Algebra and Trig. I. The graph of
|
|
- Margery Stephens
- 6 years ago
- Views:
Transcription
1 Algebra and Trig. I 4.5 Graphs of Sine and Cosine Functions The graph of The graph of. The trigonometric functions can be graphed in a rectangular coordinate system by plotting points whose coordinates satisfy the function. Thus, we graph by listing some points on the graph because the period of the sine function is 2π, we will graph the function on the interval [0,2π]. The rest of the graph is made up of repetitions of this period Period When the values of a function regularly repeat themselves, we say that the function is periodic. The values of the sine function repeat themselves every 2π Amplitude The maximum value of y on the graph of is, the amplitude. If then the curve is stretched, if then the curve is shrunk. The amplitude (half the distance between the maximum and minimum values of the function) will be, since distance is always positive. x 0 π 2π y=sinx In plotting these points we will use approximations like instead of we will use 0.87 and instead of approximating π we will mark off the x-axis in terms of π. 1 P a g e
2 The zeros of y = sin x are at the multiples of π. And it is there that the graph crosses the x-axis. But what is the maximum value of the graph, and what is its minimum value? Sin(x) has a maximum value of 1 at, and has a minimum value of -1 at, and at all angles that are coterminal to them. Here is the x-axis marked off in terms of π, with the y-axis as the approximations of sin(x). Now just connect the curve Amplitude Period 2π 1 cycle The graph of the sine function allows us to visualize some of the properties of the sine function: 1. The range of is (i.e amplitude) 2. The domain is 3. The period is 2π. The graph s pattern repeats in every interval of length 2π. 4. The function is an odd function: this can be seen by noticing that the graph is symmetric with respect to the origin. 2 P a g e
3 When wanting to graph variations of it is helpful in noticing some key points of the graph of. Draw the picture of the sine function below: The key points to notice are the X-intercepts Maximum point Minimum Point One complete cycle of the sine curve includes 3 x-intercepts, one maximum and 1 minimum point. The x-intercepts are at the beginning, middle and end of the cycle. The x-intercepts are at (0,0) (π, 0) (2π, 0) The curve reaches its maximum point of the way through the cycle. The curve reaches its minimum point of the way through the cycle. Thus the key points in graphing sine functions are obtained by dividing the period into 4 equal parts (quarters). 3 P a g e
4 The x-coordinates of the five key points are as follows: Graphing Variations of 1. Identify the amplitude and the period 2. Find the values of x for the five key points the three x- intercepts, the maximum point and the minimum point. Start with the value of x where the cycle begins and add quarter periods that is to find successive values of x 3. Find the values of y for the five key points by evaluating the function at each value of x from step Connect the five key points with a smooth curve and graph one complete cycle of the given function 5. Extend the graph in step 4 to the left or right as desired. 4 P a g e
5 Example Determine the amplitude of for. Then graph 1. = Period = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 5 P a g e
6 Example Determine the amplitude of for. Then graph 1. = Period = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 6 P a g e
7 What to do if the function is of the form, where B is the coefficient of x and. The B indicates the number of periods in an interval of length. The graph of y=sin(bx) Since the graph of has period, then the constant B in indicates the number of periods in an interval of length. (In.) For example, if -- that means there are 2 periods in an interval of length. If B= 3 y = sin 3x -- there are 3 periods in that interval: While if B = ½ y = sin ½x -- there is only half a period in that interval: 7 P a g e
8 The constant B thus signifies how frequently the function oscillates; so many radians per unit of x. Amplitudes and Periods: The graph amplitude = has Amplitude period = Period: The graph of is the graph of horizontally shrunk by a factor of if and horizontally stretched by a factor of if 8 P a g e
