Chapter 1 Quadratic Equations in One Unknown (I)


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1 Tin Ka Ping Secondary School F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) Real Number System A Integers B nal Numbers C Irrational Numbers D Real Numbers 1. Solving Quadratic by the Factor Method A Roots of Quadratic B Factor Method 1. Solving Quadratic by the Quadratic Formula A Method of Taking Square Roots B Quadratic Formula 1. Solving Quadratic by the Graphical Method 1.5 Problems Leading to Quadratic Understand the hierarchy of the real number system. Discuss the conversion between recurring decimals and fractions. Understand the general form of a quadratic equation in one unknown and its roots. Solve the quadratic equation ax + bx + c = 0 using the factor method. Solve the quadratic equation in the form (x + m) = n by taking square roots. Understand the quadratic formula. Solve quadratic equations using the quadratic formula. Solve the equation ax + bx + c = 0 by plotting the graph of y = ax + bx + c and reading the xintercepts. Understand the approximate nature of the roots obtained by using the graphical method. Solve reallife problems that involve quadratic equations. Teachers should introduce different kinds of numbers in the real number system. Real Number System Illustrate the conversion of recurring decimals into fractions with Teachers should introduce the general form of quadratic equations ax + bx + c = 0. Quadratic in One Teachers should introduce the idea of the roots of Unknown a quadratic equation. Illustrate how to solve quadratic equation by the Solving Quadratic factor method with by the Factor Method Demonstrate some quadratic equations that can be Drilling Program: solved using identities. Solving Quadratic Teachers may encourage students to read the by the Factor Method Investigation Corner on p Demonstrate some quadratic equations that can be solved by taking square roots on both sides with Solving Quadratic by the Quadratic Formula Teachers may demonstrate how to derive the quadratic formula. Illustrate how to use the quadratic formula to solve quadratic equations with Teachers may use Maths Dialogue on p.1.6 to let students explore how to solve quadratic equations involving fractions. Teachers may use examples to illustrate that the quadratic equation ax + bx + c = 0 can be solved graphically by reading the xintercept(s) of the graph of y = ax + bx + c. Teachers should stress that the roots found by the graphical method are approximate values only. Teachers may use IT Activity 1.1 on p to let students explore how to solve a quadratic equation graphically. Teachers should summarize the relationship between the number of xintercepts of the graph of y = ax + bx + c and the nature of roots of ax + bx + c = 0. Teachers may use Maths Dialogue on p.1. to let students learn how to choose an appropriate range of x when solving a quadratic equation graphically. Illustrate how to solve some number problems, geometric problems and reallife problems by formulating quadratic equations with Solving Quadratic by the Graphical Method IT Activity 1.1: Solving a quadratic equation graphically Problems Leading to Quadratic 1
2 Chapter Quadratic in One Unknown (II) 5.1 Nature of Roots of a Quadratic Equation A The Discriminant of a Quadratic Equation B Graph of a Quadratic Equation Understand the relation between the discriminant of a quadratic equation and the nature of its roots. Understand the relation between the discriminant of ax + bx + c = 0 and the number of xintercepts of the graph of y = ax + bx + c. Teachers should explain the relations among the discriminant of a quadratic equation, the nature of its roots and the number of xintercepts of its graph. Nature of Roots of a Quadratic Equation Graph of a quadratic equation and its discriminant 5. Forming a Quadratic Equation with Given Roots NF. Relations between Roots and Coefficients. Complex Number System A Complex Numbers NF B Operations of Complex Numbers Form a quadratic equation with given roots. (The given roots are confined to real numbers.) Form a quadratic equation using the sum and product of its roots. NF Understand the relations between the roots and coefficients of a quadratic equation. Learn how to form a quadratic equation whose roots are related to the roots of another quadratic equation. Introduce the historical development of complex numbers. Introduce the definition of i. Introduce the system of complex numbers a + bi. Define and perform addition, subtraction, multiplication and division of complex numbers. Illustrate with examples how to form quadratic equations with given roots by reversing the process of solving quadratic equations by the factor method. Illustrate with examples how to form quadratic equations using the sum and product of its roots. Teachers should introduce the relation between the sum / product of roots of a quadratic equation and its coefficients. Illustrate how to find the values of expressions which are in terms of the roots of a quadratic equation. Teachers may use the examples to illustrate how to form a quadratic equation whose roots are related