CRASH COURSE IN PRECALCULUS
|
|
- Diane Weaver
- 7 years ago
- Views:
Transcription
1 CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai
2 Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole Chapter 5-7.
3 LECTURE 6. TRIGONOMETRY: PART I This lecture is the first part of reviewing high school trigonometry: Pythagorean theorem, the definitions of trigonometric and inverse trigonometric functions and some of their properties.
4 Trigonometry of Right Triangle Angle Measure: If a circle of radius 1 (so its circumference is 2π) with vertex of an angle at its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle. See where the second graph shows that the length s of an arc that subtends a central angle of θ radians in a circle of radius r is s = rθ
5 Trigonometry of Right Triangle RELATIONSHIP BETWEEN DEGREES AND RADIANS 180 = π rad, 1 rad = ( 180 π ) 1 = π 180 rad Examples 1. Express 30 in radians. 30 = 30 ( π 180 ) rad = π 6 rad. 2. Express 45 in radians. 45 = 45 ( π 180 ) rad = π 4 rad. 3. Express π 3 rad in degrees. π 3 rad = (π 3 ) (180 π ) = 60. A note on terminology: We often use a phrase such as a 30 degree angle to mean an angle whose measure is 30. Also for an angle θ, we write θ = 30 or θ = π/6 to mean the measure of θ is 30 or π/6rad. When no unit is given, the angle is assume to be in radians.
6 Area of a Circular Sector In a disc of radius r, the area of a sector with central angle of θ radians is A = πr 2 θ 2π = 1 2 πr 2.
7 Trigonometry of Right Triangle Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then c 2 = a 2 + b 2.
8 Trigonometric Function: Right Triangle Approach Trigonometric Ratios for 0 < θ < π 2 sin θ = b c cos θ = a c tan θ = b a csc θ = c b sec θ = c a cot θ = a b
9 Examples 1. Suppose 0 < θ < π 2 and cos θ = 1. Find the values of 3 sin θ, tan θ, cot θ, sec θ and csc θ. By we see that the length of the opposite side is ( 3 ) = = sin θ, tan θ = 3 4, cot θ = 2 2, csc θ = 3 2 and sec θ = 3 4
10 Examples 2. Suppose 0 < θ < π and tan θ = 2. Find the values of 2 sin θ, cos θ, cot θ, sec θ and csc θ. Use We see the length of the hypotenuse side is 5. Hence sin θ = 2 5, cos θ = 1 5, cot θ = 1 2, sec θ = 5 and csc θ = 5 2
11 Trigonometric Function: Unit Circle Approach The unit circle is the circle of radius 1 centered at the origin in the xy-plane. Its equation is x 2 + y 2 = 1. Suppose θ is a real number. Mark off a distance θ along the unit circle, starting at the point (1, 0) and moving in a counterclockwise direction if θ > 0 or in a clockwise direction if θ < 0. In this way we arrive at a point P(x, y) on the unit circle. The point P(x, y) obtained in this way is called the terminal point determined by the real number θ.
12 Trigonometric Function: Unit Circle Approach,
13 Trigonometric Function: Unit Circle Approach Trigonometric Functions for θ R sin θ = y cos θ = x tan θ = y (x 0) x csc θ = 1 y (y 0) sec θ = 1 x (x 0) cot θ = x (y 0) y Note that, when 0 < θ < π 2, a = x, b = y and c = 1, the unit circle approach is identical with the right triangle approach in the defining these trigonometric functions, but the domains of these trigonometric functions are larger using the unit circle approach.
