Basic Graphs of the Sine and Cosine Functions

Size: px
Start display at page:

Download "Basic Graphs of the Sine and Cosine Functions"

Transcription

1 Chapter 4: Graphs of the Circular Functions 1 TRIG-Fall 2011-Jordan Trigonometry, 9 th edition, Lial/Hornsby/Schneider, Pearson, 2009 Section 4.1 Graphs of the Sine and Cosine Functions Basic Graphs of the Sine and Cosine Functions Example 1 y = sin x x y 0 π/6 π/4 π/3 /2 3 /2 2 Sketch the basic sine curve on the interval [0, 2 ]. y x Memorize the basic sine graph from 0 to 2. The domain of the basic sine function is (-, ) and the range is [-1, 1]. 3 Five key points of the basic sine function: (0, 0),, 1, (π, 0),, 1, (2π, 0) 2 2 Example 2 y = cos x x 0 π/6 π/4 π/3 /2 3 /2 2 y Sketch the basic cosine curve on the interval [0, 2 ]. y x Memorize the basic cosine graph from 0 to 2. The domain of the basic cosine function is (-, ) and the range is [-1, 1]. 3 Five key points of the basic cosine function: (0, 1),, 0, (π, -1),, 0, (2π, 1) 2 2

2 Amplitude 2 The amplitude of y = a sin bx or y = a cos bx is a. The amplitude of the sine and cosine curves is defined to be half the distance between the maximum and minimum values, which is not necessarily the same as the maximum height if the graph has been shifted up or down. The graph of y = a sin x or y = a cos x, with a 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range will be a, a. If a > 1, the graph is vertically stretched. If a < 1, the graph is vertically shrunk. If a is negative, the graph is a reflection with respect to the x-axis. Period The period of y = a sin bx or y = a cos bx is 2. b The period of a graph is the length of one cycle of the graph. X-scale The x-scale is found by dividing the period of the function by 4. The x-scale is the distance on the x-axis between the key points.

3 Guidelines for Sketching Variations of the Sine and Cosine Graphs, y = a sin bx or y = a cos bx (b > 0) 3 You should have the shapes of the basic sine and cosine graphs memorized between 0 and 2π. Find the amplitude. Find the period. Determine if there is an x-axis reflection (a < 0). Find the x-scale. Make your first mark on the x-axis at 0 and then make four more marks on the x-axis. Add the x-scale to 0 and then add the x-scale to each successive answer three times to label the markings on the x-axis. The fourth sum should equal the period. Label these marks on the graph. Make four more marks on the x-axis either to the right or left as required and label the last mark. The amplitude will determine how to mark the y-axis as well as how high and low to draw the curve. If the value of a is negative, remember to reflect the graph with respect to the x-axis. Draw the general shape of the curve through the x and y-axis markings. Example 3 Sketch two full periods of the graph of y = -4cos 3x. State the amplitude, x-axis reflection, period, and five key points. Example 4 Sketch two full periods of the graph of y x 1 sin. State the amplitude, 2 4 x-axis reflection, period, and five key points. Correlating Online Graphs to Paper/pencil Graphs Do all the graphing problems on paper as presented in the lecture notes, including all the requested information. When your paper graph is complete, you can graph the function in your calculator and adjust the window settings to match the ones presented in the multiple choice format. This will help determine the correct online answer since their graphs may not be scaled in the same way as our paper ones will be.

