GRAPHING TRIGONOMETRIC FUNCTIONS
|
|
- Nora Gabriella Cole
- 5 years ago
- Views:
Transcription
1 GRAPHING TRIGONOMETRIC FUNCTIONS Section.6B Precalculus PreAP/Dual, Revised 7 viet.dang@humbleisd.net 8//8 : AM.6B: Graphing Trig Functions
2 REVIEW OF GRAPHS 8//8 : AM.6B: Graphing Trig Functions
3 A. Equation: = A trig function B C + D B. A is the amplitude. a: verticall stretches b a factor of a,. a : Verticall compresses b a factor of /a C. B is the period or frequenc. equation: π for sine and cosine, π for tangent B B. B: phase compresses b a factor of π B 3. B : phasel stretches b a factor of b D. C is the phase shift TRANSFORMATIONS.If there no GCF taken out, divide the coefficient E. D is the vertical shift F. Frequenc is defined as the number of ccles per second 8//8 : AM.6B: Graphing Trig Functions 3
4 STEPS A. Identif A, B, C, and D from the equation, = A trig B C + D B. Identif the phase shift. : π B or π B (for Tan and Cot onl) C. Use the period to identif the spacing. Anchor Point Equation: D. Start with the phase shift as the middle of the trig table (at the origin) and appl the spacing before and after 8//8 : AM.6B: Graphing Trig Functions
5 = sin BASIC TABLE POINTS = cos = tan C C C 8//8 : AM.6B: Graphing Trig Functions 5
6 = csc BASIC TABLE POINTS = sec = cot C C C 8//8 : AM.6B: Graphing Trig Functions 6
7 EXAMPLE ( ) = AtrigB C + D A =, B =, C =, D = Graph = sin + in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range = sin() C + = + = 3 sin () + Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 7 Up None = = (, ),
8 EXAMPLE Graph = sin + in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range = sin () + π/ π 3π/ π 3 8//8 : AM.6B: Graphing Trig Functions 8 Amplitude Vertical Shift Spacing (A.P.) Domain Range Up None = = (, ),
9 EXAMPLE ( ) = AtrigB C + D A =, B =, C =, D = Graph = cos θ in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range = cos(θ) C + = + = 3 cosθ / / / cosθ / 3/ / Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 9 Down None = = (, ) 3,
10 EXAMPLE Graph = cos θ in one period and identif amplitude, period, vertical shift, phase shift, domain (entire graph), and range cosθ / π/ π 3π/ π 3/ / 3 8//8 : AM.6B: Graphing Trig Functions Amplitude Vertical Shift Spacing (A.P.) Domain Range Down None = = (, ) 3,
11 YOUR TURN Graph = sin t + from π, π and identif amplitude, period, vertical shift, phase shift, domain, and range = sin t + π/ π 3π/ π 5/ 3/ Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions Up None (, ) 3 5,
12 EXAMPLE 3 = AtrigB ( C) + D A =, B =, C =, D = Given = tan + from π, π and amplitude, period, vertical shift, phase shift, domain, and range Amplitude DNE = tan() = C + = = tan tan + 3 Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions Up None = = ( ) (, ),,
13 EXAMPLE 3 Given = tan + from π, π and amplitude, period, vertical shift, and phase shift Amplitude DNE tan + π/ π/ π/ 3 π/ Vertical Shift Spacing (A.P.) Domain Range Up None = = (, ) (, ) 8//8 : AM.6B: Graphing Trig Functions 3
14 EXAMPLE ( ) = AtrigB C + D A =, B =, C =, D = Given = csc + from π, π and amplitude, period, vertical shift, phase shift, domain, and range Amplitude DNE = csc() C 3 = csc / / csc + 3/ / Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions Up None = = + n 3,,
15 EXAMPLE Given = csc + from π, π and amplitude, period, vertical shift, phase shift, domain, and range Amplitude DNE π/ π/ 3π/ π csc + 3/ / Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 5 Up None = = + n 3,,
16 YOUR TURN Given = sec from π, π and identif period, vertical 3 shift, phase shift, domain, and range Amplitude Vertical Shift Spacing (A.P.) Domain Range DNE 8//8 : AM.6B: Graphing Trig Functions 6 Down None + n,, 3 3
17 EXAMPLE 5 = AtrigB C + D ( ) Given = 5 sin π 6 phase shift, and points to graph in one period +, identif amplitude, period, vertical shift, A B C D = 5, =, =, = 6 Amplitude = 5sin Vertical Shift Up 8//8 : AM.6B: Graphing Trig Functions 7
18 EXAMPLE 5 Given = 5 sin π +, identif amplitude, period, vertical shift, 6 phase shift, and points to graph in one period = 5sin + 6 B is OUTSIDE of the parenthesis = AtrigB C + D ( ) A = 5, B =, C =, D = 6 Right 6 8//8 : AM.6B: Graphing Trig Functions 8
19 EXAMPLE 5 = AtrigB C + D ( ) A = 5, B =, C =, D = 6 Given = 5 sin π +, identif amplitude, period, vertical shift, 6 phase shift, and points to graph in one period Anchor Points = = 8//8 : AM.6B: Graphing Trig Functions 9