9 Example Determine the amplitude and period of. Then graph for 1. = Overall Period = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 9 P a g e
10 The graph of The graph of is obtained by horizontally shifting the graph so that the starting point of the cycle is shifted from. If then shift right. If then shift left. The number is called the phase shift. Amplitudes and Periods and Staring Points: The graph has Starting Point Amplitude Period: amplitude = period = staring point: The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive. 10 P a g e
11 Example Determine the amplitude, period and phase shift of. Then graph one period of the function. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 11 P a g e
12 The graph of. The graph of. The trigonometric functions can be graphed in a rectangular coordinate system by plotting points whose coordinates satisfy the function. Thus, we graph by listing some points on the graph because the period of the cosine function is 2π, we will graph the function on the interval [0,2π]. The rest of the graph is made up of repetitions of this period x 0 π 2π y=cosx You can also use degree measurements: The zeros of that the graph crosses the x-axis. are at the multiples of. And it is there 12 P a g e
13 Here is the x-axis marked off in terms of, with the y-axis as as the approximations of cos(x). Notice that the graph of cox(x) is the graph of sin(x) shifted left by -2π - π 2π Amplitude Period 2π 1 cycle The graph of the cosine function allows us to visualize some of the properties of the cosine function: 1. The range of is (i.e amplitude) 2. The domain is 3. The period is 2π. The graph s pattern repeats in every interval of length 2π. 4. The function is an even function: this can be seen by noticing that the graph is symmetric with respect to the y-axis. Amplitudes and Periods: The graph amplitude = period = has Amplitude Period: 13 P a g e
14 Example Determine the amplitude and period of. Then graph the function for. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 14 P a g e
15 The graph of The graph of is obtained by horizontally shifting the graph so that the starting point of the cycle is shifted from. If then shift right. If then shift left. The number is called the phase shift. Amplitudes and Periods: The graph has Starting Point Amplitude x-axis amplitude = period = Period: starting point: The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive. 15 P a g e
16 Example Determine the amplitude, period and phase shift of. Then graph one period of the function. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 16 P a g e
17 Vertical Shifts and Sinusoidal Graphs We now look at sinusoidal graphs of The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation The constant D in the above formula causes vertical shifts in the graphs of. If D is positive, the shift is D units upward. If D is negative, the shift is D units downward. These vertical shifts result in sinusoidal graphs oscillating about the horizontal line rather than about the x-axis, thus the maximum y is and the minimum is Period: Amplitudes and Periods: The graph amplitude = has Starting Point Amplitude d units period = x-axis starting point: 17 P a g e
18 Example Graph one period of. 1. = Overall Period = Phase Shift = 2. Quarter Length 3. Find x-coordinates 4. Find y-values for x-coordinates 5. Graph the function 18 P a g e
19 Velocity of Air Flow (liters per second) Modeling Periodic Behavior The below graph shows one complete normal breathing cycle. The cycle consists of inhaling and exhaling. It takes place every five seconds. Velocity of air flow is positive when we inhale and negative when we exhale. It is measured in liters per second. If y represents velocity of air flow after x seconds, find a function of the form that models air flow in a normal breathing cycle. Inhaling Time (sec) Exhaling Period 5 seconds 19 P a g e
Section 5.2 Graphs of the Sine and Cosine Functions
Section 5.2 Graphs of the Sine and Cosine Functions We know from previously studying the periodicity of the trigonometric functions that the sine and cosine functions repeat themselves after 2 radians.