to the roots of another quadratic equation. Teachers may use To Learn More on p.. to teach how to obtain useful information of the graph of y = ax + bx + c, using the sum and the product of roots of a quadratic equation ax + bx + c = 0. Teachers may encourage students to read the Investigation Corner on p..5. Teachers may use To Learn More on p..0 to introduce the powers of i following the pattern i, 1, i, 1, i, 1, i, 1, Make sure students can identify real part and imaginary part in a complex number. Further discuss the hierarchy of the number system when complex numbers are introduced. Illustrate the rules of addition, subtraction, multiplication and division of complex numbers with Forming a Quadratic Equation with Given Roots Sum and Product of Roots Introduction to Complex Numbers Drilling Program: Operations of Complex Numbers Operations of Complex Numbers
3 Chapter and Graphs Introduction to A Concepts of B Different Representations of. Notation of a Function A Basic Notation B Values of a Function. Some Common and their Graphs A Constant B Linear C Quadratic NF. Optimum Values of Quadratic A Quadratic in the Form y = a(x h) + k NF B Finding the Optimum Values of Quadratic by the Algebraic Method NF C Practical Problems in Finding Optimum Values of Quadratic Recognize the intuitive concept of functions. Recognize the concepts of independent and dependent variables. Recognize the concepts of domain, codomain and range of a function. Recognize the algebraic, tabular and graphical representations of a function. Recognize the notation of a function. Find the values of functions. Understand the features of the graph of a constant function. Understand the features of the graph of a linear function. Understand the features of the graph of a quadratic function. Understand the optimum value of the quadratic function y = a(x h) + k and the features of its graph. Understand the method of completing the square. NF Find the optimum value of the quadratic function y = ax + bx + c by the algebraic method. NF Solve practical problems relating to the optimum value of a quadratic function. NF Teachers should introduce the concept of functions. Concept of a Function Teachers should ensure that students can Teaching Tool: distinguish independent variables from dependent Determination of variables, and can distinguish domain from range and codomain. Teachers should use examples to illustrate the algebraic, tabular and graphical representations of a function. Make sure students can determine whether y is a function of x, given the relationship between x and y. Teachers should introduce the notation of a function. Illustrate how to find the values of functions with Illustrate the operations of functions with Teachers should introduce the features of the graphs of a constant function and a linear function. Teachers should introduce the concepts of xintercept and yintercept. Teachers may use Activity.1 (or IT Activity.1 on p..69) to let students explore the features of the graph of a quadratic function in the form y = ax + bx + c. Illustrate how to find the features of quadratic graphs with Teachers may use Activity. (or IT Activity. on p..70) to let students investigate the features of the graph of a quadratic function in the form y = a(x h) + k. Teachers can introduce the idea of completing the square. Teachers should demonstrate the conversion of function y = ax + bx + c to the form y = a(x h) + k. Illustrate how to solve problems relating to the optimum value of a quadratic function with Teachers may encourage students to attempt the Investigation Corner on p..68. Notation of a Function Some Common and their Graphs Constant & Linear Some Common and their Graphs Quadratic Animation: Features of Quadratic Graphs Teaching Tool: Graph of a Quadratic Function IT Activity.1: Graphs of quadratic functions in the form y = ax + bx + c Comparing graphs of linear functions and quadratic functions Quadratic in the Form y = a(x h) + k Finding the Optimum Values of Quadratic by the Algebraic Method IT Activity.: Graphs of quadratic functions in the form y = a(x h) + k
4 Chapter of Straight Lines of Straight Lines A Pointslope Form B Twopoint Form C Slopeintercept Form D of Special Straight Lines E Further Problems on of Straight Lines. General Form of Equation of a Straight Line Understand and apply the pointslope form to find equations of straight lines. Understand and apply the twopoint form to find equations of straight lines. Understand and apply the slopeintercept form to find equations of straight lines. Find the equations of oblique lines passing through the origin, horizontal lines and vertical lines. Learn the techniques in solving problems involving equations of straight lines. Understand the general form of equation of a straight line. Explore the properties of a straight line from its equation in general form.. Possible Intersection of Solve problems involving intersection of Straight Lines straight lines on the coordinate plane. A Finding the Coordinates of Determine the number of intersections of two the Intersection of straight lines from their equations. Nonparallel lines B