1 Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC EQUATIONS Version : 3.2 Date: Examples 4, 6, 8, 9 and 10 are copyrighted to their respective owners and used with their permission.
2 Mathematics Revision Guides Solving Trigonometric Equations Page 2 of 17 Trigonometric identities. There are two very important trigonometric identities. For all angles cos 2 + sin 2 = 1 (This is the Pythagorean identity) The square on the hypotenuse in the triangle above is 1. The sum of the squares on the other two sides = cos 2 + sin 2 The length of the opposite side to A = sin and the length of the adjacent to A = cos. Since the tangent is the opposite divided by the adjacent, it also follows that tan sin cos These two identities are important when simplifying expressions or solving various types of equations.
3 Mathematics Revision Guides Solving Trigonometric Equations Page 3 of 17 SOLVING TRIGONOMETRIC EQUATIONS Any trigonometric equation of the form sin x = a (where -1 acos x = a (where -1 a or tan x = a can have an infinite variety of solutions. To solve such equations using a calculator, it must be remembered that only one of an infinite number of possible values will be given we must find all solutions within the range of the question. The graph above shows several of the possible solutions of sin x = 0.5. Apart from the one of 30 given on the calculator (the principal value), the possible solutions also include (180-30) or 150 because of the reflective symmetry of the graph about the line x = 90. Moreover, as the graph has a repeating period of 360, other solutions are 390, 510 and so on in the positive x-direction, and 210, -330 and so forth in the negative x-direction. We can also use CAST diagrams to show the two solutions in the range 0-360, and we can add or subtract multiples of 360 as desired for any further values, should the question ask for them. Positive angles are measured anticlockwise from the origin; negative ones clockwise. In general, if sin x = a, then : sin (180-x) = a sin (180n + x) = a for even integer n ; sin (180n - x) = a for odd integer n. Therefore, for instance, sin (360+x), sin (540-x) and sin (-180-x) are also equal to sin x. In radians, the solutions are given as /6 and ( /6) or 5/6. The graph repeats every 2radians, and other solutions are 13/6, 17/6 (positive x-direction) and -7/6, -11/6 (negative x-direction). In general, if sin x = a, then : sin (-x) = a sin (n+ x) = a for even integer n ; sin (n - x) = a for odd integer n. Therefore, for instance, sin (2x) sin (3-x) and sin (--x) are also equal to sin x.
4 Mathematics Revision Guides Solving Trigonometric Equations Page 4 of 17 y y = x -0.5 y = cos x -1 Line of symmetry: x = 0 x = 180 x = 360 x = 540 x = 720 The cosine function also shows similar periodicity. Several of the possible solutions of cos x = 0.5 are shown in the graph above. Apart from the one of 60 given on the calculator(the principal value), the possible solutions also include 300 because the graph has reflective symmetry about the line x = 180. Again, there is a repeating period of 360, and so other solutions are 420, 660 and so on in the positive x-direction, and -60, 300 ۥ, -420 and so forth in the negative x-direction. The CAST diagrams to show the two solutions in the range Again, we can add or subtract multiples of 360 if required. The relationship between cos 60 and cos -60 is evident from the left-hand diagram. In general, if cos x = a, then : cos (-x) = a or cos (360-x) = a cos (360n + x) = a for any integer n cos (360n - x) = a for any integer n. Therefore, for instance, cos (360+x), cos (720-x) and cos (-360-x) are also equal to cos x. In radians, the solutions are given as /3 and /3 (or 5/3). The graph repeats every 2radians, and other solutions are 5/3, 7/3 (positive x-direction) and -5/3, -7/3 (negative x-direction). In general, if cos x = a, then : cos (-x) = a or cos (2-x) = a cos (2n+ x) = a for any integer n cos (2n - x) = a for any integer n. Therefore, for instance, cos (2+x), cos (4x) and cos (-2-x) are also equal to cos x.