20 EXAMPLE 5 Given = 5 sin π +, identif amplitude, period, vertical shift, 6 phase shift, and points to graph in one period Values = AtrigB C + D ( ) A = 5, B =, C =, D = 6 = sin Y = 5sin() Y = 5sin() + π/6 C 3 5π/ + = + 6 8π/ π/3 + = + π/ + = + π/ 7π/ Anchor Point: 8//8 : AM.6B: Graphing Trig Functions
21 EXAMPLE 6 = AtrigB C + D ( ) Given = 3 cos + π phase shift, and points to graph in one period = cos Y = 3cos(), identif amplitude, period, vertical shift, Y = 3co s π/ 3 π/ 3 π 3π/ 3 Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 3 Down Left, ( ),, 3 3
22 EXAMPLE 7 Given = sin π + 5, identif amplitude, period, vertical shift, and phase shift and points from π, π = sin + 5 ( ) = AtrigB C + D ( ) 8//8 : AM.6B: Graphing Trig Functions
23 EXAMPLE 7 Given = sin π + 5, identif amplitude, period, vertical shift, and phase shift and points from π, π = sin + 5 Amplitude: : = ( C ) + D : 5 : Vertical Shift: Down 5 : Left 8//8 : AM.6B: Graphing Trig Functions 3
24 EXAMPLE 7 Given = sin π + 5, identif amplitude, period, vertical shift, and phase shift and points from π, π Amplitude: : Vertical Shift: Down 5 : Left : : : π ππ Phase : Shift: Left Left /π /π = sin = sin sin = cos (π + = cos (π = = ) cos co(π s( π+ ) + = ) = co 5cos s( (π + ) + + ) ) 5 5 /π /π 5 π/ π/ π/ /π π/ π/ π/ /π π π/ π/ /π π π/ π/ /π 5 3π/ 3π/ 3π/ /π 3π/ 3π/ 3π/ /π 9 π π π /π.6b: Graphing Trig Functions π π π /π 5 8//8 : AM
25 YOUR TURN Given = sin 3 + π 5, identif amplitude, period, vertical shift, 3 phase shift, and points to graph in one period = sin Y = sin() Y = sin() 5 π/3 5 π/3 3 π 5 5π/3 7 8π/3 5 Amplitude Vertical Shift Spacing (A.P.) Domain Range 8//8 : AM.6B: Graphing Trig Functions 5 3 Down 5 Left 3 3, ( ),, 3 3
26 EXAMPLE 8 Graph = cos 3 + π in one period and identif amplitude, 3 period, vertical shift, phase shift, domain, and range = cos Spacing : 3 : : = Left 3 3 = 3 = 6 = cos = cos 3 8//8 : AM.6B: Graphing Trig Functions 6 π/3 C π/6 π/6 π/3 π/3 π/6 π/6 π/3
27 EXAMPLE 8 Graph = cos 3 + π in one period and identif amplitude, 3 period, vertical shift, phase shift, domain, and range Amplitude ; Reflected Vertical Shift Domain Range 3 None Left 3 (, ) 8//8 : AM.6B: Graphing Trig Functions 7,
28 EXAMPLE 9 Graph = sin + π from, π and identif amplitude, period, vertical shift, phase shift, domain, and range = Asin B C + D = sin ( ) ( ) : Spacing : = : = Left = = sin = sin + π + π π/ C π/8 π/8 π/ 3 8//8 : AM.6B: Graphing Trig Functions 8 8
29 EXAMPLE 9 Graph = sin + π from, π and identif amplitude, period, vertical shift, phase shift, domain, and range = sin + π π/ π/8 π/8 π/ 3 3 Amplitude Vertical Shift Anchor Points Domain Range 8//8 : AM.6B: Graphing Trig Functions 9 3 Down Left 8 (, ) 3,
30 EXAMPLE Given = tan + π from π, π and amplitude, period, vertical shift, phase shift, domain, and range : = = tan + Spacing : = : Left = tan + π/ = tan + π = tan + π π 3π/ π/ C π/ 3 8//8 : AM.6B: Graphing Trig Functions 3
31 EXAMPLE Given = tan + π from π, π and identif period, vertical shift, phase shift, domain, and range = tan + π π 3π/ π/ π/ 3 Amplitude Vertical Shift Anchor Points Domain Range DNE 8//8 : AM.6B: Graphing Trig Functions 3 Down Left (, ) ; n (,, )
32 EXAMPLE Given = tan + π from π, π and amplitude, period, vertical shift, phase shift, domain, and range : = = tan + Spacing : = : Left = tan + π/ = tan + π = tan + π π 3π/ π/ C π/ 3 8//8 : AM.6B: Graphing Trig Functions 3
33 YOUR TURN Given = sec + π + from π, π and identif period, vertical shift, phase shift, domain, and range Amplitude DNE Vertical Shift Anchor Points Domain Up Left ( ), ; n Range (, 3, ) 8//8 : AM.6B: Graphing Trig Functions 33
34 ASSIGNMENT Worksheet 8//8 : AM.6B: Graphing Trig Functions 3
Name: 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 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 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 informationHONORS PRECALCULUS Prove the following identities- ( ) x x x x x x. cos x cos x cos x cos x 1 sin x cos x 1 sin x
HONORS PRECALCULUS Prove the following identities-.) ( ) cos sin cos cos sin + sin sin + cos sin cos sin cos.).) ( ) ( sin) ( ) ( ) sin sin + + sin sin tan + sec + cos cos cos cos sin cos sin cos cos cos
More information5-5 Multiple-Angle and Product-to-Sum Identities