Determining the Number of Intersections of Two Straight Lines Teachers may use Activity.1 to demonstrate how to find the equation of a straight line from its slope and a point on it. Introduce the pointslope form of a straight line. Demonstrate with examples how to use the pointslope form to find the equations of straight lines. Introduce the twopoint form of a straight line. Demonstrate with examples how to use the twopoint form to find the equations of straight lines. Introduce the slopeintercept form of a straight line. Demonstrate with examples how to use the slopeintercept form to find the equations of straight lines. Illustrate how to find the equations of special straight lines. Illustrate how to solve problems involving equations of straight lines. Teachers may use To Learn More on p..0 to introduce the intercept form of a straight line. Teachers should introduce the general form of the equation of a straight line. Show students how to find the slope, yintercept and xintercept from the general form of the equation of a straight line. Teachers may use Maths Dialogue on p..1 to let students explore how to find the ranges of unknown values from the graph of a straight line. Revise the techniques in solving simultaneous linear equations in two unknowns. Introduce the geometric meaning of the number of intersections of two straight lines on the coordinate plane. Pointslope Form of of Straight Lines Twopoint Form of of Straight Lines Slopeintercept Form of of Straight Lines of Special Straight Lines Drilling Program: of Straight Lines Animation: of Straight Lines IT Activity.1: Explore the meanings of m and c in the graph of y = mx + c General Form of Equation of a Straight Line Properties of equations of straight lines in the general form Possible Intersection of Straight Lines Explore the number of intersections between two straight lines
5 Chapter 5 More about Polynomials Revision on Polynomials A Monomials and Polynomials B Addition, Subtraction and Multiplication of Polynomials Review the concepts of monomials and polynomials, and the terminologies involved. Review the basic operations (addition, subtraction and multiplication) of polynomials. Make sure students have the basic knowledge of polynomials before learning the next section. Revision on Polynomials Division of Polynomials A Method of Long Division B Division Algorithm Understand and manipulate long division of polynomials up to simple quadratic divisor. Understand and apply division algorithm. Teachers should demonstrate the method of long division with Make sure students understand the idea of division algorithm. Teachers may use Maths Dialogue on p. 5.1 to let students know that the coefficients of the terms in the quotient and the remainder need not be integers. Division of Polynomials 5. Remainder Theorem Understand and apply the remainder theorem. Remind students that function notations can be used to denote polynomials. Remainder Theorem Teachers may use Activity 5.1 on p to Drilling Program: let students explore the relationship between Remainder Theorem the remainder of f(x) (x a) and the value of f(a). Teachers may use the division algorithm to introduce the remainder theorem. Remind students that they must define f (x) before applying the remainder theorem. Illustrate the use of remainder theorem with 5. Factor Theorem A Factor Theorem B Factorizing Polynomials by Factor Theorem NF 5.5 H.C.F. and L.C.M. of Polynomials NF 5.6 nal and their Manipulations A Multiplication and Division of nal B Addition and Subtraction of nal C Further Manipulations of nal Understand and apply the factor theorem. Understand and apply the converse of the factor theorem. Use the factor theorem to factorize polynomials up to degree. Understand the concepts of the highest common factor (H.C.F.) and the lowest common multiple (L.C.M.) of polynomials. NF Learn how to find the H.C.F. and L.C.M. of polynomials. NF Learn the meaning of rational functions. Learn how to perform addition, subtraction, multiplication and division of rational functions. NF Teachers may use the remainder theorem to introduce the idea of the factor theorem. Factor Theorem Illustrate the use of factor theorem and its converse with Factorizing Polynomials Teachers may use Activity 5. on p. 5.0 to by Factor Theorem let students explore the method to find the linear factors of a polynomial. Illustrate how the factor theorem can be applied to factorize a cubic polynomial. Teachers may encourage students to attempt the Maths Dialogue on p. 5.. Teachers may encourage students to attempt the Investigation Corner on p. 5.6 to explore a fast method to determine whether a 5digit number is divisible by 11. Quick review with students on the methods to find the H.C.F. and L.C.M. of two numbers. Illustrate how to find the H.C.F. and L.C.M. of polynomials with Help students revise the manipulation of simple algebraic fractions. Illustrate the application of the factor theorem and other methods for factorizing polynomials in the manipulation of rational functions. H.C.F. and L.C.M. of Polynomials nal and their Manipulations 5