4 Section 4.2 Functions Translations of the Graphs of the Sine and Cosine 4 Memorize the Basic Shapes of the Following Graphs Phase Shift The phase shift of y = a sin (bx + c) + d or y = a cos (bx + c) + d is found by setting bx + c = 0 and solving for x. The phase shift of a trig function is the horizontal translation (shift) of the graph and will give you the left or starting endpoint of one cycle of the graph. The phase shift can also be found by setting the expression inside the trigonometric function between 0 and 2π, 0 bx + c 2π, and solving for x,. The left side of the solved inequality is the phase shift and represents the beginning of one cycle and the right side of the solved inequality is the end of one cycle. Vertical Translations y = f(x) + d is a vertical shift of the graph of y = f(x). If d > 0, then the graph is shifted up from the original graph. If d < 0, then the graph is shifted down from the original graph. Example 1 Find the amplitude, x-axis reflection, period, phase shift, and vertical 1 translation of the graph of the following function: y 3cos x 2 2

5 Example 2 Find the amplitude, x-axis reflection, period, phase shift, and vertical translation of the graph of the following function: y 2 sin 3x 5 5 Guidelines for Sketching Variations of the Sine and Cosine Graphs, y = a sin (bx + c) + d or y = a cos (bx + c) + d You should have the shapes of the basic sine and cosine graphs memorized between 0 and 2π. Find the amplitude. Find the period. Determine if there is an x-axis reflection. Find the phase shift. Find the vertical translation. Find the x-scale. Set up and solve the inequality 0 bx + c 2π to determine the beginning and end of one cycle. The left side of the solved inequality should equal the phase shift. Make your first mark on the x-axis at the phase shift. Add the x-scale to the phase shift and then add the x-scale to each successive answer three times to label the markings on the x-axis. The fourth sum should equal the right side of the solved inequality. Make four more marks on the x-axis either to the right or left as required and label the last mark. If you are going to the right, the last mark will be the mark at the end of the first cycle added to the period. If you are going to the left, the last mark will be the phase shift minus the period. If there is a vertical translation, mark it on the y-axis. Dash in a horizontal line through the y-axis marking. This is the vertically translated x-axis. Use the amplitude and count that many units above and below the vertical translation. This will determine how high and low to draw the curve. If the value of a is negative, remember to reflect the graph with respect to the x-axis. Draw the general shape of the curve through the x and y-axis markings.

6 6 Example 3 Sketch two full periods of the graph of y 3sin x. State the 4 amplitude, x-axis reflection, period, phase shift, vertical translation, and five key points. Example 4 Sketch two full periods of the graph of y 1 2cos(4x ). State the amplitude, x-axis reflection, period, phase shift, vertical translation, and five key points.

7 Section 4.3 Graphs of the Tangent and Cotangent Functions 7 Tangent Function Use your graphing calculator to graph the basic function, y = tan x. Use window settings of Xmin = -π/2, Xmax = 3π/2, Xscl = π/4, Ymin = -4, Ymax = 4, Yscl = 1. The period of the basic tangent function is π. Cotangent Function Use your graphing calculator to graph the basic function, y = cot x. Use y = (cos x)/(sin x) so that the calculator table values are correct. Use window settings of Xmin = 0, Xmax = 2π, Xscl = π/4, Ymin = -4, Ymax = 4, Yscl = 1. The period of the basic cotangent function is π. Memorize the Basic Shapes of the Following Graphs

8 Guidelines for Sketching Graphs of Tangent and Cotangent Functions, y = a tan (bx + c) + d or y = a cot (bx + c) + d 8 Find the period of the graph by using the formula b. Find the x-scale by dividing the period by 4. Tangent Function: Bound the expression inside the tangent function between and and solve for x. This gives the beginning and the end of one cycle. 2 2 Cotangent Function: Bound the expression inside the cotangent function between 0 and π and solve for x. This gives the beginning and the end of one cycle. Add the x-scale to the beginning of the cycle and then add the x-scale to each successive answer three times to determine the markings on the x-axis. The fourth sum should equal the end of the first cycle. Find the end of the second cycle by adding the period to the end of the first cycle. The vertical asymptotes of both functions are found at the endpoints of each cycle. If there is a vertical translation, mark it on the y-axis and draw a dashed horizontal line through the marking to denote the translated x-axis. The midpoint between two consecutive vertical asymptotes is an x-intercept. The absolute value of a will determine the marking on the y-axis. If there is a vertical translation, count up and down from the translated x-axis. If a < 0, the graph is reflected with respect to the x-axis. Plot three points between each asymptote. Set bx + c = 0 to find the phase shift. Warning: The phase shift of the tangent function will not necessarily equal the beginning of the first cycle.. State the period, x-axis reflection, phase shift, vertical translation, left asymptote, right asymptote, left key point, middle key point, and right key point. Example 1 Sketch two full periods of the graph of y tan 3x