Find the values of sin 2, cos 2, and tan 2 for the given value and interval. 1. cos =, (270, 360 ) Since on the interval (270, 360 ), one point on the terminal side of θ has x-coordinate 3 and a distance
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 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 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 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 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 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 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 informationTrigonometry, Exam 2 Review, Spring (b) y 4 cos x
Trigonometr, Eam Review, Spring 8 Section.A: Basic Sine and Cosine Graphs. Sketch the graph indicated. Remember to label the aes (with numbers) and to carefull sketch the five points. (a) sin (b) cos Section.B:
More informationHonors Algebra 2 w/ Trigonometry Chapter 14: Trigonometric Identities & Equations Target Goals
Honors Algebra w/ Trigonometry Chapter 14: Trigonometric Identities & Equations Target Goals By the end of this chapter, you should be able to Identify trigonometric identities. (14.1) Factor trigonometric
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 informationArkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3
Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan Review Problems for Test #3 Exercise 1 The following is one cycle of a trigonometric function. Find an equation of this graph. Exercise
More informationMath Section 4.3 Unit Circle Trigonometry
Math 0 - Section 4. Unit Circle Trigonometr An angle is in standard position if its verte is at the origin and its initial side is along the positive ais. Positive angles are measured counterclockwise
More informationTrig Graphs. What is a Trig graph? This is the graph of a trigonometrical function e.g.
Trig Graphs What is a Trig graph? This is the graph of a trigonometrical function e.g. sin, cos or tan How do we draw one? We make a table of value using the calculator. Tr to complete the one below (work
More informationYou found trigonometric values using the unit circle. (Lesson 4-3)
You found trigonometric values using the unit circle. (Lesson 4-3) LEQ: How do we identify and use basic trigonometric identities to find trigonometric values & use basic trigonometric identities to simplify
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 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 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 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 informationMAC 1114 REVIEW FOR EXAM #2 Chapters 3 & 4
MAC 111 REVIEW FOR EXAM # Chapters & This review is intended to aid you in studying for the exam. This should not be the only thing that you do to prepare. Be sure to also look over your notes, textbook,
More informationMATH Week 10. Ferenc Balogh Winter. Concordia University
MATH 20 - Week 0 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics for Calculus, 5th Edition, Thomson All figures and videos
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 informationChapter 14 Trig Graphs and Reciprocal Functions Algebra II Common Core
Chapter 14 Trig Graphs and Reciprocal Functions Algebra II Common Core LESSON 1: BASIC GRAPHS OF SINE AND COSINE LESSON : VERTICAL SHIFTING OF SINUSOIDAL GRAPHS LESSON 3 : THE FREQUENCY AND PERIOD OF A
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 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 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 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 informationMath 180 Chapter 6 Lecture Notes. Professor Miguel Ornelas
Math 180 Chapter 6 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 6.1 Section 6.1 Verifying Trigonometric Identities Verify the identity. a. sin x + cos x cot x = csc
More informationAlgebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes. Name: Date: Block:
Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Mrs. Grieser Name: Date: Block: Trig Functions in a Circle Circle with radius r, centered around origin (x 2 + y 2 = r 2 ) Drop
More information13.4 Chapter 13: Trigonometric Ratios and Functions. Section 13.4
13.4 Chapter 13: Trigonometric Ratios and Functions Section 13.4 1 13.4 Chapter 13: Trigonometric Ratios and Functions Section 13.4 2 Key Concept Section 13.4 3 Key Concept Section 13.4 4 Key Concept Section
More informationTrig Identities Packet