5 Mathematics Revision Guides Solving Trigonometric Equations Page 5 of 17 The symmetrical pairing of solutions in the range 0 x (or 0 xin radians) is common to both the sine and cosine graphs, for all angles on either side of the graphs lines of symmetry. (An exception occurs when sin x and cos x = ±1 ; then there is only one solution.in that range.) Thus sin 115 = sin 65 (both equidistant from line of symmetry at 90 ), and cos 155 = cos 205 (both equidistant from line of symmetry at 180 ). The tangent function, on the other hand, has a period of 180 rather than 360 as in the sine and cosine functions. Also, there is no linear symmetry, so there is no doubling-up of solutions (as in sine and cosine examples). The graph shows several solutions of the equation tan x = 1. Apart from the one of 45 given on the calculator, there are other possible solutions because the graph has a repeating period of 180. Those solutions are 225, 405 and so on in the positive x-direction, and 135, -315 and so forth in the negative x-direction. The CAST diagrams for the tangent function are simpler in form than the others you merely add multiples of 180 to obtain all the solutions needed. In general, if tan x = a, then tan (180n + x) = a for any integer n. In radians, the calculator solution is /4. The graph repeats every radians, and other solutions are 5/4, 9/4 (positive x-direction) and -3/4, -7/4 (negative x-direction). In general, if tan x = a, then tan (n + x) = a for any integer n.
6 Mathematics Revision Guides Solving Trigonometric Equations Page 6 of 17 Example (1): Solve sin x = 0.85 for 0 x, giving answers in degrees to one decimal place. The principal value is 58.2, but care must be taken to ensure that all solutions within the range are given. The substitution formulae can be used, but a sketch graph or CAST diagram will probably be easier in finding extra solutions. The two important lines of symmetry here are x = 90 and x = 450. Since 58.2 is 31.8 less than 90, the solution on the other side of that line of symmetry must be 31.8 greater than 90, or Alternatively we can use sin (180-x) = sin x, and = CAST diagram illustration: Having obtained the two solutions above, it is a simple matter of adding and subtracting multiples of 360 as required. Subtracting 360 is no help as there will be no new values found within the range, but adding 360 will give two other solutions: ( ) or ( ) or the solutions of sin x = 0.85 for 0 x0 are 58.2, 121.8, and
7 Mathematics Revision Guides Solving Trigonometric Equations Page 7 of 17 Example (2): Solve sin x = -0.8 for 0 x360, giving answer in degrees to one decimal place. The principal value, and the one given on a calculator, is -53.1, from which we can derive the other solutions. Note that this solution is not in the quoted range, and so we must add an appropriate multiple of 360 (the period of sin x) to it. Here, adding 360 gives one solution, i.e y 1 y = sin x x y = -0.8 Line of symmetry: x = 270 The important line of symmetry here is x = 270. Since is 36.9 greater than 270, the solution on the other side of that line of symmetry must be 36.9 less than 270, or the solutions of sin x = -0.8 for 0 x360 are and Again we could have used sin (180-x) = sin x, and (-53.1 ) = 233.1, although it is less obvious from the diagram. CAST diagram illustration:
8 Mathematics Revision Guides Solving Trigonometric Equations Page 8 of 17 Example (3): Solve cos x = for 0 x720, giving answer in degrees to one decimal place. The two important lines of symmetry here are x = 180 and x = 540. Here, the principal value, and the one given on a calculator, is Since is 48.7 less than 180, the solution on the other side of that line of symmetry must be 48.7 greater than 180, or We could have used cos (360-x) = cos x, and = Having obtained the two solutions above, it is a simple matter of adding and subtracting multiples of 360 as required. Subtracting 360 is no help as there will be no new values found within the range, but adding 360 will give two other solutions, i.e and the solutions of cos x = for 0 x0 are 131.3, 228.7, and CAST diagram illustration:
9 Mathematics Revision Guides Solving Trigonometric Equations Page 9 of 17 Example (4): Solve cos x = 0.75 for 0 x, giving answers in radians to 3 decimal places. The principal value is c, but to find the other solution, we can make use of the lines of symmetry at x = 0 and x = 2Since the principal value is greater than zero, the other required value must be less than 2or c. (This is the same as using the identity cos (2-x) = cos x.) We could have also used the line of symmetry at x = and worked the value as + ( ), leading to the same conclusion. (This, and the substitution identity sin (-x) = sin x, are perhaps easier to use when solving sin x = k or cos x = k in radians.). the solutions of cos x = 0.75 for 0 xare c and c. CAST diagram illustration:
10 Mathematics Revision Guides Solving Trigonometric Equations Page 10 of 17 Solution of equations of the form tan x = k is easier: we just add or subtract multiples of 180 (or multiples of if using radians) to the principal value. Example (5): Solve tan x = 1.5 for - x, giving the answer in radians to 3 decimal places. The principal value is c, but because the tangent function repeats itself every radians, there is another solution at ( ) c or c. CAST diagram illustration: Note that the negative angle is denoted by a clockwise arrow.
