Topic outline

  • Changes in schools


    This text book is part of the reform of the school curriculum in Rwanda; that is changes in what is taught in schools and how it is taught. It is hoped that this will make what you learn in school useful to you when you leave school.

    In the past, the main thing in schooling has been to learn knowledge – that is facts and ideas about each subject. Now, the main idea is that you should be able to use the knowledge you learn by developing skills or competencies. These skills or competencies include the ability to think for yourself, to be able to communicate with others and explain what you have learnt, and to be creative, that is, developing your own ideas, not just following those of the teacher and the text book. You should also be able to find out information and ideas for yourself rather than just relying on what the teacher or text book tells you.

    Activity-based learning

    This book has a variety of activities for you to do as well as information for you to read. These activities present you with material or things to do which will help you to learn things and find out things for yourself. You already have a lot of knowledge and ideas based on the experiences you have had and your life within your own community. Some of the activities, therefore, ask you to think about the knowledge and ideas you already have.

    In using this book, therefore, it is essential that you do all the activities. You will not learn properly unless you do these activities. They are the most important part of the book.In some ways, this makes learning more of a challenge. It is more difficult to think for yourself than to copy what the teacher tells you. But if you take up this challenge, you will become a better person and become more successful in your life.


    Group work

    You can learn a lot from other people in your class. If you have a problem, it can often be solved by discussing it with others. Many of the activities in this book, therefore, involve discussion or other activities in groups or pairs. Your teacher will help to organise these groups and may arrange the classroom so that you are always sitting in groups facing each other. You cannot discuss properly unless you are facing each other.

    Research

    One of the objectives of the competence based curriculum is to help you find things out for yourself. Some activities, therefore, ask you to do research using books in the library, the internet if your school has this, or other sources such as newspapers and magazines. This means that you will develop the skills of learning for yourself when you leave school. Your teacher will help you if your school does not have a good library or internet.

    Icons

    To guide you, each activity in the book is marked by a symbol or icon to show you what kind of activity it is. The icons are as follows:







    • Introduction

      As we saw it in senior 4, trigonometry studies relationship involving lengths and angles of a triangle. The techniques in trigonometry are used for finding relevance in navigation particularly satellite systems and astronomy, naval and aviation industries, oceanography, land surveying and in cartography (creation of maps). Now, those are the scientific applications of the concepts in trigonometry, but most of the mathematics we study would seem (on the surface) to have little real-life application. Trigonometry is really relevant in our day to day activities.


      1.1. Trigonometric formulae






      1.1.1. Addition and subtraction formulae

            From activity 1.1:The addition and subtraction formulae are:

         

      Addition and subtraction formulae are useful when finding trigonometric number of some angles.







      1.1.2. Double angle formulae



      From activity 1.2, we have2

      cos2 x+ sin2x = 1

      This relation is called the fundamental relation of trigonometry.

      From this relation, we can write



      The above four formulae are known as double angle formulae


      Example 1.4

      From double angle formulae and fundamental relation of trigonometry, prove that


      Solution



      1.1.3. Half angle formulae









      1.1.4. Transformation of product into sum






      1.1.5. Transformation of sum into product


      From activity 1.5, the formulae for transforming sum in product are:



      The solutions of a trigonometric equation for which are called principle solutions while the expression (involving integer k) of solution containing all values of the unknown angle is called the general solution of the trigonometric equation. When the interval of solution is not given, you are required to find the general solution.When solving trigonometric equations, the following identities are helpful:






      1.2.2. Solving the equation of the form







      1.3.2. Refraction of light


      In optics, light changes speed as it moves from one medium to another (for example, from air into the glass of the prism). This speed change causes the light to be refracted and to enter the new medium at a different angle (Huygens principle).
      A prism is a transparent optical element with flat, polished surfaces that refract light.
                                 