14 Trigonometric Function: Special Triangles Special Triangles θ in θ in rad sin θ cos θ tan θ csc θ sec θ cot θ π π π π
15 Trigonometric Function: Special Triangles
16 Fundamental Identities of Trigonometric Function csc θ = 1 sin θ Reciprocal Identities sec θ = 1 cos θ cot θ = tan θ = sin θ cot θ = cos θ cos θ sin θ Pythagorean Identities 1 tan θ sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ
17 Fundamental Identities of Trigonometric Function Even-Odd Identities sin( θ) = sin θ cos( θ) = cos θ tan( θ) = tan θ csc( θ) = csc θ sec( θ) = sec θ cot( θ) = cot θ
18 Fundamental Identities of Trigonometric Function Cofunction Identities sin ( π 2 θ) = cos θ tan (π 2 θ) = cot θ sec (π θ) = csc θ 2 cos ( π 2 θ) = sin θ cot (π 2 θ) = tan θ csc (π θ) = secθ 2
19 Examples on Applications of Trigonometric Identities 1. Simplify tan θ + cos θ 1 + sin θ. tan θ + cos θ 1 + sin θ = sin θ cos θ + cos θ 1 + sin θ = sin θ(1 + sin θ) + cos2 θ cos θ(1 + sin θ) = sin θ + sin2 θ + cos 2 θ 1 + sin θ = cos θ(1 + sin θ) cos θ(1 + sin θ) = 1 cos θ = sec θ 2. Prove 1 sin θ 1 + sin θ = (sec θ tan θ)2. 1 sin θ 1 + sin θ = 1 sin θ 1 + sin θ 1 sin θ 2 (1 sin θ)2 sin θ = 1 sin θ 1 sin 2 = (1 θ cos θ ) = ( 1 cos θ sin θ cos θ ) 2 = (sec θ tan θ) 2.
20 Trigonometric Graphs A function f is periodic if there is a positive p such that f (θ + p) = f (θ) for every θ. The least such positive number is the period of f. If f has period p, then the graph of f on any interval of length p is called one complete period of f. Periodic Properties of Sine and Cosine The functions sine and cosine have period 2π: sin(θ + 2π) = sin θ cos(θ + 2π) = cos θ
21 Trigonometric Graphs Periodic Properties of tangent and Cotangent The functions tangent and cotangent have period π: tan(θ + π) = tan θ cot(θ + π) = cot θ
22 Trigonometric Graphs Periodic Properties of Secant and Cosecant The functions secant and cosecant have period 2π: sec(θ + 2π) = sec θ csc(θ + 2π) = csc θ
23 Examples 1. sin 2π 3 = sin (π 2 + π 6 ) = cos ( π 6 ) = cos π 6 = cos π = cos ( π 2 + π 2 ) = sin ( π 2 ) = sin π 2 = cot 5π 4 = cot [π 2 + (π 2 + π 4 )] = tan [ (π 2 + π 4 )] = tan ( π 2 + π 4 ) = cot ( π 4 ) = cot π 4 = tan 3π 2 = tan (2π π 2 ) = tan ( π 2 ) = tan π 2 = sec 5π 6 = sec (2π π 6 ) = sec ( π 6 ) = sec π 6 = csc 31π 6 = csc (5π + π 6 ) = csc (π + π 6 ) = sec [ (π 2 + π 6 )] = sec ( π 2 + π 6 ) = csc ( π 6 ) = csc π 6 = 1 sin π 6 = = 2.
24 Domains and Ranges of Trigonometric Functions sin R [ 1, 1] cos R [ 1, 1] tan R {kπ + π 2 k Z} R cot R {kπ k Z} R sec R {kπ k Z} (, 1] [1, ) csc R {kπ + π k Z} (, 1] [1, ) 2
25 Signs of Trigonometric Functions Quadrant Positive Functions Negative Functions I all none II sin, csc cos, sec, tan, cot III tan, cot sin, csc, cos, sec IV cos, sec sin, csc, tan, cot
26 Inverse Trigonometric Functions The Inverse Sine Function is the function defined by sin 1 [ 1, 1] [ π 2, π 2 ] sin 1 x = y sin y = x The inverse sine function is also called arcsine, denoted by arcsin.
27 Inverse Trigonometric Functions Examples i sin = π 6. 3 ii arcsin ( 2 ) = π 3. iii sin 1 2 =? Because 2 > 1, 2 is not in the domain of arcsin, so sin 1 2 is not defined. iv arcsin (sin π 4 ) = π 4. v sin 1 (sin 7π 6 ) = sin 1 ( 1 2 ) = π 6.
28 Inverse Trigonometric Functions The Inverse Cosine Function is the function defined by cos 1 [ 1, 1] [0, π] cos 1 x = y cos y = x The inverse cosine function is also called arccosine, denoted by arccos.
29 Inverse Trigonometric Functions The Inverse Tangent Function is the function defined by tan 1 R ( π 2, π 2 ) tan 1 x = y tan y = x The inverse tangent function is also called arctangent, denoted by arctan.
