Know the following (or better yet, learn a couple and be able to derive the rest, quickly, from your knowledge of the trigonometric functions):


 Augusta Bruce
 2 years ago
 Views:
Transcription
1 OCR Core Module Revision Sheet The C exam is hour 0 minutes long. You are allowed a graphics calculator. Before you go into the exam make sureyou are fully aware of the contents of theformula booklet you receive. Also be sure not to panic; it is not uncommon to get stuck on a question (I ve been there!). Just continue with what you can do and return at the end to the question(s) you have found hard. If you have time check all your work, especially the first question you attempted...always an area prone to error. Trigonometry J.M.S. We define tanθ sinθ cosθ. This identity is very useful in solving equations like sinθ cosθ = 0 which yields tanθ =. The solutions of this in the range 0 θ 60 are θ = 6.4 and θ = 4.4 to one decimal place. Know the following (or better yet, learn a couple and be able to derive the rest, quickly, from your knowledge of the trigonometric functions): θ sinθ cosθ tanθ undefined Be able to sketch sinθ, cosθ and tanθ in both degrees and radians. By considering a right angled triangle (or a point on the unit circle) we can derive the important result sin θ+cos θ. This is useful in solving certain trigonometric equations. Worked example; solve = cos θ +sinθ for 0 θ 60. = cos θ+sinθ = ( sin θ)+sinθ get rid of cos θ, 0 = sin θ +sinθ quadratic in sinθ, 0 = sin θ sinθ factorise as normal, 0 = (sinθ+)(sinθ ). So we just solve sinθ = and sinθ =. Therefore θ = 0 or θ = 0 or θ = 90. The above relation is also useful in converting between the different trigonometric functions. For example if cosθ = 6 7 then, to find sinθ, do not use cos on your calculator J.M.Stone
2 and then sin the answer. Instead sin θ +cos θ =, sin θ =, sinθ = ± 49 = ± 7. Without further information you must keep both the positive and negative solution. If a question tells you that the angle is acute, obtuse or reflex then you must visualise the appropriate graph and interpret. For example given that sinθ = and that θ is obtuse, find the value of cosθ. By the argument above you will find that 8 cosθ = ± = ±. However, given an obtuse angle (90 < θ < 80 ) the cosine graph is negative, so the final answer should be cosθ =. You must be careful when you see things like tanxsinx = tanx. It isso tempting to divide both sides by tanx to yield sinx =. But you must bring everything to one side and factorise; tanxsinx tanx = 0 tanx(sinx ) = 0. The full set of solutions can then be by solving tanx = 0 and sinx = 0. [It is completely analogous to x = x. If we divide by x we find x =, but we know this has missed the solution x = 0. However when we factorise we find x(x ) = 0 and both solutions are found.] Given a trigonometric equation it is always best first to isolate the trigonometric function on its own; for example 9cos(...)+ = 7 cos(...) = 9. For complicated trigonometric equations where you are not just cos ing, sin ing or tan ing a single variable (x, θ, t or the like), it is often easiest to make a substitution. For example to solve cos(x+0) = 4 in the range 0 x 60 the desired substitution is clearly u = x + 0, but you must remember to also convert the range also (many students forget this) so: cos(x+0) = 4 0 x 60, cosu = 4 0 u 70, u =...,...,...,.... However, we don t want solutions in u, so we need to use x = u 0 on each u solution to get x =...,...,..., J.M.Stone
3 Sequences You must be comfortable with notation. It works as follows; you put in the number at the bottom of the and then keep summing until you reach the top number. For example: 8 (i+) = = 7, i=4 n (i +i) = (+)+(4+)+(9+)+ +(n +n). i= A sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence. Sequences are sometimes defined recursively. For example the sequence a n+ = a n + with a = 0 defines the sequence 0,,6,9... We know that this is an arithmetic sequence which can also be defined deductively by a n = 0+(n ). An arithmetic sequence increases or decreases by a constant amount. The letter a always denotes the first term and d is the difference between the terms (negative for a decreasing sequence!). The nth term is denoted a n and satisfies the important relationship a n = a+(n )d. For example if told the third term of a sequence is 0 and the seventh term is 4 then we can use the above equation to find the a and d. 0 = a+( )d 4 = a+(7 )d 4d = 4 d = 6 a =. The sum of the n