More information5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs
Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2
More informationhttp://www.math.utah.edu/~palais/sine.html http://www.ies.co.jp/math/java/trig/index.html http://www.analyzemath.com/function/periodic.html http://math.usask.ca/maclean/sincosslider/sincosslider.html http://www.analyzemath.com/unitcircle/unitcircle.html
More information2.4 Translating Sine and Cosine Functions
www.ck1.org Chapter. Graphing Trigonometric Functions.4 Translating Sine and Cosine Functions Learning Objectives Translate sine and cosine functions vertically and horizontally. Identify the vertical
More informationSection 5.2 Graphs of the Sine and Cosine Functions
A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in
More informationSection 8.4: The Equations of Sinusoidal Functions
Section 8.4: The Equations of Sinusoidal Functions In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation. Transformed
More informationSection 7.6 Graphs of the Sine and Cosine Functions
4 Section 7. Graphs of the Sine and Cosine Functions In this section, we will look at the graphs of the sine and cosine function. The input values will be the angle in radians so we will be using x is
More informationGraphing Sine and Cosine
The problem with average monthly temperatures on the preview worksheet is an example of a periodic function. Periodic functions are defined on p.254 Periodic functions repeat themselves each period. The
More information5.3-The Graphs of the Sine and Cosine Functions
5.3-The Graphs of the Sine and Cosine Functions Objectives: 1. Graph the sine and cosine functions. 2. Determine the amplitude, period and phase shift of the sine and cosine functions. 3. Find equations
More information2.5 Amplitude, Period and Frequency
2.5 Amplitude, Period and Frequency Learning Objectives Calculate the amplitude and period of a sine or cosine curve. Calculate the frequency of a sine or cosine wave. Graph transformations of sine and
More informationGraph of the Sine Function
1 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE 6.3 GRAPHS OF THE SINE AND COSINE Periodic Functions Graph of the Sine Function Graph of the Cosine Function Graphing Techniques, Amplitude, and Period
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle. We noticed how the x and y
More informationThe Sine Function. Precalculus: Graphs of Sine and Cosine
Concepts: Graphs of Sine, Cosine, Sinusoids, Terminology (amplitude, period, phase shift, frequency). The Sine Function Domain: x R Range: y [ 1, 1] Continuity: continuous for all x Increasing-decreasing
More informationSection 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions
Section 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions In this section, we will look at the graphs of the other four trigonometric functions. We will start by examining the tangent
More informationSection 7.1 Graphs of Sine and Cosine
Section 7.1 Graphs of Sine and Cosine OBJECTIVE 1: Understanding the Graph of the Sine Function and its Properties In Chapter 7, we will use a rectangular coordinate system for a different purpose. We
More information5.3 Trigonometric Graphs. Copyright Cengage Learning. All rights reserved.
5.3 Trigonometric Graphs Copyright Cengage Learning. All rights reserved. Objectives Graphs of Sine and Cosine Graphs of Transformations of Sine and Cosine Using Graphing Devices to Graph Trigonometric
More informationSection 8.4 Equations of Sinusoidal Functions soln.notebook. May 17, Section 8.4: The Equations of Sinusoidal Functions.
Section 8.4: The Equations of Sinusoidal Functions Stop Sine 1 In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation.
More information1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle
Pre- Calculus Mathematics 12 5.1 Trigonometric Functions Goal: 1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle Measuring Angles: Angles in Standard
More informationthe input values of a function. These are the angle values for trig functions
SESSION 8: TRIGONOMETRIC FUNCTIONS KEY CONCEPTS: Graphs of Trigonometric Functions y = sin θ y = cos θ y = tan θ Properties of Graphs Shape Intercepts Domain and Range Minimum and maximum values Period
More informationTrigonometric Equations
Chapter Three Trigonometric Equations Solving Simple Trigonometric Equations Algebraically Solving Complicated Trigonometric Equations Algebraically Graphs of Sine and Cosine Functions Solving Trigonometric
More informationWARM UP. 1. Expand the expression (x 2 + 3) Factor the expression x 2 2x Find the roots of 4x 2 x + 1 by graphing.
WARM UP Monday, December 8, 2014 1. Expand the expression (x 2 + 3) 2 2. Factor the expression x 2 2x 8 3. Find the roots of 4x 2 x + 1 by graphing. 1 2 3 4 5 6 7 8 9 10 Objectives Distinguish between
More information6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function.
Math 160 www.timetodare.com Periods of trigonometric functions Definition A function y f ( t) f ( t p) f ( t) 6.4 & 6.5 Graphing Trigonometric Functions = is periodic if there is a positive number p such
More information3. Use your unit circle and fill in the exact values of the cosine function for each of the following angles (measured in radians).