9 x Example 2 Sketch two full periods of the graph of y 2 cot 2. State the period, x-axis reflection, phase shift, vertical translation, left asymptote, right asymptote, left key point, middle key point, and right key point Summary of Basic Shapes of Graphs to be Memorized

10 Section 4.4 Graphs of the Secant and Cosecant Functions 10 Cosecant Function Use your graphing calculator to graph the basic function, y = csc x. Let y = 1/sin x Use window settings of Xmin = -2π, Xmax = 2π, Xscl = π/2, Ymin = -4, Ymax = 4, Yscl = 1. Secant Function Use your graphing calculator to graph the basic function, y = sec x. Let y = 1/cos x Use window settings of Xmin = -2π, Xmax = 2π, Xscl = π/2, Ymin = -4, Ymax = 4, Yscl = 1. Guidelines for Sketching Graphs of Cosecant and Secant Functions Graph the corresponding reciprocal function as a guide, using a dashed curve. Include all data that was used in the preceding section, including translations and period changes. Sketch the vertical asymptotes everywhere there is an x-intercept. Sketch the graph of the desired function by drawing the U-shaped branches between the adjacent asymptotes. Both the basic cosecant and secant functions will have period 2π just as the basic sine and cosine functions do. Example 1 Sketch two full periods of the graph of y csc x. State the 3 amplitude of sine, x-axis reflection, period, phase shift, vertical translation, and five key points of sine. Example 2 Sketch two full periods of the graph of y 2sec 3x. State the 2 amplitude of cosine, x-axis reflection, period, phase shift, vertical translation, and five key points of cosine.

11 Section 4.5 Harmonic Motion 11 Simple Harmonic Motion Consider a mass on a spring. The system is said to be in equilibrium when the mass is at rest. The point of rest is called the origin of the system. The distance above the equilibrium point is the positive direction and the distance below the equilibrium point is the negative direction. If the mass is lifted a distance a and released, the mass will oscillate up and down in a periodic motion. If there is NO FRICTION, the motion repeats itself in a certain period of time. This is called simple harmonic motion. The distance a is called the displacement from the origin. The number of times the mass oscillates in 1 unit of time is called the frequency, F, of the motion, measured in number of cycles per unit of time. The time one oscillation takes is the period, P, of the motion. The maximum displacement from the equilibrium position is called the amplitude of the motion. The frequency and the period are related by the formulas 1 F and P 1 P F

12 Equations Modeling Simple Harmonic Motion 12 If the displacement from the origin is at a maximum at time t = 0, simple harmonic motion can be modeled by s( t) acos t If the displacement from the origin is zero at time t = 0, simple harmonic motion can be modeled by s( t) a sin t In either case, a is the amplitude, the period is P 2, the frequency is F 2, s(t) is the position of the spring, and t is the time. Example 1 Suppose that an object is attached to a coiled spring. It is pulled down a distance of 5 inches from its equilibrium position, and then released. The time for one complete oscillation is 4 seconds. a) Give an equation that models the position of the object at time t. b) Determine the position at t = 1.5 seconds. c) Find the frequency. L Example 2 The period P of a pendulum and its length L are related by P 2 32 where P is in seconds and L is in feet. How long should a pendulum be to have a period of 5 seconds? Example 3 The position of a weight attached to a spring is s( t) 6cos 4t inches after t seconds. a) What is the maximum height that the weight rises above the equilibrium position? b) When does the weight first reach its maximum height? c) What are the frequency and period?