11 Mathematics Revision Guides Solving Trigonometric Equations Page 11 of 17 Summary. (The principal solution is the one displayed on the calculator.) The rules are designed to find solutions in the range -180 x < 180 or - x < This range has the virtue of having the principal solution coincide with the calculator display. If a different range is stipulated, it is only a matter of adding multiples of 360 or 2 to the solutions so obtained (when solving equations of the form sin x = k or cos x = k), or multiples of 180 or (when solving equations of the form tan x = k). In each case, n is an integer. Solving sin x = k. Value of k Principal soln. Companion solution Additional solutions 0 x = 0 or 0 rad (none) Add/subtract 180n or n 1 x = 90 or /2 (none) Add/subtract 360n or 2n -1 x = -90 or -/2 (none) Add/subtract 360n or 2n positive 0 < x < 90 or 0 < x </ x or - x Add/subtract 360n or 2n negative -90 < x < 0 or -/2 < x < x or -- x Add/subtract 360n or 2n Solving cos x = k. Value of k Principal soln. Companion solution Additional solutions 0 x = 90 or /2 (none) Add/subtract 180n or n 1 x = 0 or 0 rad (none) Add/subtract 360n or 2n -1 x = -180 or - (none) Add/subtract 360n or 2n positive 0 < x < 90 or 0 < x </2 - x Add/subtract 360n or 2n negative 90 < x < 180 or /2 < x < - x Add/subtract 360n or 2n Solving tan x = k. Value of k Principal soln. Companion solution Additional solutions 0 x = 0 or 0 rad (none) Add/subtract 180n or n positive 0 < x < 90 or 0 < x </2 (none) Add/subtract 180n or n negative -90 < x < 0 or -/2 < x < (none) Add/subtract 180n or n
12 Mathematics Revision Guides Solving Trigonometric Equations Page 12 of 17 Sometimes we need to use the idea of transformations to solve slightly more complex trig equations. Example (6): Use the results from Example (4) to solve cos 2x = 0.75 for 0 x, giving your answers in radians to three decimal places. Here the limits for x are 0 x, but we are asked to solve for 2x. To ensure that no values are omitted, we must substitute A for 2x and multiply the limiting values by 2 to get the transformed limit of 0 A2 From Example (4), we see that the solutions of cos A = 0.75 for 0 Aare c and c. To convert the A-values back to x-values, we must divide by 2. the solutions of cos 2x = 0.75 for 0 xare c and c. Example (7): Solve tan(2x + 60 ) = 1 for 0 x360. By letting A stand for 2x + 60, we first transform the x-limits to A-limits: 0 x A (x-values doubled, 60 added). When drawing sketch graphs or CAST diagrams, draw them with respect to the transformed variable A, and not the original variable x. The substitution back to x-values must be done afterwards. The principal value of A where tan A = 1 is 45, but that value is not within the A-limits. We therefore keep adding multiples of 180 to obtain A = 225, 405, 585 and 765. Since we doubled the x-values and then added 60 to get the A-values, we must perform the inverse operations to change the A-values back to x-values i.e. subtract 60 and then halve the result. Therefore A = 225 x = ½(225-60) = Similarly A = 405 x = 172.5, A = 585 x = and finally A = 765 x = The solutions in the range are therefore x = 82.5, 172.5, and