Graphing Sine and Cosine Functions Desmos Activity 1. Use your unit circle and fill in the exact values of the sine function for each of the following angles (measured in radians). sin 0 sin π 2 sin π
More informationAmplitude, Reflection, and Period
SECTION 4.2 Amplitude, Reflection, and Period Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the amplitude of a sine or cosine function. Find the period of a sine or
More informationChapter #2 test sinusoidal function
Chapter #2 test sinusoidal function Sunday, October 07, 2012 11:23 AM Multiple Choice [ /10] Identify the choice that best completes the statement or answers the question. 1. For the function y = sin x,
More informationUnit 3 Unit Circle and Trigonometry + Graphs
HARTFIELD PRECALCULUS UNIT 3 NOTES PAGE 1 Unit 3 Unit Circle and Trigonometry + Graphs (2) The Unit Circle (3) Displacement and Terminal Points (5) Significant t-values Coterminal Values of t (7) Reference
More informationPractice Test 3 (longer than the actual test will be) 1. Solve the following inequalities. Give solutions in interval notation. (Expect 1 or 2.
MAT 115 Spring 2015 Practice Test 3 (longer than the actual test will be) Part I: No Calculators. Show work. 1. Solve the following inequalities. Give solutions in interval notation. (Expect 1 or 2.) a.
More informationYou analyzed graphs of functions. (Lesson 1-5)
You analyzed graphs of functions. (Lesson 1-5) LEQ: How do we graph transformations of the sine and cosine functions & use sinusoidal functions to solve problems? sinusoid amplitude frequency phase shift
More informationUnit 5 Graphing Trigonmetric Functions
HARTFIELD PRECALCULUS UNIT 5 NOTES PAGE 1 Unit 5 Graphing Trigonmetric Functions This is a BASIC CALCULATORS ONLY unit. (2) Periodic Functions (3) Graph of the Sine Function (4) Graph of the Cosine Function
More informationPlease grab the warm up off of the chair in the front of the room and begin working!
Please grab the warm up off of the chair in the front of the room and begin working! add the x! #2 Fix to y = 5cos (2πx 2) + 9 Have your homework out on your desk to be checked. (Pre requisite for graphing
More informationName: Date: Group: Learning Target: I can determine amplitude, period, frequency, and phase shift, given a graph or equation of a periodic function.
Pre-Lesson Assessment Unit 2: Trigonometric Functions Periodic Functions Diagnostic Exam: Page 1 Name: Date: Group: Learning Target: I can determine amplitude, period, frequency, and phase shift, given
More informationof the whole circumference.
TRIGONOMETRY WEEK 13 ARC LENGTH AND AREAS OF SECTORS If the complete circumference of a circle can be calculated using C = 2πr then the length of an arc, (a portion of the circumference) can be found by
More informationGraphs of sin x and cos x
Graphs of sin x and cos x One cycle of the graph of sin x, for values of x between 0 and 60, is given below. 1 0 90 180 270 60 1 It is this same shape that one gets between 60 and below). 720 and between
More information5.4 Graphs of the Sine & Cosine Functions Objectives
Objectives 1. Graph Functions of the Form y = A sin(wx) Using Transformations. 2. Graph Functions of the Form y = A cos(wx) Using Transformations. 3. Determine the Amplitude & Period of Sinusoidal Functions.