6 Chapter 6 Exponential NF 6.1 Laws of nal Indices A Radicals B nal Indices Understand the definitions of radicals and rational indices. Understand and use the laws of rational indices. Learn how to solve equations in the form m x n c. Help students revise the laws of integral indices learnt in junior forms using the Basic Knowledge Review on p. 6.. Teachers should introduce the concept of radicals and rational indices. Teachers should ensure students understand that the laws of integral indices can be extended to rational indices. Illustrate how to apply the laws of indices to solve equations in the form m x n c with Laws of nal Indices Drilling Program: Laws of Indices 6. Exponential Learn how to solve exponential equations by using the laws of indices. Teachers should introduce the concept of exponential equations. Exponential Illustrate how to apply the laws of indices to solve exponential equations with 6. Exponential and their Graphs A Exponential B Graphs of Exponential Understand the exponential functions and their properties. Recognize the features of the graphs of exponential functions. Teachers should introduce the concept of exponential functions and their properties. Teachers may use Activity 6.1 or IT Activity 6.1 to let students explore the features of the graphs of exponential functions. Teachers may use a graph plotting software to demonstrate the relationship between the graphs of y = a x and y 1 for a > 1. a Illustrate how to solve problems related to exponential functions and their graphs with Teachers may encourage students to attempt the Investigation Corner on p x Exponential and their Graphs IT Activity 6.1: Features of the graphs of y = a x for a > 1 and 0 < a < 1 6
7 Chapter 7 Logarithmic NF Common A Definition of Common B Properties of Common C Logarithmic Understand the definition of common logarithms. Learn the properties of common logarithms. Apply the properties of common logarithms to solve problems. Solve logarithmic equations. Solve exponential equations by converting them into logarithmic equations. Teachers may use the WarmUp Activity on p.7. to prepare students for learning the definition of common logarithms. Teachers should introduce the definition of common logarithms. Teachers may use Activity 7.1 to let students explore the properties of common logarithms. Teachers may derive the properties of common logarithms from the laws of indices. Illustrate how to apply the properties of common logarithms to solve related problems with Teachers should introduce the concept of logarithmic equations. Teachers may use Maths Dialogue on p.7.1 to clarify that log is not a common factor and cannot be simply crossed out. Demonstrate how to apply the properties of common logarithms to solve exponential and logarithmic equations with Common Properties of common logarithms Logarithmic.5 7. Applications of Common A Sound Intensity Level B Richter Scale C Logarithmic Transformation D Other Applications Appreciate the applications of logarithms in real life situations such as measuring the sound intensity and the magnitude of an earthquake, logarithmic transformation and other applications. Teachers should use various real life situations to illustrate the applications of common logarithms. Applications of Common Video: Applications of Animation: Logarithmic Transformation 7. to an Arbitrary Base A Definition of to an Arbitrary Base B Properties of to an Arbitrary Base C Logarithmic Understand the definition of logarithms to an arbitrary base. Learn the properties of logarithms to an arbitrary base. Apply the properties of logarithms to an arbitrary base to solve problems. Solve logarithmic equations to an arbitrary base. Teachers should emphasize that common logarithm is not the only type of logarithms. Its definition and properties can be extended to an arbitrary base a, where a > 0 and a 1. Teachers should introduce the basechange formula of logarithms. Illustrate how to apply the properties of logarithms to solve related problems with to an Arbitrary Base 7. Graphs of Logarithmic and their Features A The Graphs of y = log a x where a > 1 B The Graphs of y = log a x where 0 < a < 1 C Relationship between Graphs of Exponential and Logarithmic Understand the logarithmic functions and their properties. Recognize the features of the graphs of logarithmic functions. Understand the relationship between y = a x and y = log a x. Teachers may use Activity 7. or IT Activity 7.1 to let students explore the features of the graphs of logarithmic functions. Teachers should use the tabular and graphical representation of exponential functions and logarithmic functions to discuss their relationship. Teachers may use Extra IT Activity to demonstrate the relationship between the graphs of y = a x and y = log a x. Graphs of Logarithmic and their Features IT Activity 7.1: Features of the graphs of y = log a x for a > 1 and 0 < a < 1 Relationship between the graphs of y = a x and y = log a x 7.5 Historical Development of Appreciate the development of the concept of the Concept of logarithms. A The Logarithm Tables B The Antilogarithm Tables C The Slide Rule Teachers can introduce the logarithm tables, the antilogarithm tables and the slide rule. Historical Development of the Concept of 1 7