      The degree of bending of the light’s path depends on the angle that the incident beam of light makes with the surface, and on the ratio between the refractive indices of the two media (Snell’s law).If the light is traveling from a rarer region (lower n) to a denser region (higher n), it will bend towards the normal but if it is traveling from a denser region (higher n) to a rarer region (lower n), it will bend away from the normal.











      • Introduction
        Consider a scientist doing an experiment; he/she is collecting data every day. Put u1 to be the data collected the first day, u2 be the data collected the second day, u3 be the data collected the third day, and so on..., and un be the data collected after n days. Clearly, we are generating a set of numbers with a very special characteristic. There is an order in the numbers; that is, we actually have the first number, the second number and so on... A sequence is an ordered list of numbers.










        2.1.2. General term of an arithmetic sequence





        2.1.3. Arithmetic means


        2.1.4. Arithmetic series





        2.1.5. Harmonic sequences


        Harmonic sequence is a sequence of numbers in which the reciprocals of the terms are in arithmetic sequence.


        Remark


        To find the term of harmonic sequence, convert the sequence into arithmetic sequence then do the calculations using the arithmetic formulae. Then take the reciprocal of the answer in arithmetic sequence to get the correct term in harmonic sequence.








        2.2. Geometric sequences

        2.2.1. Definition



        Sequences of numbers that follow a pattern of multiplying a fixed number from one term to the next are called geometric sequences or geometric progression.






        2.2.2. General term of a geometric sequence







        2.2.3. Geometric means




        2.2.4. Geometric series




        2.2.5. Infinity geometric series






        2.3. Applications

        There are many applications of sequences. Sequences are useful in our daily lives as well as in higher mathematics. For example; the interest portion of monthly payments made to pay off an automobile or home loan, and the list of maximum daily temperatures in one area for a month are sequences. Sequences are used in calculating interest, population growth, half-life and decay in radioactivity ...



        A child building a tower with blocks uses 15 for the bottom row. Each row has 2 fewer blocks than the previous row. Suppose that there are 8 rows in the tower.
        a) How many blocks are used for the top row?
        b) What is the total number of blocks in the tower?


        An insect population is growing in such a way that each new generation is 1.5 times as large as the previous generation. Suppose there are 100 insects in the first generation,
        a) How many will there be in the fifth generation?
        b) What will be the total number of insects in the five generations?




        Another important application of sequences is their use in compound interest and simple interest.











        • Introduction

          Many quantities increase or decrease with time in proportion to the amount of the quantity present. Some examples are human population, bacteria in culture, drug concentration in the bloodstream, radioactivity and the values of certain kinds of investments. For example, the director of education is more concerned with the rate at which the school population is increasing or decreasing than with what the population is now, because he/she has to plan for the future and ensure that there are enough (and not too many) school places available to meet demand each year.

          We will find how exponential or logarithmic equations are very useful in studies of the growth and decay of such quantities. Hence, exponential and logarithmic equations are really relevant in our day to day activities.


          3.1. Exponential and logarithmic functions



          Remember that only a one to one function is invertible. To find the inverse of the function y = ax, where a is a positive real number different from 1, we make x the subject of the formula by introducing a new function called logarithm and write x=loga which is read “x is logarithm of y in base a”.

          The graph of a logarithmic function is found by reflecting the graph of the corresponding exponential function in the line y=x. Thus, from activity 1, the curve of y= 2x and g(x) are inverse to each other. Thus,  g(x) = log2x.

          Since g(x)  = log2x  is the inverse of y=2x, the curve of g(x)  = log2x is the image of the curve of y=2x with respect to the first bisector, y=x. Then, the coordinates of the points for y=2x are reversed to obtain the coordinates of the points for g(x)  = log2x.

          Note that the words power, index, exponent and logarithm are synonymous; they are four different words to describe exactly the same thing.







          3.2. Exponential and logarithmic equations




          You should find an "ln" button on your calculator which will evaluate logarithms to base e and "log" button to evaluate logarithms to base 10.