More informationMath 1205 Trigonometry Review
Math 105 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x + y =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of
More informationCopyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1
8.3-1 Transformation of sine and cosine functions Sections 8.2 and 8.3 Revisit: Page 142; chapter 4 Section 8.2 and 8.3 Graphs of Transformed Sine and Cosine Functions Graph transformations of y = sin
More informationUnit 6 Test REVIEW Algebra 2 Honors
Unit Test REVIEW Algebra 2 Honors Multiple Choice Portion SHOW ALL WORK! 1. How many radians are in 1800? 10 10π Name: Per: 180 180π 2. On the unit circle shown, which radian measure is located at ( 2,
More informationSECTION 1.5: TRIGONOMETRIC FUNCTIONS
SECTION.5: TRIGONOMETRIC FUNCTIONS The Unit Circle The unit circle is the set of all points in the xy-plane for which x + y =. Def: A radian is a unit for measuring angles other than degrees and is measured
More informationWhat is a Sine Function Graph? U4 L2 Relate Circle to Sine Activity.pdf
Math 3 Unit 6, Trigonometry L04: Amplitude and Period of Sine and Cosine AND Translations of Sine and Cosine Functions WIMD: What I must do: I will find the amplitude and period from a graph of the sine
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More informationPrecalculus Second Semester Final Review
Precalculus Second Semester Final Review This packet will prepare you for your second semester final exam. You will find a formula sheet on the back page; these are the same formulas you will receive for
More informationPREREQUISITE/PRE-CALCULUS REVIEW
PREREQUISITE/PRE-CALCULUS REVIEW Introduction This review sheet is a summary of most of the main topics that you should already be familiar with from your pre-calculus and trigonometry course(s), and which
More informationChapter 4 Trigonometric Functions
Chapter 4 Trigonometric Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Section 8 Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry
More information1 Graphs of Sine and Cosine
1 Graphs of Sine and Cosine Exercise 1 Sketch a graph of y = cos(t). Label the multiples of π 2 and π 4 on your plot, as well as the amplitude and the period of the function. (Feel free to sketch the unit
More informationUnit 8 Trigonometry. Math III Mrs. Valentine
Unit 8 Trigonometry Math III Mrs. Valentine 8A.1 Angles and Periodic Data * Identifying Cycles and Periods * A periodic function is a function that repeats a pattern of y- values (outputs) at regular intervals.
More informationSection 5.1 Angles and Radian Measure. Ever Feel Like You re Just Going in Circles?
Section 5.1 Angles and Radian Measure Ever Feel Like You re Just Going in Circles? You re riding on a Ferris wheel and wonder how fast you are traveling. Before you got on the ride, the operator told you
More informationHow 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 informationName: Period: Date: Math Lab: Explore Transformations of Trig Functions
Name: Period: Date: Math Lab: Explore Transformations of Trig Functions EXPLORE VERTICAL DISPLACEMENT 1] Graph 2] Explain what happens to the parent graph when a constant is added to the sine function.
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More informationGraphing Trig Functions. Objectives: Students will be able to graph sine, cosine and tangent functions and translations of these functions.
Graphing Trig Functions Name: Objectives: Students will be able to graph sine, cosine and tangent functions and translations of these functions. y = sinx (0,) x 0 sinx (,0) (0, ) (,0) /2 3/2 /2 3/2 2 x
More informationLesson 8.3: The Graphs of Sinusoidal Functions, page 536
. The graph of sin x repeats itself after it passes through 360 or π. 3. e.g. The graph is symmetrical along the x-axis, with the axis of symmetry being at 90 and 70, respectively. The graph is rotationally
More informationPrecalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor
Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor As we studied last section points may be described in polar form or rectangular form. Likewise an equation may be written using either
More informationIntroduction to Trigonometry. Algebra 2
Introduction to Trigonometry Algebra 2 Angle Rotation Angle formed by the starting and ending positions of a ray that rotates about its endpoint Use θ to represent the angle measure Greek letter theta
More information2. Be able to evaluate a trig function at a particular degree measure. Example: cos. again, just use the unit circle!