Advanced Math Name Trig Identities Packet = = = = = = = = cos 2 θ + sin 2 θ = sin 2 θ = cos 2 θ cos 2 θ = sin 2 θ + tan 2 θ = sec 2 θ tan 2 θ = sec 2 θ tan 2 θ = sec 2 θ + cot 2 θ = csc 2 θ cot 2 θ = csc
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 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 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 informationUnit 5. Algebra 2. Name:
Unit 5 Algebra 2 Name: 12.1 Day 1: Trigonometric Functions in Right Triangles Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Theta Opposite Side of an Angle Adjacent Side of
More informationTRANSFORMING TRIG FUNCTIONS
Chapter 7 TRANSFORMING TRIG FUNCTIONS 7.. 7..4 Students appl their knowledge of transforming parent graphs to the trigonometric functions. The will generate general equations for the famil of sine, cosine
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 informationTrigonometric Identities. Copyright 2017, 2013, 2009 Pearson Education, Inc.
5 Trigonometric Identities Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 5.3 Sum and Difference Identities Difference Identity for Cosine Sum Identity for Cosine Cofunction Identities Applications
More informationTrigonometry Review Tutorial Shorter Version
Author: Michael Migdail-Smith Originally developed: 007 Last updated: June 4, 0 Tutorial Shorter Version Avery Point Academic Center Trigonometric Functions The unit circle. Radians vs. Degrees Computing
More informationAlgebra and Trig. I. The graph of
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
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 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 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 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 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 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 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 informationPreCalc: Chapter 6 Test Review
Name: Class: Date: ID: A PreCalc: Chapter 6 Test Review Short Answer 1. Draw the angle. 135 2. Draw the angle. 3. Convert the angle to a decimal in degrees. Round the answer to two decimal places. 8. If
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 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 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 informationThe reciprocal identities are obvious from the definitions of the six trigonometric functions.
The Fundamental Identities: (1) The reciprocal identities: csc = 1 sec = 1 (2) The tangent and cotangent identities: tan = cot = cot = 1 tan (3) The Pythagorean identities: sin 2 + cos 2 =1 1+ tan 2 =
More information# 1,5,9,13,...37 (hw link has all odds)
February 8, 17 Goals: 1. Recognize trig functions and their integrals.. Learn trig identities useful for integration. 3. Understand which identities work and when. a) identities enable substitution by
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 informationGeometry Problem Solving Drill 11: Right Triangle
Geometry Problem Solving Drill 11: Right Triangle Question No. 1 of 10 Which of the following points lies on the unit circle? Question #01 A. (1/2, 1/2) B. (1/2, 2/2) C. ( 2/2, 2/2) D. ( 2/2, 3/2) The
More information4-3 Trigonometric Functions on the Unit Circle
Find the exact values of the five remaining trigonometric functions of θ. 33. tan θ = 2, where sin θ > 0 and cos θ > 0 To find the other function values, you must find the coordinates of a point on the
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 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 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 informationTrig/AP Calc A. Created by James Feng. Semester 1 Version fengerprints.weebly.com
Trig/AP Calc A Semester Version 0.. Created by James Feng fengerprints.weebly.com Trig/AP Calc A - Semester Handy-dandy Identities Know these like the back of your hand. "But I don't know the back of my
More informationTrigonometric Integrals Section 5.7
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Trigonometric Integrals Section 5.7 Dr. John Ehrke Department of Mathematics Spring 2013 Eliminating Powers From Trig Functions
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 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 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 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 informationAlgebra and Trig. I. In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position.