terms of an arithmetic sequence is given by S = n (First+Last) = n (a+(n )d). For example the sum of the first 0 terms of a sequence is 0 and the first term is 4. What is the difference? Binomial Theorem S = n (a+(n )d) 0 = 0 (8+(0 )d) d =. Binomial expansion allows us to expand (a + b) n for any integer n. Best explained by means of an example; expand (x y).. Begin by considering prototype expansion of (a+b).. So (a+b) = ( ) 0 a + ( ) a 4 b+ ( ) a b + ( ) a b + ( ) 4 ab 4 + ( ) b.. Calculate binomial coefficients either on calculator or by drawing a mini Pascal s Triangle to give (a+b) = a +a 4 b+0a b +0a b +ab 4 +b. If you want to multiply instead then use notation. For example n i = 4 n n! i= J.M.Stone
4 4. Next notice that in our case a = x and b = y and substitute in to get (x y) = (x) +(x) 4 ( y)+0(x) ( y) +0(x) ( y) +(x)( y) 4 +( y).. Tidying up we get (x y) = x 80x 4 y +80x y 40x y +0xy 4 y. It is worth noting that when the expansion is of the form (something another thing) n, then the signs will alternate. Also of note is the way each individual component is constructed. For example; find the ( x coefficient in the expansion of ( x) 7. The component with x is given by 7 ) () ( x) = 04x, so the coefficient is 04. ( ) n = n C r = r n!. For example r!(n r)! Know that (+x) n expands thus: ( n (+x) n = + ) x+ ( ) =!!! = 4 ( ) ( ) = 0. ( ) n x + = +nx+ n(n ) (This is particularly useful in Core 4.) Sine & Cosine Rules The sine rule states for any triangle sina a ( ) n x + +x n x + n(n )(n ) x + +x n.! = sinb b = sinc. c The cosine rule states that a = b +c bccosa. Practice both sine and cosine rules on page 9. By considering half of a general parallelogram we can show that the area of any triangle is given by A = absinc. You must be good at bearing problems which result in triangles. Remember to draw lots of North lines and remember also that they are all parallel; therefore you can use Corresponding, Alternate and Allied angle theorems... revise your GCSE notes! Bearings are measured clockwise from North and must contain three digits. For example Integration. 0.. Calculus is the combined study of differentiation and integration (and their relationship). A good description is that calculus is the study of change in the same way that geometry is the study of shapes. Integration is the reverse of differentiation. That is if dy dx = f(x) then y = f(x)dx. For example if dy dx = x then y = x dx = 4 x4 +c. The general rule is therefore ax n dx = axn+ n+ +c. ydx is an indefinite integral because there are no limits on the integral sign. When evaluating these integrals never forget an arbitrary constant added on at the end. For example 6x dx = x +c. 4 J.M.Stone
5 b a ydx is a definite integral and is the area between the curve and the xaxis from x = a to x = b. Areas under the xaxis are negative. (For areas between the curve and the yaxis switch the x and the y and use q p xdy between y = p and y = q.) To find the area between two curves between x = a and x = b evaluate b a (top bottom)dx. For example, given that y = x + x, find the area under the curve from x = and x =. x+ x dx = x +x dx [ = = ] x +x ( + = 0 8. ) ( ) + To calculate integrals where one of the limits is infinite ( or ), proceed as normal until you input the or into the integral. Then you must not write such things as, +,,, and the like. You must think what these things would equal and just write down the number; in the previous four cases you would get 0, 0, and. (If, in the C exam, you think that when you put in you get an infinite answer, chances are you ve made a mistake somewhere.) For example: 8 x dx = 8 x dx [ = 4 ] x = (0) ( 4 ) =. Geometric Sequences A geometric sequence is one where the terms are multiplied by a constant amount. For example,,4,8,6,...,[ n ] is a geometric sequence with a = and r =. The n th term is given by a n = ar n. So for the above example the 0th term is a 0 = 9 = 488. The sum of n terms of a geometric sequence is given by ( r n ) S = a or (equivalently) by r ( ) r n S = a. r For example sum the first 0 terms of 4,,,,...,[4 ( n ]. ) This is given by ) 0 ) S = 4(( = J.M.Stone