Study Guide for PART II of the Fall 18 MAT187 Final Exam NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will be
More informationChapter 7 Repetitive Change: Cyclic Functions
Chapter 7 Repetitive Change: Cyclic Functions 7.1 Cycles and Sine Functions Data that is periodic may often be modeled by trigonometric functions. This chapter will help you use Excel to deal with periodic
More informationFunctions Modeling Change A Preparation for Calculus Third Edition
Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1 CHAPTER
More informationSection 8.1 Radians and Arc Length
Section 8. Radians and Arc Length Definition. An angle of radian is defined to be the angle, in the counterclockwise direction, at the center of a unit circle which spans an arc of length. Conversion Factors:
More information2009 A-level Maths Tutor All Rights Reserved
2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents radians 3 sine, cosine & tangent 7 cosecant, secant & cotangent
More informationGraphs of other Trigonometric Functions
Graphs of other Trigonometric Functions Now we will look at other types of graphs: secant. tan x, cot x, csc x, sec x. We will start with the cosecant and y csc x In order to draw this graph we will first
More informationCHAPTER 14 ALTERNATING VOLTAGES AND CURRENTS
CHAPTER 4 ALTERNATING VOLTAGES AND CURRENTS Exercise 77, Page 28. Determine the periodic time for the following frequencies: (a) 2.5 Hz (b) 00 Hz (c) 40 khz (a) Periodic time, T = = 0.4 s f 2.5 (b) Periodic
More informationPythagorean Identity. Sum and Difference Identities. Double Angle Identities. Law of Sines. Law of Cosines
Review for Math 111 Final Exam The final exam is worth 30% (150/500 points). It consists of 26 multiple choice questions, 4 graph matching questions, and 4 short answer questions. Partial credit will be
More informationSecondary Math Amplitude, Midline, and Period of Waves
Secondary Math 3 7-6 Amplitude, Midline, and Period of Waves Warm UP Complete the unit circle from memory the best you can: 1. Fill in the degrees 2. Fill in the radians 3. Fill in the coordinates in the
More informationMath 1330 Precalculus Electronic Homework (EHW 6) Sections 5.1 and 5.2.
Math 0 Precalculus Electronic Homework (EHW 6) Sections 5. and 5.. Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in
More informationMath 102 Key Ideas. 1 Chapter 1: Triangle Trigonometry. 1. Consider the following right triangle: c b
Math 10 Key Ideas 1 Chapter 1: Triangle Trigonometry 1. Consider the following right triangle: A c b B θ C a sin θ = b length of side opposite angle θ = c length of hypotenuse cosθ = a length of side adjacent
More informationTrigonometry. An Overview of Important Topics
Trigonometry An Overview of Important Topics 1 Contents Trigonometry An Overview of Important Topics... 4 UNDERSTAND HOW ANGLES ARE MEASURED... 6 Degrees... 7 Radians... 7 Unit Circle... 9 Practice Problems...
More informationCalculus for the Life Sciences
Calculus for the Life Sciences Lecture Notes Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego
More informationReady To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine
14A Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine Find these vocabulary words in Lesson 14-1 and the Multilingual Glossary. Vocabulary periodic function cycle period amplitude frequency
More information4-4 Graphing Sine and Cosine Functions
Describe how the graphs of f (x) and g(x) are related. Then find the amplitude of g(x), and sketch two periods of both functions on the same coordinate axes. 1. f (x) = sin x; g(x) = sin x The graph of
More informationGRAPHING TRIGONOMETRIC FUNCTIONS
GRAPHING TRIGONOMETRIC FUNCTIONS Section.6B Precalculus PreAP/Dual, Revised 7 viet.dang@humbleisd.net 8//8 : AM.6B: Graphing Trig Functions REVIEW OF GRAPHS 8//8 : AM.6B: Graphing Trig Functions A. Equation:
More informationTrigonometry Review Page 1 of 14
Trigonometry Review Page of 4 Appendix D has a trigonometric review. This material is meant to outline some of the proofs of identities, help you remember the values of the trig functions at special values,
More informationMATH 1113 Exam 3 Review. Fall 2017
MATH 1113 Exam 3 Review Fall 2017 Topics Covered Section 4.1: Angles and Their Measure Section 4.2: Trigonometric Functions Defined on the Unit Circle Section 4.3: Right Triangle Geometry Section 4.4:
More informationUnit 5 Investigating Trigonometry Graphs
Mathematics IV Frameworks Student Edition Unit 5 Investigating Trigonometry Graphs 1 st Edition Table of Contents INTRODUCTION:... 3 What s Your Temperature? Learning Task... Error! Bookmark not defined.