Module 4 Graphs of the Circular Functions

Module 4 Graphs of the Circular Functions MAC 1114 Module 4 Graphs of the Circular Functions Learning Objectives Upon completing this module, you should be able to: 1. Recognize periodic functions. 2. Determine the amplitude and period, when given

More information

Chapter 4. Trigonometric Functions. 4.6 Graphs of Other. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 4. Trigonometric Functions. 4.6 Graphs of Other. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter 4 Trigonometric Functions 4.6 Graphs of Other Trigonometric Functions Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Understand the graph of y = tan x. Graph variations of y =

More information

This is called the horizontal displacement of also known as the phase shift.

This is called the horizontal displacement of also known as the phase shift. sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for

More information

1. GRAPHS OF THE SINE AND COSINE FUNCTIONS

1. GRAPHS OF THE SINE AND COSINE FUNCTIONS GRAPHS OF THE CIRCULAR FUNCTIONS 1. GRAPHS OF THE SINE AND COSINE FUNCTIONS PERIODIC FUNCTION A period function is a function f such that f ( x) f ( x np) for every real numer x in the domain of f every

More information

Graphs of Other Trig Functions

Graphs of Other Trig Functions Graph y = tan. y 0 0 6 3 3 3 5 6 3 3 1 Graphs of Other Trig Functions.58 3 1.7 undefined 3 3 3 1.7-1 0.58 3 CHAT Pre-Calculus 3 The Domain is all real numbers ecept multiples of. (We say the domain is

More information

Notice there are vertical asymptotes whenever y = sin x = 0 (such as x = 0).

Notice there are vertical asymptotes whenever y = sin x = 0 (such as x = 0). 1 of 7 10/1/2004 6.4 GRAPHS OF THE OTHER CIRCULAR 6.4 GRAPHS OF THE OTHER CIRCULAR Graphs of the Cosecant and Secant Functions Graphs of the Tangent and Cotangent Functions Addition of Ordinates Graphs

More information

Graphs of the Circular Functions. Copyright 2017, 2013, 2009 Pearson Education, Inc.

Graphs of the Circular Functions. Copyright 2017, 2013, 2009 Pearson Education, Inc. 4 Graphs of the Circular Functions Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 4.3 Graphs of the Tangent and Cotangent Functions Graph of the Tangent Function Graph of the Cotangent Function Techniques

More information

Section 5.3 Graphs of the Cosecant and Secant Functions 1

Section 5.3 Graphs of the Cosecant and Secant Functions 1 Section 5.3 Graphs of the Cosecant, Secant, Tangent, and Cotangent Functions The Cosecant Graph RECALL: 1 csc x so where sin x 0, csc x has an asymptote. sin x To graph y Acsc( Bx C) D, first graph THE

More information

4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS

4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS 4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch

More information

MAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand. Overview

MAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand. Overview MAT 115: Precalculus Mathematics Constructing Graphs of Trigonometric Functions Involving Transformations by Hand Overview Below are the guidelines for constructing a graph of a trigonometric function

More information

Section 6.2 Graphs of the Other Trig Functions

Section 6.2 Graphs of the Other Trig Functions Section 62 Graphs of the Other Trig Functions 369 Section 62 Graphs of the Other Trig Functions In this section, we will explore the graphs of the other four trigonometric functions We ll begin with the

More information

Unit 3 Trig II. 3.1 Trig and Periodic Functions

Unit 3 Trig II. 3.1 Trig and Periodic Functions Unit 3 Trig II AFM Mrs. Valentine Obj.: I will be able to use a unit circle to find values of sine, cosine, and tangent. I will be able to find the domain and range of sine and cosine. I will understand

