Period of Trigonometric Functions

Size: px
Start display at page:

Download "Period of Trigonometric Functions"

Transcription

1 Period of Trigonometric Functions In previous lessons we have learned how to translate any primary trigonometric function horizontally or vertically, and how to Stretch Vertically (change Amplitude). In this unit we are going to learn how to Stretch Horizontally. In most Real World applications that involve trigonometry, what we are doing is determining the trigonometric function s Period. In order to master the techniques explained here it is very important that you undertake plenty of practice exercises so that they become second nature. After reading this unit, you should be able to complete the following: Determine the period for Sine, Cosine, and Tangent Sketch the function on blank grid Sketch the function on fixed grid Recall: The three basic trigonometric functions have periods as demonstrated below: Sine function -> period is radians or 360. Cosine function -> period is radians or 360. Tangent function -> period is radians or 180. The basic graphs of these 3 trigonometric functions are:

2 The length of one complete cycle of a trigonometric function is called the Period. Typically we use x=0 as the starting point for the graph. When the length of the period f t sin kt, is not the default, the functions will be written in the format similar to: g x coskx, or h tank. Where the constant k aids us in determining the period for the function. The Period (wavelength) of sin(kt) and cos(kt) If k>0, then the graph of f t sinkt or f t cos kt between 0 and, and each functions has a period of: Period= k makes k complete cycles Note: we are working in radians, else 360 k

3 The Period of tan(kt) If k>0, then the graph of f t tan kt and, and each functions has a period of: makes k complete cycles between 0 Period= k Note: we are working in radians, else 360 k Example: Determine the period of each of the functions: a) y sin 3 x f t 3 k tan b) cos t c) Solutions: a) period = 3 We have 3 complete cycles between 0 and b) period = 1

4 ½ cycle between 0 and c) period = 3 3 Note: A calculator may not produce an accurate graph of trigonometric functions f t sin 50t has 50 with a large k value. For example the graph of complete cycles between 0 and, but some calculators have problems showing this (try it).

5 Graphing The Sine and Cosine Trigonometric Functions by Hand When we are asked to graph trigonometric functions by hand, two types of questions are usually presented to you: The first being that you have full control. A blank piece of graph paper is provided and you are to sketch your graph on the grid. You may be asked to one cycle or a set number of cycles, so use logic to guide you. It is best to use 1 blank spaces or a multiple of 1 blank spaces for your cycle. Use logic to determine which is best to provide the number of required cycles. With 1 spaces (or multiple of 1) you can easily find the intervals (or quarters) that will correspond to the 3 0,,,, values of either the sine or cosine function. locations that would have a value of {-1, 0, 1} The second being that the grid with its domain is already provided for you. Here you have to conform to its restrictions when graphing the function. There is a bit more work here, but a fun puzzle to solve. Type 1: When you are given a blank grid. These are the easiest to graph for you are given a blank grid and have full control, so make it easy for yourself. If you are given no domain restrictions and you need to draw one complete cycle. Step 1: Determine the Period Step : Let 1 spaces on the grid represent the period. Step 3: Determine the ¼ period, ½ period, ¾ period values for your function (these will be 3 spaces apart on your grid). Step : Plot your sin or cos graph by using (0, 1, 0, -1, 0, 1, 0, -1) amplitude points and follow the pattern. If you need more than one cycle, see how many multiples of 1 you can use on your grid. Use logic to assist you, as you will need to break each cycle in to quarters, so you may have to use 8 grid spaces, or grid spaces per cycle. Some questions may require you to plot both positive and neagative domain values. If this is the case ensure you provide a symmetric axis location. Example: Graph f t sin t when given a blank grid Solution: Determine the period:

6 Now horizontal grid markers will be at ¼ period, ½ period, ¾ period ( Q1) Q) Q3) Since this is a sine function. The f(t) values will (starting at t=0) follow the pattern {0, 1, 0, -1, 0} at the quarter period t-values. 3 Plot the following 5 points: 0,0,,1,,0,, 1,,0 Now connect the dots to complete the graph. Example: Graph y cos x 10 when given a blank grid. Solution: Determine the Period: Now horizontal grid markers will be at ¼ period, ½ period, ¾ period