30 Inverse Trigonometric Functions The Inverse Cotangent Function is the function defined by cot 1 R (0, π) cot 1 x = y cot y = x The inverse cotangent function is also called arccotangent, denoted by arccot.
31 Inverse Trigonometric Functions The Inverse Secant Function is the function defined by sec 1 {x R x 1} [0, π 2 ) (π 2, π] sec 1 x = y sec y = x The inverse cosecant function is also called arcsecant, denoted by arcsec.
32 Inverse Trigonometric Functions The Inverse Cosecant Function is the function defined by csc 1 {x R x 1} [ π 2, 0) (0, π 2 ] csc 1 x = y csc y = x The inverse cosecant function is also called arccosecant, denoted by arccsc.
33 Inverse Trigonometric Functions Examples 1. sin 1 cos 2π 3 = sin = π cos 1 tan ( π 4 ) = cos 1 ( 1) = π. 3. tan arcsin 1 2 = tan π 6 = csc cos == csc π 4 = 2.
Find 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 informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
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 informationRight Triangle Trigonometry
Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned
More informationSection 5-9 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
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 informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
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
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationTrigonometry LESSON ONE - Degrees and Radians Lesson Notes
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:
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus 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 information4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles
4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred
More informationRIGHT TRIANGLE TRIGONOMETRY
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
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 informationFunction Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015
Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
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 informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More informationSolutions to Exercises, Section 5.1
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
More information6.1 Basic Right Triangle Trigonometry
6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at
More information1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic
1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single
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 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 informationDear Accelerated Pre-Calculus Student:
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
More informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
More informationChapter 5 Resource Masters
Chapter Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois StudentWorks TM This CD-ROM includes the entire Student Edition along with the Study Guide, Practice,
More informationalternate interior angles
alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
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 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 informationTrigonometry for AC circuits
Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationChapter 5: Trigonometric Functions of Angles
Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has
More informationTrigonometry. An easy way to remember trigonometric properties is:
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.
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 informationLesson 1: Exploring Trigonometric Ratios
Lesson 1: Exploring Trigonometric Ratios Common Core Georgia Performance Standards MCC9 12.G.SRT.6 MCC9 12.G.SRT.7 Essential Questions 1. How are the properties of similar triangles used to create trigonometric
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 informationGraphing Trigonometric Skills
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
More informationINVERSE TRIGONOMETRIC FUNCTIONS. Colin Cox
INVERSE TRIGONOMETRIC FUNCTIONS Colin Cox WHAT IS AN INVERSE TRIG FUNCTION? Used to solve for the angle when you know two sides of a right triangle. For example if a ramp is resting against a trailer,
More informationDOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2
DOE-HDBK-1014/2-92 JUNE 1992 DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2 U.S. Department of Energy Washington, D.C. 20585 FSC-6910 Distribution Statement A. Approved for public release; distribution
More informationUnit 6 Trigonometric Identities, Equations, and Applications
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
More information1. Introduction circular definition Remark 1 inverse trigonometric functions
1. Introduction In Lesson 2 the six trigonometric functions were defined using angles determined by points on the unit circle. This is frequently referred to as the circular definition of the trigonometric
More informationEvaluating trigonometric functions
MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
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 informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
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 informationTrigonometric Functions
Trigonometric Functions 13A Trigonometry and Angles 13-1 Right-Angle Trigonometry 13- Angles of Rotation Lab Explore the Unit Circle 13-3 The Unit Circle 13-4 Inverses of Trigonometric Functions 13B Applying
More informationDavid Bressoud Macalester College, St. Paul, MN. NCTM Annual Mee,ng Washington, DC April 23, 2009
David Bressoud Macalester College, St. Paul, MN These slides are available at www.macalester.edu/~bressoud/talks NCTM Annual Mee,ng Washington, DC April 23, 2009 The task of the educator is to make the
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 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 informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : ANALYTICAL METHODS FOR ENGINEERS Unit code: A/60/40 QCF Level: 4 Credit value: 5 OUTCOME 3 - CALCULUS TUTORIAL DIFFERENTIATION 3 Be able to analyse and model engineering situations and solve problems
More informationTrigonometry Review Workshop 1
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
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
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
More informationLesson Plan. Students will be able to define sine and cosine functions based on a right triangle
Lesson Plan Header: Name: Unit Title: Right Triangle Trig without the Unit Circle (Unit in 007860867) Lesson title: Solving Right Triangles Date: Duration of Lesson: 90 min. Day Number: Grade Level: 11th/1th
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
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
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 informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.