13 Mathematics Revision Guides Solving Trigonometric Equations Page 13 of 17 Example (8): Solve sin (3x - 40) = -0.3 for 0 x0 giving answers in degrees to one decimal place. Again, we must substitute the x-limits of 0 x0 with A limits. By using A = 3x- 40, the A-limits transform to 40 A 00. The graph below uses the A-limits. The principal value of A is here 17.5,. which is inside the A-limits. This value is to the left of the line of symmetry at A= 90, so another solution will be to the right, i.e. at A = (We could also have used sin (180-x) = sin x, and (-17.5 ) = ) CAST diagram working: Note that the negative angle is labelled with a clockwise arrow. To complete the full set of solutions for A in the required range, we add multiples of 360 to the two above values. In fact the only additional one is A = ( ) = These three solutions will then need converting back from A-values to x-values by the inverse operation of x A 40, so when A = -17.5, x = Similarly, when A = 197.5, x = 79.2 and when A = 342.5, x = Hence the solutions of sin (3x - 40) = -0.3 for 0 x0 are 7.5, 79.2 and
14 Mathematics Revision Guides Solving Trigonometric Equations Page 14 of 17 Quadratic equations in the trigonometric functions are solved in the same way, but care must be taken when factorising them. Note also that the graphs accompanying the solutions are for illustration only you will not be asked to sketch them! Example (9) : Solve the equation 2cos 2 x - cos x = 0 for - x. This factorises at once into (cos x) (2 cos x 1) = 0. cos x = 0 when x = 2 or x = cos x = 1, or cos x = 0.5, when x = 3 or x = -3. The solutions to the equation are therefore x = 2 and 3 (illustrated below). Important: we cannot simply cancel out cos x from the equation as follows: 2cos 2 x - cos x = 0 2cos 2 x = cos x 2cos x = 1 x = 3. The final division of both sides by cos x has led to a loss of the solutions satisfying cos x = 0, i.e. x = 2.
15 Mathematics Revision Guides Solving Trigonometric Equations Page 15 of 17 Example (10) : Solve the equation cos 2 x - cos x - 1 = 0 for 0 x2. Give the answer in radians to three decimal places. This quadratic does not factorise and so the general formula must be used. Substitute x = cos x, a = 1, b = -1 and c = -1 into the formula. x b b 2 4ac 2a cos x. These are the possible solutions, and are and to 3 decimal places. The first value can be rejected, since the cosine function cannot take values outside the range 1 cos x 1. The only solutions are those where cos x = The principal value of x is c. Since cos (2-x) = cos x, another solution would be ( ) c or c. the solutions of cos 2 x - cos x - 1 = 0 where 0 x2are c and c to 3 decimal places.