7 1 1 3 Q1) 0 5 Q) 0 10 Q3) 0 15 Use these values to label your grid. Since this is a cosine function. The y-values will (starting at x=0) follow the pattern {1, 0, -1, 0, 1} at the quarter period x-values. Plot the following 5 points: 0,1, 5,0, 10, 1, 15,0, 0,1 Type : When you are given a labelled grid. These are harder to do; but by following the technique below you will find these are actually fun to plot. Remember the teacher or textbook will provide questions that fit nicely into the grid that they provide. This makes your job very easy. If possible, try to provide 1 blank spaces for the cycle (1 is divisible by, 3,, 6 and thus gives you more room to play). We will be using the quarters of the cycle, thus if 1 spaces for each cycle does not fit into the grid, try 8, or even. Again have fun with the logic to make the graph fit into the set grid.

8 Step 1: Determine blank space width by domain divided by 1. Step : Step 3: Step : Step 5: Step 6: Determine the period. If your teacher set up the question correctly, the period should be a multiple of blank grid squares. Count the numbers of grid squares for a complete cycle. Use period blank square width. Take Step s value and divide by, this new result will be the numbers of squares each quarter of the function will take for the pattern (, 0, 1, 0, - 1, 0, 1, 0, ). Mark these quarter place markers on your grid starting at 0. Place the (, 0, 1, 0, -1, 0, 1, 0, ) at each marked of spot on your grid, remember each quarter will be Step number of spaces. You should label your grid at the ¼ period, ½ period, ¾ period values for your function. Example: Graph x y sin 3 when domain is 0 x Solution: Our blank space width: domain The period of the graph is: 3 The number of squares the period occupies is: period black square width 3 6 The number of spaces per quarter is 1. Therefore every grid spaces the function will complete a full cycle, with 1 space providing the quarter function point at the { 0, 1, 0, -1, 0, } pattern. This gives points at: 1 3 0,0,,1,,0,, 1,,

9 Let s graph it demonstrating the k=3 complete cycles. Example x y cos 3 when domain is x and the grid is provided as below. Solution: Since we have 60 spaces on the horizontal axis, and we would prefer 1, we will use 5 spaces per major tick. Our blank space width : domain The period of the graph is: 6 1 3

10 The number of squares a complete cycle occupies is: period 6 6 space width Therefore every grid spaces (with the grid only providing 1 spaces) the function will complete a cycle with 6 spaces providing the quarter function point interval for the { 0, 1, 0, -1, 0, } pattern. In this question we will only be able to graph 1 point (the ¼ period value). Since this is a cosine function, start at x=0 with (0,1) then its next point will be 6 grid 3,0,0. space from 0 at the location 6 Let s graph it. Example Graph f sin for on the following grid. 6

11 Solution: The grid spacing does not look nice. We have 18 major horizontal tick marks and 5 space-per-tick mark (giving a total of 90 horizontal spaces). Let s first find the period of our graph. Period = Now since we have 18 horizontal ticks over a domain of 3, the width of each horizontal tick is Oh no, it looks like we have a problem. The period of the horizontal tick spacing of does not work well with 5. We could use our calculator to approximate the 6 locations, or remember that the question is set up to work on the given grid (most times). What about those 5 space-per-tick mark for a total of 90 spaces, can we use them? The width of each space is Now how many space does it take for the period of? 5 period 30 Number of spaces per cycle = 5 space width 5 30 Hey, this is a nice number to work with. Each quarter cycle has a length of 6 spaces. Since this is a Sine Function, it starts at (0,0) and each 6 spaces will follow the usual Sine pattern. We can use the horizontal tick width of the graph for reference to improve readability. to place labels on 6

How to Graph Trigonometric Functions

How to Graph Trigonometric Functions 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

More information

2.2 Derivative as a Function

2.2 Derivative as a Function 2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x

More information

Trigonometry 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 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 information

Graphs of Polar Equations

Graphs of Polar Equations Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate

More information

Introduction Assignment

Introduction 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 information

Evaluating trigonometric functions

Evaluating 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 information

G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

More information

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Prentice 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 information

Trigonometry Hard Problems

Trigonometry 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 information

Algebra and Geometry Review (61 topics, no due date)

Algebra 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 information

SOLVING TRIGONOMETRIC EQUATIONS

SOLVING 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 information

MCR3U - Practice Test - Periodic Functions - W2012

MCR3U - Practice Test - Periodic Functions - W2012 Name: Date: May 25, 2012 ID: A MCR3U - Practice Test - Periodic Functions - W2012 1. One cycle of the graph of a periodic function is shown below. State the period and amplitude. 2. One cycle of the graph

More information

Finding Equations of Sinusoidal Functions From Real-World Data

Finding Equations of Sinusoidal Functions From Real-World Data Finding Equations of Sinusoidal Functions From Real-World Data **Note: Throughout this handout you will be asked to use your graphing calculator to verify certain results, but be aware that you will NOT

More information

Pre-Calculus Math 12 First Assignment

Pre-Calculus Math 12 First Assignment Name: Pre-Calculus Math 12 First Assignment This assignment consists of two parts, a review of function notation and an introduction to translating graphs of functions. It is the first work for the Pre-Calculus

More information

PRE-CALCULUS GRADE 12

PRE-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 information

0 Introduction to Data Analysis Using an Excel Spreadsheet

0 Introduction to Data Analysis Using an Excel Spreadsheet Experiment 0 Introduction to Data Analysis Using an Excel Spreadsheet I. Purpose The purpose of this introductory lab is to teach you a few basic things about how to use an EXCEL 2010 spreadsheet to do

More information

a cos x + b sin x = R cos(x α)

a cos x + b sin x = R cos(x α) a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this

More information

GeoGebra. 10 lessons. Gerrit Stols

GeoGebra. 10 lessons. Gerrit Stols GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter

More information

Trigonometric Functions: The Unit Circle

Trigonometric 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 information

Tutorial 2: Using Excel in Data Analysis

Tutorial 2: Using Excel in Data Analysis Tutorial 2: Using Excel in Data Analysis This tutorial guide addresses several issues particularly relevant in the context of the level 1 Physics lab sessions at Durham: organising your work sheet neatly,

More information

Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

Parallel 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 information

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Geometry 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 information

Algebra. 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. 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 information

PROVINCE OF THE EASTERN CAPE EDUCATION

PROVINCE OF THE EASTERN CAPE EDUCATION PROVINCE OF THE EASTERN CAPE EDUCATION DIRECTORATE: CURRICULUM FET PROGRAMMES LESSON PLANS TERM 3 MATHEMATICS GRADE 10 FOREWORD The following Grade 10, 11 and 12 Lesson Plans were developed by Subject

More information

Using Excel to Execute Trigonometric Functions

Using Excel to Execute Trigonometric Functions In this activity, you will learn how Microsoft Excel can compute the basic trigonometric functions (sine, cosine, and tangent) using both radians and degrees. 1. Open Microsoft Excel if it s not already

More information

Advanced Math Study Guide

Advanced Math Study Guide Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular

More information

Higher Education Math Placement

Higher 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 information

Trigonometry Review Workshop 1

Trigonometry 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 information

http://school-maths.com Gerrit Stols

http://school-maths.com Gerrit Stols For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It

More information

Week 13 Trigonometric Form of Complex Numbers

Week 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 information

MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:

MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module: 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

More information

Unit 6 Trigonometric Identities, Equations, and Applications

Unit 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 information

Biggar 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 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 information

How To Solve The Pythagorean Triangle

How 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 information

2.5 Transformations of Functions

2.5 Transformations of Functions 2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [

More information

Dear Accelerated Pre-Calculus Student:

Dear 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 information

Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring

Right 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 information

Right Triangles 4 A = 144 A = 16 12 5 A = 64

Right 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 information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.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 information

Section 5-9 Inverse Trigonometric Functions

Section 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 information

Geometry Notes PERIMETER AND AREA

Geometry Notes PERIMETER AND AREA 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

More information

Semester 2, Unit 4: Activity 21

Semester 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 information

Objective: Use calculator to comprehend transformations.