          Basic rules for logarithms







          3.3. Applications


          Exponential growth




          Exponential decay


          A population whose rate of decrease is proportional to the size of the population at any time obeys a law of the forms  P =Ae -kt .The negative sign on exponent indicates that the population is decreasing. This is known as exponential decay.

          If a quantity has an exponential growth model, then the time required for it to double in size is called the doubling time. Similarly, if a quantity has an exponential decay model, then the time required for it to reduce in value by half is called the halving time. For radioactive elements, halving time is called half-life.


          Doubling time and half time









          A man deposits 800,000 Frw into his savings account on which interest is 15% per annum. If he makes no withdrawals, after how many years will his balance exceed 8 million Frw?













          • Introduction


            The techniques in trigonometry are used for finding relevance in navigation particularly satellite systems and astronomy, naval and aviation industries, oceanography, land surveying, and in cartography (creation of maps). Now, those are the scientific applications of the concepts in trigonometry, but most of the mathematics we study would seem (on the surface) to have little real-life application. Trigonometry is really relevant in our day to day activities. The following pictures show us the areas where this science finds use in our daily activities. In this unit, we will see how we can use this to resolve problems we might encounter.


            4.1. Generalities on trigonometric functions and their inverses

            4.1.1. Domain and range of six trigonometric functions



            Cosine and Sine

            sinx  and cosx are functions which are defined for all positive and negative values of x even for x =0 Thus, the domain of sinx and cosx is the set of real numbers. The range of sinx and cosx is [ -1,1].




            Tangent and cotangent





            Secant and cosecant





            4.1.2. Domain and range of inverses of trigonometric functions





            Remark


            The inverses of the trigonometric functions are not functions, they are relations. The reason they are not functions is that for a given value of x , there is an infinite number of angles at which the trigonometric functions take on the value of x . Thus, the range of the inverses of the trigonometric functions must be restricted to make them functions. Without these restricted ranges, they are known as the inverse trigonometric relations.


            From these relations, we obtain the following important result:




            Inverse tangent



            Inverse secant




            Inverse cotangent and inverse cosecant




            4.1.3. Parity of trigonometric functions




            Even function



            Odd function




            4.1.4. Period of trigonometric functions




            We call P a period of the function. The smallest positive period is called the fundamental period (also primitive period, basic period, or prime period) of f. A function with period P repeats on intervals of length P, and these intervals are referred to as periods.Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function is periodic with period P if its graph is invariant under translation in the x-direction by a distance of P.




            The most important examples of periodic functions are the trigonometric functions.
            Any function which is not periodic is called aperiodic.





            We have seen that sine and cosine are both periodic and have the same period. When we add them up, subtract them, multiply them, etc we get functions that are also periodic.

            To see this, let us assume that f(x+kP) =  f(x) is true for all real x, k integers.

            Simply multiplying each side by some constant does not change the equation and adding or subtracting some constant to each side does not change periodicity.If we have two functions f(x) and f(g) with the same period, P say, we can throw them together any way we want.

            Let h(x) = f(x)+g(x), at any value a of x:
            h(a)= f(a)+g(a) = c 
            h (a + kP) = f(a+kP)+g(a+kP)= c

            Thus,h(a) =  h(a+kP)
            This is different for functions that don’t have the same fundamental period.

            Let us say that we have two periodic functions f(x) and g(x) with period P and Q respectively:

            f(x+kP) = f(x) is true for all real x, k integers.
            g(x + kQ) = g (x) is true for all real x, k integers.

            Now, we cannot construct that nice h(x) as we did before because we have different periods.


            Consider the following case:

            If a function repeats every 2 units, then it will also repeat every 6 units. So if we have one function with fundamental period 2, and another function with fundamental period 8, we have got no problem because 8 is a multiple of 2, and both functions will cycle every 8 units.

            So, if we can patch up the periods to be the same,we know that if we combine them, we will get a function with the patched up period.