Algebra and Trig. I 4.4 Trigonometric Functions of Any Angle In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position. IN this section we
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 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 informationAlgebra2/Trig Chapter 10 Packet
Algebra2/Trig Chapter 10 Packet In this unit, students will be able to: Convert angle measures from degrees to radians and radians to degrees. Find the measure of an angle given the lengths of the intercepted
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 informationMATH 130 FINAL REVIEW version2
MATH 130 FINAL REVIEW version2 Problems 1 3 refer to triangle ABC, with =. Find the remaining angle(s) and side(s). 1. =50, =25 a) =40,=32.6,=21.0 b) =50,=21.0,=32.6 c) =40,=21.0,=32.6 d) =50,=32.6,=21.0
More informationModule 5 Trigonometric Identities I
MAC 1114 Module 5 Trigonometric Identities I Learning Objectives Upon completing this module, you should be able to: 1. Recognize the fundamental identities: reciprocal identities, quotient identities,
More informationMATH STUDENT BOOK. 12th Grade Unit 5
MATH STUDENT BOOK 12th Grade Unit 5 Unit 5 ANALYTIC TRIGONOMETRY MATH 1205 ANALYTIC TRIGONOMETRY INTRODUCTION 3 1. IDENTITIES AND ADDITION FORMULAS 5 FUNDAMENTAL TRIGONOMETRIC IDENTITIES 5 PROVING IDENTITIES
More informationExercise 1. Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ.
1 Radian Measures Exercise 1 Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ. 1. Suppose I know the radian measure of the
More informationFerris Wheel Activity. Student Instructions:
Ferris Wheel Activity Student Instructions: Today we are going to start our unit on trigonometry with a Ferris wheel activity. This Ferris wheel will be used throughout the unit. Be sure to hold on to
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 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 informationChapter 8. Analytic Trigonometry. 8.1 Trigonometric Identities
Chapter 8. Analytic Trigonometry 8.1 Trigonometric Identities Fundamental Identities Reciprocal Identities: 1 csc = sin sec = 1 cos cot = 1 tan tan = 1 cot tan = sin cos cot = cos sin Pythagorean Identities:
More informationMath 104 Final Exam Review
Math 04 Final Exam Review. Find all six trigonometric functions of θ if (, 7) is on the terminal side of θ.. Find cosθ and sinθ if the terminal side of θ lies along the line y = x in quadrant IV.. Find
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 informationPrerequisite Knowledge: Definitions of the trigonometric ratios for acute angles
easures, hape & pace EXEMPLAR 28 Trigonometric Identities Objective: To explore some relations of trigonometric ratios Key Stage: 3 Learning Unit: Trigonometric Ratios and Using Trigonometry Materials
More informationSection 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 informationDay 62 Applications of Sinusoidal Functions after.notebook. January 08, Homework... Worksheet Sketching in radian measure.
Homework... Worksheet Sketching in radian measure.doc 1 1. a) b) Solutions to the Worksheet... c) d) 2. a)b) 2 Developing Trigonometric Functions from Properties... Develop a trigonometric function that
More information13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. SOLUTION: 2. If, find cos θ.
Find the exact value of each expression if 0 < θ < 90 1. If cot θ = 2, find tan θ. 2. If, find cos θ. Since is in the first quadrant, is positive. Thus,. 3. If, find sin θ. Since is in the first quadrant,
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 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 informationTrigonometric Functions. Copyright 2017, 2013, 2009 Pearson Education, Inc.
1 Trigonometric Functions Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean
More information1 Trigonometric Identities
MTH 120 Spring 2008 Essex County College Division of Mathematics Handout Version 6 1 January 29, 2008 1 Trigonometric Identities 1.1 Review of The Circular Functions At this point in your mathematical
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 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 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 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 informationChapter 3, Part 4: Intro to the Trigonometric Functions
Haberman MTH Section I: The Trigonometric Functions Chapter, Part : Intro to the Trigonometric Functions Recall that the sine and cosine function represent the coordinates of points in the circumference
More informationMath 10/11 Honors Section 3.6 Basic Trigonometric Identities
Math 0/ Honors Section 3.6 Basic Trigonometric Identities 0-0 - SECTION 3.6 BASIC TRIGONOMETRIC IDENTITIES Copright all rights reserved to Homework Depot: www.bcmath.ca I) WHAT IS A TRIGONOMETRIC IDENTITY?
More information( x "1) 2 = 25, x 3 " 2x 2 + 5x "12 " 0, 2sin" =1.
Unit Analytical Trigonometry Classwork A) Verifying Trig Identities: Definitions to know: Equality: a statement that is always true. example:, + 7, 6 6, ( + ) 6 +0. Equation: a statement that is conditionally
More informationTriangle Definition of sin θ and cos θ
Triangle Definition of sin θ and cos θ Then Consider the triangle ABC below. Let A be called θ. A HYP (hpotenuse) θ ADJ (side adjacent to the angle θ ) B C OPP (side opposite to the angle θ ) (SOH CAH
More information