6 If the ratio (r) lies between and (i.e. < r < ) then there exists a sum to infinity given by S = a r. Therefore S for the above example is S = 4 terms is very close to S. Exponentials & Logarithms = 8. We can see that the sum to 0 (In these notes if I write logx I mean log 0 x. If I mean a different base, I will write it explicitly as log a x. When we see log a x we say log to the base a of x.) We define a logarithm to be the solution (in x) to the equation a x = b. It is written x = log a b. The fundamental relationship is therefore a x = b x = log a b. From weseethat log a bmeans: Thenumberahastoberaisedto, tomakeb. Therefore some simple logarithms can be calculated without a calculator: log 8 = the number has to be raised to, to make 8 =, log = the number 0 has to be raised to, to make 0000 = 4, log 9 = the number 9 has to be raised to, to make =, ( = 9 = 9 ) log a a = the number a has to be raised to, to make a =. We see from that logarithms pluck out powers from equations. Therefore if you ever see an equation with the unknown in the power, then that is the clue that you will need to use logarithms. For example to solve 7 x = we discover 7 x =, x = log 7, x = log 7+. However we need to build on because not all equations are this simple (e.g. x = 7 x+ ) and not all calculators can calculate log 7. You can also use to eliminate logarithms from an equation. Given an equation of the form log a ( ) = b, you can eliminate the logarithm instantly to get ( ) = a b. A good way to remember this is Girvan s Bullying Base. So if we have log x = 8, then the bullying base knocks the log out of the way and moves to the other side and squeezes up the 8 to put it in its place; therefore x = 8. Logarithms and exponentials (powers) are the inverse functions of each other (as can be seen from if one puts one into the other). Therefore So if loga =.4 then a = 0.4. From a colleague I respect; Mr Girvan log 0 0 x = x and 0 log 0 x = x. 6 J.M.Stone
7 There are some rules that can be derived from that must be learnt. They are (for all bases): log(ab) = loga+logb log = 0 log( a b ) = loga logb log aa = log(a n ) = nloga log( a ) = loga log a b = log cb log c a. Whenweneedtosolveanequationwheretheunknownisintheexponentsuchas x = 8 take log 0 of both sides and simplify: Factors and Remainders x = 8 log 0 ( x ) = log 0 8 (x )log 0 = log 0 8 x = log 08 log 0 x = ( log0 8 log 0 + x =. (sf). Need to know how to divide any polynomial by a linear factor of the form ax b. For example divide x +x +x 6 by x. (Always devote a column to each power of x.) x +4x + x +x +x +x 6 +x x +4x +x +4x 8x +x 6 +x +6 ) So the remainder is +6. Therefore x +x +x 6 = (x )(x +4x+)+6. If the remainder is zero, then the divisor is said to be a factor of the original polynomial. The Factor Theorem states: (x a) is a factor of f(x) f(a) = 0. [More generally (but used less often in exams) is: (ax b) is a factor of f(x) f( b a ) = 0.] For example if x +ax +8x 4 has (x ) as a factor, find a. From factor theorem we know f() = 0, so we discover +a +8 4 = 0, and therefore a =. 7 J.M.Stone
8 The Remainder Theorem states: When f(x) is divided by (x a) the remainder is f(a). [More generally (but used less often in exams) is: When f(x) is divided by (ax b) the remainder is f( b a ).] Notice that the factor theorem is a subsetof theremainder theorem. In the factor theorem all remainders are zero, by definition. For example if told that when f(x) = x +x x 7 is divided by x the remainder is, we know f() =. Worked example: f(x) = x + x + kx. The remainder when f(x) is divided by (x ) is four times the remainder when f(x) is divided by (x+). Find k. We know Radians f() = 4 f( ) + +k = 4[ ( ) + ( ) k ] k =. There are (by definition) π radians in a circle. So 60 = π. To convert from degrees π to radians we use the conversion factor of 80. For example to convert 4 to radians we calculate 4 π 80 = π 80 4 rad. From radians to degrees we use its reciprocal π. When using radians the formulae for arc length and area of a sector of a circle become simpler. They are s = rθ and A = r θ. The Trapezium Rule The area under any curve can be approximated by the Trapezium Rule. The governing formula is given by (and contained in the formula booklet you will have in the exam) b a ydx h[y 0 +y n +(y +y + +y n )], where h is the width of each trapezium, y 0 and y n are the end heights and y +y + + y n are the internal heights. By considering the shape of the graph in the interval over which you are approximating it should be clear whether your estimate of the area is an over or underestimate of the true area For example if you were to estimate π 0 sinxdx (above, left) using the trapezium rule, due to the shape of the curve, the trapezia would all fall below the curve, so we would obtain an underestimate. However, with 0 x dx (above, right) we would obtain an overestimate. 8 J.M.Stone