5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationSample Problems. 10. 1 2 cos 2 x = tan2 x 1. 11. tan 2 = csc 2 tan 2 1. 12. sec x + tan x = cos x 13. 14. sin 4 x cos 4 x = 1 2 cos 2 x
Lecture Notes Trigonometric Identities page Sample Problems Prove each of the following identities.. tan x x + sec x 2. tan x + tan x x 3. x x 3 x 4. 5. + + + x 6. 2 sec + x 2 tan x csc x tan x + cot x
More informationTriangle Trigonometry and Circles
Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of
More informationSelf-Paced Study Guide in Trigonometry. March 31, 2011
Self-Paced Study Guide in Trigonometry March 1, 011 1 CONTENTS TRIGONOMETRY Contents 1 How to Use the Self-Paced Review Module Trigonometry Self-Paced Review Module 4.1 Right Triangles..........................
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 informationTRIGONOMETRY FOR ANIMATION
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
More informationWORKBOOK. MATH 30. PRE-CALCULUS MATHEMATICS.
WORKBOOK. MATH 30. PRE-CALCULUS MATHEMATICS. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributor: U.N.Iyer Department of Mathematics and Computer Science, CP 315, Bronx Community College, University
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 informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More information25 The Law of Cosines and Its Applications
Arkansas Tech University MATH 103: Trigonometry Dr Marcel B Finan 5 The Law of Cosines and Its Applications The Law of Sines is applicable when either two angles and a side are given or two sides and an
More informationPRE-CALCULUS GRADE 12
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.
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 informationHigh School Geometry Test Sampler Math Common Core Sampler Test
High School Geometry Test Sampler Math Common Core Sampler Test Our High School Geometry sampler covers the twenty most common questions that we see targeted for this level. For complete tests and break
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More information9 Right Triangle Trigonometry
www.ck12.org CHAPTER 9 Right Triangle Trigonometry Chapter Outline 9.1 THE PYTHAGOREAN THEOREM 9.2 CONVERSE OF THE PYTHAGOREAN THEOREM 9.3 USING SIMILAR RIGHT TRIANGLES 9.4 SPECIAL RIGHT TRIANGLES 9.5
More informationFunctions and their Graphs
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
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationQuestion Bank Trigonometry
Question Bank Trigonometry 3 3 3 3 cos A sin A cos A sin A 1. Prove that cos A sina cos A sina 3 3 3 3 cos A sin A cos A sin A L.H.S. cos A sina cos A sina (cosa sina) (cos A sin A cosa sina) (cosa sina)
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 informationhow to use dual base log log slide rules
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
More informationIntroduction Assignment
PRE-CALCULUS 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 informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More information(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its
(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:
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More informationTHE COMPLEX EXPONENTIAL FUNCTION
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
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More informationCourse outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems)
Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Functions, domain and range Domain and range of rational and algebraic
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 0-1 Sets There are no state-mandated Precalculus 0-2 Operations
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
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)
More informationTrigonometry Hard Problems
Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.
More informationWith the Tan function, you can calculate the angle of a triangle with one corner of 90 degrees, when the smallest sides of the triangle are given:
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
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationThe Deadly Sins of Algebra
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.
More informationRoof Framing Geometry & Trigonometry for Polygons
Roof Framing Geometry & Trigonometry for Polygons section view polygon side length polygon center inscribed circle central angle plan angle working angle plan view plan view projection B angle B exterior
More informationClick here for answers. f x CD 1 2 ( BC AC AB ) 1 2 C. (b) Express da dt in terms of the quantities in part (a). can be greater than.
CHALLENGE PROBLEM CHAPTER 3 A Click here for answers. Click here for solutions.. (a) Find the domain of the function f x s s s3 x. (b) Find f x. ; (c) Check your work in parts (a) and (b) by graphing f
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 informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More information