More informationIn Exercises 1-12, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function.
0.5 Graphs of the Trigonometric Functions 809 0.5. Eercises In Eercises -, graph one ccle of the given function. State the period, amplitude, phase shift and vertical shift of the function.. = sin. = sin.
More informationMathematics UNIT FIVE Trigonometry II. Unit. Student Workbook. Lesson 1: Trigonometric Equations Approximate Completion Time: 4 Days
Mathematics 0- Student Workbook Unit 5 Lesson : Trigonometric Equations Approximate Completion Time: 4 Days Lesson : Trigonometric Identities I Approximate Completion Time: 4 Days Lesson : Trigonometric
More information1 Mathematical Methods Units 1 and 2
Mathematical Methods Units and Further trigonometric graphs In this section, we will discuss graphs of the form = a sin ( + c) + d and = a cos ( + c) + d. Consider the graph of = sin ( ). The following
More information4.4 Graphs of Sine and Cosine: Sinusoids
350 CHAPTER 4 Trigonometric Functions What you ll learn about The Basic Waves Revisited Sinusoids and Transformations Modeling Periodic Behavior with Sinusoids... and why Sine and cosine gain added significance
More informationExtra Practice for Section I: Chapter 4
Haberman MTH 112 Extra Practice for Section I: Chapter You should complete all of these problems without a calculator in order to prepare for the Midterm which is a no-calculator exam. 1. Find two different
More informationTHE SINUSOIDAL WAVEFORM
Chapter 11 THE SINUSOIDAL WAVEFORM The sinusoidal waveform or sine wave is the fundamental type of alternating current (ac) and alternating voltage. It is also referred to as a sinusoidal wave or, simply,
More informationAlgebra 2/Trigonometry Review Sessions 1 & 2: Trigonometry Mega-Session. The Unit Circle
Algebra /Trigonometry Review Sessions 1 & : Trigonometry Mega-Session Trigonometry (Definition) - The branch of mathematics that deals with the relationships between the sides and the angles of triangles
More informationChapter 2: Functions and Graphs Lesson Index & Summary
Section 1: Relations and Graphs Cartesian coordinates Screen 2 Coordinate plane Screen 2 Domain of relation Screen 3 Graph of a relation Screen 3 Linear equation Screen 6 Ordered pairs Screen 1 Origin
More informationName: Which equation is represented in the graph? Which equation is represented by the graph? 1. y = 2 sin 2x 2. y = sin x. 1.
Name: Print Close Which equation is represented in the graph? Which equation is represented by the graph? y = 2 sin 2x y = sin x y = 2 sin x 4. y = sin 2x Which equation is represented in the graph? 4.
More informationTrigonometric identities
Trigonometric identities An identity is an equation that is satisfied by all the values of the variable(s) in the equation. For example, the equation (1 + x) = 1 + x + x is an identity. If you replace
More information5-5 Multiple-Angle and Product-to-Sum Identities
Find the values of sin 2, cos 2, tan 2 1 cos for the given value interval, (270, 360 ) Since on the interval (270, 360 ), one point on the terminal side of θ has x-coordinate 3 a distance of 5 units from
More informationMATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) (sin x + cos x) 1 + sin x cos x =? 1) ) sec 4 x + sec x tan x - tan 4 x =? ) ) cos
More informationMathematics Lecture. 3 Chapter. 1 Trigonometric Functions. By Dr. Mohammed Ramidh
Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions By Dr. Mohammed Ramidh Trigonometric Functions This section reviews the basic trigonometric functions. Trigonometric functions are important because
More informationTrig functions are examples of periodic functions because they repeat. All periodic functions have certain common characteristics.