Core Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
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
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
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
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationMark Scheme (Results) Summer 2013. GCE Core Mathematics 2 (6664/01R)
Mark (Results) Summer 01 GCE Core Mathematics (6664/01R) Edexcel and BTEC Qualifications Edexcel and BTEC qualifications come from Pearson, the world s leading learning company. We provide a wide range
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationWeek 13 Trigonometric Form of Complex Numbers
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
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationTOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM
Content Area: Mathematics Course Title: Precalculus Grade Level: High School Right Triangle Trig and Laws 34 weeks Trigonometry 3 weeks Graphs of Trig Functions 34 weeks Analytic Trigonometry 56 weeks
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
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
More informationStage 1 Higher Revision Sheet
Stage 1 Higher Revision Sheet This document attempts to sum up the contents of the Higher Tier Stage 1 Module. There are two exams, each 25 minutes long. One allows use of a calculator and the other doesn
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationMATH 2300 review problems for Exam 3 ANSWERS
MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationMTN Learn. Mathematics. Grade 10. radio support notes
MTN Learn Mathematics Grade 10 radio support notes Contents INTRODUCTION... GETTING THE MOST FROM MINDSET LEARN XTRA RADIO REVISION... 3 BROADAST SCHEDULE... 4 ALGEBRAIC EXPRESSIONS... 5 EXPONENTS... 9
More informationNotes and questions to aid Alevel Mathematics revision
Notes and questions to aid Alevel Mathematics revision Robert Bowles University College London October 4, 5 Introduction Introduction There are some students who find the first year s study at UCL and
More informationOxford Cambridge and RSA Examinations
Oxford Cambridge and RSA Examinations OCR FREE STANDING MATHEMATICS QUALIFICATION (ADVANCED): ADDITIONAL MATHEMATICS 6993 Key Features replaces and (MEI); developed jointly by OCR and MEI; designed for
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationThinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks
Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More information106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.
5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function
More informationNew HigherProposed OrderCombined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC
New HigherProposed OrderCombined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiationbut not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs
More informationSOLVING TRIGONOMETRIC EQUATIONS
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
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationUnderstanding Basic Calculus
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
More information14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style
Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationComplex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers
Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, June 14, 013 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationInfinite Algebra 1 supports the teaching of the Common Core State Standards listed below.
Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School
More informationExpress a whole number as a product of its prime factors. e.g
Subject OCR 1 Number Operations and Integers 1.01 Calculations with integers 1.01a Four rules Use noncalculator methods to calculate the sum, difference, product and quotient of positive and negative
More informationFind the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.
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.
More information4.1 Radian and Degree Measure
Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
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:
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationALGEBRA 2/ TRIGONOMETRY
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA 2/ TRIGONOMETRY Friday, June 14, 2013 1:15 4:15 p.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 28.......................