8 Chapter 8 More about NF Solving Simultaneous by the Algebraic Method 8. Solving Simultaneous by the Graphical Method 8. Reducible to Quadratic A Fractional B of Higher Degree C with Square Root Signs D Exponential E Logarithmic 8. Practical Problems Leading to Quadratic Learn how to solve simultaneous equations, one linear and one quadratic, by the algebraic method. Learn how to solve practical problems involving simultaneous equations. Help students revise the algebraic methods for solving simultaneous linear equations using the Basic Knowledge Review on p. 8.. Extend the idea to solve simultaneous equations, one linear and one quadratic, and illustrate with Teachers may use Maths Dialogue on p to let students compare different ways of solving simultaneous equations. Illustrate how to solve practical problems involving simultaneous equations, one linear and one quadratic, with Learn how to solve simultaneous equations, Help students revise the graphical method for solving one linear and one quadratic, by the simultaneous linear equations using the Basic graphical method. Knowledge Review on p. 8.. Understand the use and the limitations of the Extend the idea to solve simultaneous equations, one graphical method in solving equations. linear and one quadratic and illustrate with Learn how to use the discriminant to find Illustrate how the discriminant can be used to find the the number of intersections between a number of intersections between a straight line and a straight line and a quadratic curve. quadratic curve. Teachers may encourage students to attempt the Investigation Corner on p Solve equations (including fractional equations, equations of higher degree, equations with square root signs, exponential equations and logarithmic equations) which can be reduced to quadratic equations. Solve practical problems involving equations which can be reduced to quadratic equations. Help students revise the techniques of solving quadratic equations, exponential equations and logarithmic equations using the Basic Knowledge Review on p Illustrate the techniques of solving an equation by reducing it into a quadratic equation, with Remind students to check the validity of the answers obtained. Teachers may tell students that summarizing the given information using tables help understand the problem and formulate an equation. Solving Simultaneous by the Algebraic Method Solving Simultaneous by the Graphical Method Teaching Tool: Solving Simultaneous by the Graphical Method IT Activity 8.1: Explore the number of intersections between a straight line and a quadratic curve Reducible to Quadratic Practical Problems Leading to Quadratic 8
9 Chapter 9 Variations Basic Concept of Variation Understand the basic concept of variation through daily life Teachers should help students revise the concepts of rate and ratio using the Basic Knowledge Review on p. 9.. Teachers may ask students to give more examples of variations in daily life. 9. Direct Variation Understand the concept of direct Explore the algebraic and graphical representations of two quantities in direct Learn how to solve real life problems involving direct Teachers should introduce the concept of direct Teachers may give some real life examples of direct Direct Variation variation to consolidate students understanding. Teachers may use Activity 9.1 to let students learn the concept of direct variation and its graph. Illustrate how to set up an equation connecting the quantities in a direct variation and how to find the value of a quantity/an unknown in a direct Make sure students know that the graph of y = kx is a straight line passing through the origin with slope k. Help students revise percentage using the Basic Knowledge Review on p. 9.. Illustrate how to find the percentage change of one quantity when the other quantity in a direct variation changes. Illustrate how to solve real life problems involving direct Teachers may discuss with students some ambiguities about direct variation using the Maths Dialogue on p Inverse Variation Understand the concept of inverse Explore the algebraic and graphical representations of two quantities in inverse Learn how to solve real life problems involving inverse Teachers should introduce the concept of inverse Teachers may give some real life examples of inverse variation to consolidate students understanding. Teachers may use Activity 9. to let students learn the concept of inverse variation and its graph. Illustrate how to set up an equation connecting the quantities in an inverse variation and how to find the value of a quantity/an unknown in an inverse Inverse Variation Joint Variation Understand the concept of joint Learn how to solve real life problems involving joint 9.5 Partial Variation Understand the concept of partial Learn how to solve real life problems involving partial k Make sure students know that the graph of y is a x curve which does not pass through the origin. Illustrate how to find the percentage change of one quantity when the other quantity in an inverse variation changes. Illustrate