Objective: Use calculator to comprehend transformations. math111 (Bradford) Worksheet #1 Due Date: Objective: Use calculator to comprehend transformations. Here is a warm up for exploring manipulations of functions. specific formula for a function, say, Given

More information

Graphing Quadratic Functions

Graphing Quadratic Functions Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

More information

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM. 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 information

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

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 information

TRIGONOMETRY Compound & Double angle formulae

TRIGONOMETRY Compound & Double angle formulae TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae

More information

Core Maths C3. Revision Notes

Core 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 information

Graphing Trigonometric Skills

Graphing 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 information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

Trigonometry. An easy way to remember trigonometric properties is:

Trigonometry. 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 information

Basic Understandings

Basic Understandings Activity: TEKS: Exploring Transformations Basic understandings. (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential to understanding underlying

More information

SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen

SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions

More information

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Angles 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 information

Solutions to Exercises, Section 5.1

Solutions 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 information

Chapter 8 Geometry We will discuss following concepts in this chapter.

Chapter 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 information

Graphical Integration Exercises Part Four: Reverse Graphical Integration

Graphical Integration Exercises Part Four: Reverse Graphical Integration D-4603 1 Graphical Integration Exercises Part Four: Reverse Graphical Integration Prepared for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Laughton Stanley

More information

2. Select Point B and rotate it by 15 degrees. A new Point B' appears. 3. Drag each of the three points in turn.

2. Select Point B and rotate it by 15 degrees. A new Point B' appears. 3. Drag each of the three points in turn. In this activity you will use Sketchpad s Iterate command (on the Transform menu) to produce a spiral design. You ll also learn how to use parameters, and how to create animation action buttons for parameters.

More information

Lesson Plan Teacher: G Johnson Date: September 20, 2012.

Lesson Plan Teacher: G Johnson Date: September 20, 2012. Lesson Plan Teacher: G Johnson Date: September 20, 2012. Subject: Mathematics Class: 11L Unit: Trigonometry Duration: 1hr: 40mins Topic: Using Pythagoras Theorem to solve trigonometrical problems Previous

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL 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 information

With 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:

With 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 information

2 Session Two - Complex Numbers and Vectors

2 Session Two - Complex Numbers and Vectors PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

More information

www.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

www.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 information

Algebra 2: Themes for the Big Final Exam

Algebra 2: Themes for the Big Final Exam Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,

More information

Lesson 3 Using the Sine Function to Model Periodic Graphs

Lesson 3 Using the Sine Function to Model Periodic Graphs Lesson 3 Using the Sine Function to Model Periodic Graphs Objectives After completing this lesson you should 1. Know that the sine function is one of a family of functions which is used to model periodic

More information

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

1 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 information

Section 5.4 More Trigonometric Graphs. Graphs of the Tangent, Cotangent, Secant, and Cosecant Function

Section 5.4 More Trigonometric Graphs. Graphs of the Tangent, Cotangent, Secant, and Cosecant Function Section 5. More Trigonometric Graphs Graphs of the Tangent, Cotangent, Secant, and Cosecant Function 1 REMARK: Many curves have a U shape near zero. For example, notice that the functions secx and x +

More information

1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic

1. 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 information

Triangle Trigonometry and Circles

Triangle 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 information

Algebra II and Trigonometry

Algebra II and Trigonometry Algebra II and Trigonometry Textbooks: Algebra 2: California Publisher: McDougal Li@ell/Houghton Mifflin (2006 EdiHon) ISBN- 13: 978-0618811816 Course descriphon: Algebra II complements and expands the

More information

Functions and their Graphs

Functions 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 information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1 Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

More information

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were: Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell 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 information

CGE 3b 2 What s My Ratio? The Investigate the three primary trigonometric ratios for right-angled MT2.01 triangles. Summarize investigations.