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More informationAdditional Mathematics
Jawapan Additional Mathematics Additional Mathematics Paper 1 SOALAN ULANGKAJI SPM 2011 Additional Mathematics Paper 2 SOALAN ULANGKAJI SPM 2011 SOALAN ULANGKAJI SPM 2011 SOALAN ULANGKAJI SPM 2011 SOALAN
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationTrigonometry Lesson Objectives
Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the
More informationSolving Logarithmic Equations
Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide
More informationparallel to the xaxis 3f( x 4
OCR Core 3 Module Revision Sheet The C3 exam is hour 30 minutes long. You are allowed a graphics calculator. Before you go into the exam make sureyou are fully aware of the contents of theformula booklet
More informationPreCalculus Review Problems Solutions
MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry PreCalculus Review Problems Solutions Problem 1. Give equations for the following lines in both pointslope and slopeintercept form. (a) The
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
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 
More informationSection 1: How will you be tested? This section will give you information about the different types of examination papers that are available.
REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationMATHEMATICS Unit Pure Core 2
General Certificate of Education January 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Wednesday 9 January 2008 1.30 pm to 3.00 pm For this paper you must have: an 8page answer
More informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationEuler s Formula Math 220
Euler s Formula Math 0 last change: Sept 3, 05 Complex numbers A complex number is an expression of the form x+iy where x and y are real numbers and i is the imaginary square root of. For example, + 3i
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationAlgebra 2/Trigonometry Practice Test
Algebra 2/Trigonometry Practice Test Part I Answer all 27 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. For each question, write on the separate
More informationIntegration Involving Trigonometric Functions and Trigonometric Substitution
Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More informationSPECIFICATION. Mathematics 6360 2014. General Certificate of Education
Version 1.0: 0913 General Certificate of Education Mathematics 6360 014 Material accompanying this Specification Specimen and Past Papers and Mark Schemes Reports on the Examination Teachers Guide SPECIFICATION
More informationGCE Mathematics. Mark Scheme for June 2014. Unit 4722: Core Mathematics 2. Advanced Subsidiary GCE. Oxford Cambridge and RSA Examinations
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
More informationAngles and Their Measure
Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two
More informationStudent Performance Q&A:
Student Performance Q&A: AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief Reader,
More information2.2 Separable Equations
2.2 Separable Equations 73 2.2 Separable Equations An equation y = f(x, y) is called separable provided algebraic operations, usually multiplication, division and factorization, allow it to be written
More informationNational 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationANALYTICAL METHODS FOR ENGINEERS
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
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
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
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationChapter 8. Exponential and Logarithmic Functions
Chapter 8 Exponential and Logarithmic Functions This unit defines and investigates exponential and logarithmic functions. We motivate exponential functions by their similarity to monomials as well as their
More informationAn important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.
Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could set
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, June 1, 011 1:15 to 4:15 p.m., only Student Name: School Name: Print your name
More informationTechniques of Integration
CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an artform than a collection of algorithms. Many problems in applied mathematics involve the integration
More information2312 test 2 Fall 2010 Form B
2312 test 2 Fall 2010 Form B 1. Write the slopeintercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function
More informationTips for Solving Mathematical Problems
Tips for Solving Mathematical Problems Don Byrd Revised late April 2011 The tips below are based primarily on my experience teaching precalculus to highschool students, and to a lesser extent on my other
More information6. Vectors. 1 20092016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationWORKBOOK. MATH 30. PRECALCULUS MATHEMATICS.
WORKBOOK. MATH 30. PRECALCULUS MATHEMATICS. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributor: U.N.Iyer Department of Mathematics and Computer Science, CP 315, Bronx Community College, University
More information4.1: Angles and Radian Measure
4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is
More informationCHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY
CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY Specific Expectations Addressed in the Chapter Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that
More informationRELEASED. Student Booklet. Precalculus. Fall 2014 NC Final Exam. Released Items
Released Items Public Schools of North arolina State oard of Education epartment of Public Instruction Raleigh, North arolina 276996314 Fall 2014 N Final Exam Precalculus Student ooklet opyright 2014
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationMathematics PreTest Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics PreTest Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {1, 1} III. {1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationPreCalculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.
PreCalculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationMathematics. GCSE subject content and assessment objectives
Mathematics GCSE subject content and assessment objectives June 2013 Contents Introduction 3 Subject content 4 Assessment objectives 11 Appendix: Mathematical formulae 12 2 Introduction GCSE subject criteria
More information