O A 161  Trigonometry for Angular Distance Work in pairs Which of the following angles are in st