CGE 3b 2 What s My Ratio? The Investigate the three primary trigonometric ratios for right-angled MT2.01 triangles. Summarize investigations. Unit 2 Trigonometry Lesson Outline Grade 10 Applied BIG PICTURE Students will: investigate the relationships involved in right-angled triangles to the primary trigonometric ratios, connecting the ratios

More information

High School Geometry Test Sampler Math Common Core Sampler Test

High 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 information

Grade 12 Pre-Calculus Mathematics (40S) A Course for Independent Study

Grade 12 Pre-Calculus Mathematics (40S) A Course for Independent Study Grade 1 Pre-Calculus Mathematics (40S) A Course for Independent Study GRADE 1 PRE-CALCULUS MATHEMATICS (40S) A Course for Independent Study 007 Manitoba Education, Citizenship and Youth Manitoba Education,

More information

GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

More information

TRANSFORMATIONS OF GRAPHS

TRANSFORMATIONS OF GRAPHS Mathematics Revision Guides Transformations of Graphs Page 1 of 24 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C1 Edexcel: C1 OCR: C1 OCR MEI: C1 TRANSFORMATIONS OF GRAPHS Version

More information

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.

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. 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 information

SIGNAL PROCESSING & SIMULATION NEWSLETTER

SIGNAL PROCESSING & SIMULATION NEWSLETTER 1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

More information

OA3-10 Patterns in Addition Tables

OA3-10 Patterns in Addition Tables OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20

More information

Grade 7/8 Math Circles November 3/4, 2015. M.C. Escher and Tessellations

Grade 7/8 Math Circles November 3/4, 2015. M.C. Escher and Tessellations Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Tiling the Plane Grade 7/8 Math Circles November 3/4, 2015 M.C. Escher and Tessellations Do the following

More information

Drawing a histogram using Excel

Drawing a histogram using Excel Drawing a histogram using Excel STEP 1: Examine the data to decide how many class intervals you need and what the class boundaries should be. (In an assignment you may be told what class boundaries to

More information

Core Maths C2. Revision Notes

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 information

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx 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.

More information

Updates to Graphing with Excel

Updates to Graphing with Excel Updates to Graphing with Excel NCC has recently upgraded to a new version of the Microsoft Office suite of programs. As such, many of the directions in the Biology Student Handbook for how to graph with

More information

National 5 Mathematics Course Assessment Specification (C747 75)

National 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 information

Examples of Data Representation using Tables, Graphs and Charts

Examples of Data Representation using Tables, Graphs and Charts Examples of Data Representation using Tables, Graphs and Charts This document discusses how to properly display numerical data. It discusses the differences between tables and graphs and it discusses various

More information

X On record with the USOE.

X On record with the USOE. Textbook Alignment to the Utah Core Algebra 2 Name of Company and Individual Conducting Alignment: Chris McHugh, McHugh Inc. A Credential Sheet has been completed on the above company/evaluator and is

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

Gestation Period as a function of Lifespan

Gestation Period as a function of Lifespan This document will show a number of tricks that can be done in Minitab to make attractive graphs. We work first with the file X:\SOR\24\M\ANIMALS.MTP. This first picture was obtained through Graph Plot.

More information

RIGHT TRIANGLE TRIGONOMETRY

RIGHT 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 information

Dealing with Data in Excel 2010

Dealing with Data in Excel 2010 Dealing with Data in Excel 2010 Excel provides the ability to do computations and graphing of data. Here we provide the basics and some advanced capabilities available in Excel that are useful for dealing

More information

Electrical Resonance

Electrical Resonance Electrical Resonance (R-L-C series circuit) APPARATUS 1. R-L-C Circuit board 2. Signal generator 3. Oscilloscope Tektronix TDS1002 with two sets of leads (see Introduction to the Oscilloscope ) INTRODUCTION

More information

Page. Trigonometry Sine Law and Cosine Law. push

Page. Trigonometry Sine Law and Cosine Law. push Trigonometry Sine Law and Cosine Law Page Trigonometry can be used to calculate the side lengths and angle measures of triangles. Triangular shapes are used in construction to create rigid structures.

More information

Oxford Cambridge and RSA Examinations

Oxford 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 information