            What we have to do is to find the Lowest Common Multiple (LCM) of two periods.

            What about if one function has period 4 and another has period 5? We can see that in 20 units, both will cycle, so they are fine.








            4.2. Limits of trigonometric functions and their inverses


            4.2.1. Limits of trigonometric functions



















            4.3. Differentiation of trigonometric functions and their inverses

            4.3.1. Derivative of sine and cosine




            4.3.2. Derivative of tangent and cotangent






            4.3.3. Derivative of secant and cosecant



            4.3.4. Derivative of inverse sine and inverse cosine





            4.3.5. Derivative of inverse tangent and inverse cotangent






            4.3.6. Derivative of inverse secant and inverse cosecant










            4.4. Applications

            Simple harmonic motion













            • Introduction


              A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context.

              To put it really simple, vectors are basically all about directions and magnitudes. These are critical in basically all situations.

              In physics, vectors are often used to describe forces, and forces add as vectors do.Classical Mechanics: Block sliding down a ramp. You need to calculate the force of gravity (a vector down), the normal force (a vector perpendicular to the ramp), and a friction force (a vector opposite the direction of motion).


              5.1. Scalar product of two vectors



              Properties of scalar product




              5.2. Magnitude (or norm or length) of a vector



              Properties of magnitude of a vector



              Remark


              A vector is said to be a normal vector or simply the normal to a surface if it is perpendicular to that surface. Often, the normal unit vector is desired, which is sometimes known as the unit normal.

              The terms normal vector and normalised vector should not be confused, especially since unit norm vectors might be called normalised normal vectors without redundancy.




              5.3. Angle between two vectors









              5.4. Vector product





              Properties of vector product





              Mixed product





              Mixed product properties




              5.5. Applications


              5.5.1. Work done as scalar product







              5.5.2. Area of a parallelogram




              Notice: Area of a triangle







              5.5.3. Volume of a parallelepiped














              • Introduction


                A matrix is a rectangular arrangement of numbers, expressions, symbols which are arranged in rows and columns.

                Matrices play a vital role in the projection of a three dimensional image into a two dimensional image. Matrices and their inverse are used by programmers for coding or encrypting a message. Matrices are applied in the study of electrical circuits, quantum mechanics and optics. A message is made as a sequence of numbers in a binary format for communication and it follows code theory for solving. Hence, with the help of matrices, those equations are solved. Matrices are used for taking seismic surveys.

                6.1. Square matrices of order three


                6.1.1. Definitions


                6.1.2. Types of matrices
                Upper triangular matrix

                In an upper triangular matrix, the elements located below the leading diagonal are zeros.



                Lower triangular matrix

                In a lower triangular matrix, the elements above the leading diagonal are zeros.


                Diagonal matrix

                In a diagonal matrix, all the elements above and below the leading diagonal are zeros.



                Scalar matrix
                A scalar matrix is a diagonal matrix in which the leading diagonal elements are equal.



                Identity matrix or unity matrix

                An identity matrix (denoted by I) is a diagonal matrix in which the leading diagonal elements are equal to 1.


                Equality of matrices

                Two matrices are equal if the elements of the two matrices that occupy the same position are equal.






                6.1.3. Operations on matrices



                Adding matrices

                When adding two matrices of the same dimension, the resultant matrix’s elements are obtained by adding the elements that occupy the same position.




                Properties

                1. Closure
                    The sum of two matrices of order three is another matrix of order three.


                2. Associative 

                      A + (B + C) = (A + B) + C

                3. Additive identity 

                   A + 0 = A , where 0 is the zero-matrix of the same dimension.

                4. Additive inverse  A + (- A) =0 

                The opposite matrix of A is -A.

                5. Commutative

                A + B =  B + A


                6.1.4. Scalar multiplication




                Transpose matrix






                Properties of transpose of matrices




                Multiplying matrices








                Properties of matrices multiplication







                Properties of trace of matrix




                6.2. Determinants of order three


                6.2.1. Determinant



                The terms with a positive sign are formed by the elements of the principal diagonal and those of the parallel diagonals with its corresponding opposite vertex.