how to solve real life problems involving inverse Teachers should introduce the concept of joint Teachers should give examples of different forms of joint Joint Variation Illustrate how to set up an equation connecting the quantities in a joint variation and how to find the value of a quantity/an unknown in a joint Illustrate how to find the percentage change of one quantity when the other quantities in a joint variation change. Illustrate how to solve real life problems involving joint Teachers should introduce the concept of partial Teachers may use Activity 9. to let students understand the Partial Variation nature of partial variation through real life Teachers should give examples of different forms of partial Illustrate how to set up an equation connecting the quantities in a partial variation and how to find the value of a quantity/an unknown in a partial Illustrate how to find the variation constants by setting up a pair of simultaneous linear equations. Illustrate how to solve real life problems involving partial 9
10 Chapter 10 More about Trigonometry Angles of Rotation Understand the definitions of angle of A Angles of Rotation rotation and quadrant. B The Four Quadrants Recognize the concepts of angles with the same terminal side on a rectangular coordinate plane. 10. Trigonometric s of Any Angle A Definitions of Trigonometric s of Any Angle B The Signs of Trigonometric s 10. Graphs of Trigonometric A Graph of y = sin B Graph of y = cos C Graph of y = tan D Periodicity of Trigonometric Understand the definitions of trigonometric ratios, including sine ratio, cosine ratio and tangent ratio, of any angle. Recognize the values of trigonometric ratios of 0, 90, 180, 70 and 60. Understand the signs of trigonometric ratios in different quadrants and the CAST diagram. Understand the features of graphs of sine, cosine and tangent functions. From the graphs of the trigonometric functions, recognize their features including periodicity and optimum values. Help students revise the basic definitions of trigonometric ratios using the Basic Knowledge Review on p Teachers should introduce the concepts of angle of rotation, quadrants and angles with the same terminal side on the rectangular coordinate plane. Teachers should introduce the definitions of trigonometric ratios of an arbitrary angle. Teachers may use WarmUp Activity to let students realize that the coordinates of P is related to the angle of rotation before learning the definitions of trigonometric ratio of any angle. Teachers may use Activity 10.1 (or IT Activity on p ) to let students explore the signs of trigonometric ratios of angles in different quadrants. Teachers should introduce the CAST diagram. Illustrate how to find the values of the trigonometric ratios of angles in different quadrants with Teachers may use Maths Dialogue on p to help students understand the concept and fully master how to determine the signs of x, y and r when a condition is given. Teachers may use Activity 10. to let students explore the properties of sine function, such as the maximum and minimum values, the signs in different quadrants, by considering the graph of y = sin θ. Teachers should discuss with students the periodicity of trigonometric functions using their graphs. Teachers may use To Learn More on p or IT Activity 10. on p to let students plot the graph of y = sin θ by using the unit circle. Illustrate how to find the optimum values of trigonometric functions algebraically. Angles of Rotation Trigonometric s of Any Angle IT Activity 10.1: The signs of trigonometric ratios Graphs of Trigonometric IT Activity 10.: Plot the graph of y = sin θ by using the unit circle Plot graphs of trigonometric functions by using the unit circle Animation: Periodicity of Trigonometric Graphical Solutions of Trigonometric 10.5 Trigonometric Identities A Trigonometric s of (180 ) B Trigonometric s of (180 + ) C Trigonometric s of (60 ) & D Trigonometric s of (60 + ) E Trigonometric s of (90 + ) F Trigonometric s of (70 ) and (70 + ) 10.6 Solving Trigonometric by Algebraic Methods Learn how to solve trigonometric equations such as sin x = k graphically, where k is a constant. Learn the trigonometric identities for trigonometric ratios of (180 ), (60 ),, (90 + ) and (70 ). Learn to simplify trigonometric expressions and prove trigonometric identities. Learn how to solve various trigonometric equations algebraically. Illustrate the steps in solving trigonometric equations graphically with Teachers may use Activity 10. to let students explore the relationships between the trigonometric ratios of θ and (180 θ). Teachers should encourage students to make use of the CAST diagram to memorize the trigonometric identities. Illustrate the use of trigonometric identities to simplify trigonometric expression and prove other trigonometric identities with Illustrate how to solve trigonometric equations by algebraic methods with Graphical Solutions of Trigonometric Drilling Program: Trigonometric Identities Trigonometric Identities Trigonometric identities Solving Trigonometric by Algebraic Methods 10
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