Trig functions are examples of periodic functions because they repeat. All periodic functions have certain common characteristics. The sine wave is a common term for a periodic function. But not all periodic
More information7.3 The Unit Circle Finding Trig Functions Using The Unit Circle Defining Sine and Cosine Functions from the Unit Circle
7.3 The Unit Circle Finding Trig Functions Using The Unit Circle For any angle t, we can label the intersection of the terminal side and the unit circle as by its coordinates,(x,y).the coordinates x and
More informationSection 2.4 General Sinusoidal Graphs
Section. General Graphs Objective: any one of the following sets of information about a sinusoid, find the other two: ) the equation ) the graph 3) the amplitude, period or frequency, phase displacement,
More information7.1 INTRODUCTION TO PERIODIC FUNCTIONS
7.1 INTRODUCTION TO PERIODIC FUNCTIONS *SECTION: 6.1 DCP List: periodic functions period midline amplitude Pg 247- LECTURE EXAMPLES: Ferris wheel, 14,16,20, eplain 23, 28, 32 *SECTION: 6.2 DCP List: unit
More informationMAT01A1. Appendix D: Trigonometry
MAT01A1 Appendix D: Trigonometry Dr Craig 12 February 2019 Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: C-Ring 508 acraig@uj.ac.za Web: http://andrewcraigmaths.wordpress.com
More informationPrecalculus ~ Review Sheet
Period: Date: Precalculus ~ Review Sheet 4.4-4.5 Multiple Choice 1. The screen below shows the graph of a sound recorded on an oscilloscope. What is the period and the amplitude? (Each unit on the t-axis
More informationFind all the remaining sides, angles and area of the following triangles
Trigonometry Angles of Elevation and depression 1) If the angle of elevation of the top of a vertical 30m high aerial is 32, how is it to the aerial? 2) From the top of a vertical cliff 80m high the angles
More informationName Date Class. Identify whether each function is periodic. If the function is periodic, give the period
Name Date Class 14-1 Practice A Graphs of Sine and Cosine Identify whether each function is periodic. If the function is periodic, give the period. 1.. Use f(x) = sinx or g(x) = cosx as a guide. Identify
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Trigonometry Final Exam Study Guide Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a polar equation is given. Select the polar
More informationSolutions to Exercises, Section 5.6
Instructor s Solutions Manual, Section 5.6 Exercise 1 Solutions to Exercises, Section 5.6 1. For θ = 7, evaluate each of the following: (a) cos 2 θ (b) cos(θ 2 ) [Exercises 1 and 2 emphasize that cos 2
More information1 Trigonometry. Copyright Cengage Learning. All rights reserved.
1 Trigonometry Copyright Cengage Learning. All rights reserved. 1.2 Trigonometric Functions: The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives Identify a unit circle and describe
More informationPhasor. Phasor Diagram of a Sinusoidal Waveform
Phasor A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates. Generally, vectors
More informationChapter 3, Part 1: Intro to the Trigonometric Functions
Haberman MTH 11 Section I: The Trigonometric Functions Chapter 3, Part 1: Intro to the Trigonometric Functions In Example 4 in Section I: Chapter, we observed that a circle rotating about its center (i.e.,
More informationTrigonometry LESSON ONE - Degrees and Radians Lesson Notes
8 = 6 Trigonometry LESSON ONE - Degrees and Radians Example : Define each term or phrase and draw a sample angle. Angle in standard position. b) Positive and negative angles. Draw. c) Reference angle.
More informationChapter 4/5 Part 2- Trig Identities and Equations
Chapter 4/5 Part 2- Trig Identities and Equations Lesson Package MHF4U Chapter 4/5 Part 2 Outline Unit Goal: By the end of this unit, you will be able to solve trig equations and prove trig identities.
More information