16 Mathematics Revision Guides Solving Trigonometric Equations Page 16 of 17 Example (11): Find the angle(s) between 0 and 360 satisfying the equation 1-2cos 2 x + sin x = 0 This equation can be manipulated into a quadratic in sin x by replacing 2cos 2 x with 2(1 -sin 2 x) using the Pythagorean identity. 1 2(1-sin 2 x) + sin x = sin 2 x + sin x = 0 2sin 2 x + sin x 1 = 0 This is now a quadratic in sin x which factorises into (2 sin x - 1)(sin x + 1) = 0 Hence sin x = 0.5 or 1. This gives x = 30 or 150 (where sin x = 0.5); also x = 270 (where sin x = -1). Example (12): Prove that (1 cos x)(1 cos x) sin xcos x tan x. The top line of the left-hand expression can be recognised as a difference of squares result, namely 1-cos 2 x. This in turn can be replaced by sin 2 x using the Pythagorean identity. The expression on the left thus becomes obtain sin x cos x or tan x. sin 2 x sin x cos x, and dividing both sides by sin x we finally
17 Mathematics Revision Guides Solving Trigonometric Equations Page 17 of 17 Example (13): Show that the equation sin x - cos 2 x 5 = 0 has no solutions. Substituting cos 2 x by 1 -sin 2 x gives sin x (1-sin 2 x) 5 = 0 sin x 1 + sin 2 x 5 = 0 sin 2 x + sin x 6 = 0 sin x + 3) (sin x 2) = 0 sin x = 2 or 3. The quadratic in sin x factorises all right, but the solutions are impossible because the sine function cannot take values outside the range 1 sin x 1. Example (14): Solve the equation sin x = 3 cos x for angles between 0 and 360. sin x cos x We can divide the equation by cos x to give 3, or tan x 3. The principal value of x is 60, but since the tangent graph repeats every 180, another solution is x = 240.
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
Page 1 In game development, there are a lot of situations where you need to use the trigonometric functions. The functions are used to calculate an angle of a triangle with one corner of 90 degrees. By
GCE Mathematics Unit 4722: Core Mathematics 2 Advanced Subsidiary GCE Mark Scheme for June 2014 Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our
How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle
210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:
Trigonometry It is possible to solve many force and velocity problems by drawing vector diagrams. However, the degree of accuracy is dependent upon the exactness of the person doing the drawing and measuring.
FOREWORD The Botswana Examinations Council is pleased to authorise the publication of the revised assessment procedures for the Junior Certificate Examination programme. According to the Revised National
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.
EVALUATING ACADEMIC READINESS FOR APPRENTICESHIP TRAINING For ACCESS TO APPRENTICESHIP MATHEMATICS SKILL OPERATIONS WITH INTEGERS AN ACADEMIC SKILLS MANUAL for The Precision Machining And Tooling Trades
PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities
how to use dual base log log slide rules by Professor Maurice L. Hartung The University of Chicago Pickett The World s Most Accurate Slide Rules Pickett, Inc. Pickett Square Santa Barbara, California 93102
1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
CHAPTER 1 Introduction and Mathematical Concepts PREVIEW In this chapter you will be introduced to the physical units most frequently encountered in physics. After completion of the chapter you will be
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
FACTORING ANGLE EQUATIONS: For convenience, algebraic names are assigned to the angles comprising the Standard Hip kernel. The names are completely arbitrary, and can vary from kernel to kernel. On the
Mark Scheme (Results) Summer 2014 Pearson Edexcel GCSE In Mathematics A (1MA0) Higher (Calculator) Paper 2H Edexcel and BTEC Qualifications Edexcel and BTEC qualifications are awarded by Pearson, the UK
TRIGONOMETRY FOR ANIMATION What is Trigonometry? Trigonometry is basically the study of triangles and the relationship of their sides and angles. For example, if you take any triangle and make one of the
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
Mathematics programmes of study: key stage 4 National curriculum in England July 2014 Contents Purpose of study 3 Aims 3 Information and communication technology (ICT) 4 Spoken language 4 Working mathematically
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract -
MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
Tips for Solving Mathematical Problems Don Byrd Revised late April 2011 The tips below are based primarily on my experience teaching precalculus to high-school students, and to a lesser extent on my other
Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
2013 2014 Scheme of Work Subject MATHS Year 9 Course/ Year Term 1 Key Topics What will ALL students learn? What will the most able students learn? Number Written methods of calculations Decimals Rounding
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
MATH 2 Course Syllabus Spring Semester 2007 Instructor: Brian Rodas Class Room and Time: MC83 MTWTh 2:15pm-3:20pm Office Room: MC38 Office Phone: (310)434-8673 E-mail: rodas firstname.lastname@example.org Office Hours:
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a