                The terms with a negative sign are formed by the elements of the secondary diagonal and those of the parallel diagonals with its corresponding opposite vertex.








                Properties of a determinant











                6.2.2. Matrix inverse






                Properties of the Inverse Matrix






                6.3. Application

                6.3.1. System of 3 linear equations




                Alternative method: Cramer’s rule












                Alternative method











                • Introduction

                  Descriptive statistics is a set of brief descriptive coefficients that summarises a given data set, which can either be a representation of the entire population or sample. Data may be qualitative such as sex, color and so on or quantitative represented by numerical quantity such as height, mass, time and so on.

                  The measures used to describe the data are measures of central tendency and measures of variability or dispersion. Until now, we know how to determine the measures of central tendency in one variable. In this unit, we will use those measures in two quantitative variables known as double series.

                  In statistics, double series includes technique of analyzing data in two variables, when we focus on the relationship between a dependent variable-y and an independent variable-x. The linear regression method will be used in this unit. The estimation target is a function of the independent variable called the regression function which will be a function of a straight line.


                  Descriptive statistics provide useful summary of security returns when performing empirical and analytical analysis, as they provide historical account of return behavior. Although past information is useful in any analysis, one should always consider the expectations of future events.

                  Some variables are discrete, others are continuous. If the variable can take only certain values, for example the number of apples on a tree, then the variable is discrete. If however, the variable can take any decimal value (in some range), for example the heights of the children in a school, then the variable are continuous. In this unit, we will consider discrete variables.


                  7.1 .Covariance





                  In case of two variables, say x and y, there is another important result called covariance of x and y, denoted Cov(x,y), which is a measure of how these two variables change together.

                  The covariance of variables x and y is a measure of how these two variables change together. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the smaller values, i.e. the variables tend to show similar behavior, the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the smaller values of the other, i.e. the variables tend to show opposite behavior, the covariance is negative. If covariance is zero the variables are said to be uncorrelated, meaning that there is no linear relationship between them.

                  Therefore, the sign of covariance shows the tendency in the linear relationship between the variables. The magnitude of covariance is not easy to interpret.Covariance of variables x and y, where the summation of  frequencies are equal for both variables, is defined to be












                  7.2. Regression lines

                  We use the regression line to predict a value of y for any given value of x and vice versa. The “best” line would make the best predictions: the observed y-values should stray as little as possible from the line. This straight line is the regression line from which we can adjust its algebraic expressions and it is written as y=  ax  + b.




                  Shortcut method to find regression lines






                                            

                  7.3. Coefficient of correlation


                  Pearson’s coefficient of correlation or product moment coefficient of correlation











                  Method of ranking
                  Ranking can be done in ascending order or descending order.








                  7.4. Applications


                   
                      












                  • Introduction


                    Probability is a common sense for scholars and people in modern days. It is the chance that something will happen-how likely it is that some event will happen. No engineer or scientist can conduct research and development works without knowing the probability theory. Some academic fields based on the probability theory are statistics, communication theory, computer performance evaluation, signal and image processing, game theory.... Some applications of the probability theory are character recognition, speech recognition, opinion survey, missile control and seismic analysis.

                    The theory of game of chance formed the foundations of probability theory, contained at the same time the principle for combinations of elements of a finite set, and thus establishes the traditional connection between combinatorial analysis and probability theory.

                    8.1. Tree diagram



                    A tree diagram is a means which can be used to find the number of possible outcomes of experiments where each experiment occurs in a finite number of ways.

                    The outcome is written at the end of the branch and the fraction on the branch gives the probability of the outcome occurring.






                    8.2. Independent events







                    8.3. Conditional probability


                         










                    8.4. Bayes theorem and its applications

















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