More information

Unit 4 Graphs of Trigonometric Functions - Classwork

Unit 4 Graphs of Trigonometric Functions - Classwork Unit Graphs of Trigonometric Functions - Classwork For each of the angles below, calculate the values of sin x and cos x (2 decimal places) on the chart and graph the points on the graph below. x 0 o 30

More information

Unit 4 Graphs of Trigonometric Functions - Classwork

Unit 4 Graphs of Trigonometric Functions - Classwork Unit Graphs of Trigonometric Functions - Classwork For each of the angles below, calculate the values of sin x and cos x decimal places) on the chart and graph the points on the graph below. x 0 o 30 o

More information

Unit 4 Graphs of Trigonometric Functions - Classwork

Unit 4 Graphs of Trigonometric Functions - Classwork Unit Graphs of Trigonometric Functions - Classwork For each of the angles below, calculate the values of sin x and cos x ( decimal places) on the chart and graph the points on the graph below. x 0 o 30

More information

Math 1330 Section 5.3 Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions

Math 1330 Section 5.3 Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions Math 1330 Section 5.3 Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions In this section, you will learn to graph the rest of the trigonometric functions. We can use some information from

More information

Section Graphs of the Sine and Cosine Functions

Section Graphs of the Sine and Cosine Functions Section 5. - Graphs of the Sine and Cosine Functions In this section, we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations.

More information

Unit 7: Trigonometry Part 1

Unit 7: Trigonometry Part 1 100 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( α ) = d) Cosecant csc( α ) = α Adjacent Opposite b) Cosine cos( α ) = e) Secant sec( α ) = c) Tangent f) Cotangent tan(

More information

Lesson Goals. Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) Overview. Overview

Lesson Goals. Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) Overview. Overview Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Graph

More information

Precalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant Practice Problems. Questions

Precalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant Practice Problems. Questions Questions 1. Describe the graph of the function in terms of basic trigonometric functions. Locate the vertical asymptotes and sketch two periods of the function. y = 3 tan(x/2) 2. Solve the equation csc

More information

2.7 Graphing Tangent, Cotangent, Secant, and

2.7 Graphing Tangent, Cotangent, Secant, and www.ck12.org Chapter 2. Graphing Trigonometric Functions 2.7 Graphing Tangent, Cotangent, Secant, and Cosecant Learning Objectives Apply transformations to the remaining four trigonometric functions. Identify

More information

Section 5: Introduction to Trigonometry and Graphs

Section 5: Introduction to Trigonometry and Graphs Section 5: Introduction to Trigonometry and Graphs The following maps the videos in this section to the Texas Essential Knowledge and Skills for Mathematics TAC 111.42(c). 5.01 Radians and Degree Measurements

More information

Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5)

Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) Unit 6 Introduction to Trigonometry Graphing Other Trig Functions (Unit 6.5) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Graph

More information

June 6 Math 1113 sec 002 Summer 2014

June 6 Math 1113 sec 002 Summer 2014 June 6 Math 1113 sec 002 Summer 2014 Sec. 6.4 Plotting f (x) = a sin(bx c) + d or f (x) = a cos(bx c) + d Amplitude is a. If a < 0 there is a reflection in the x-axis. The fundamental period is The phase

More information

Translation of graphs (2) The exponential function and trigonometric function

Translation of graphs (2) The exponential function and trigonometric function Lesson 35 Translation of graphs (2) The exponential function and trigonometric function Learning Outcomes and Assessment Standards Learning Outcome 2: Functions and Algebra Assessment Standard Generate

More information

Math 1330 Test 3 Review Sections , 5.1a, ; Know all formulas, properties, graphs, etc!

Math 1330 Test 3 Review Sections , 5.1a, ; Know all formulas, properties, graphs, etc! Math 1330 Test 3 Review Sections 4.1 4.3, 5.1a, 5. 5.4; Know all formulas, properties, graphs, etc! 1. Similar to a Free Response! Triangle ABC has right angle C, with AB = 9 and AC = 4. a. Draw and label

More information

Verifying Trigonometric Identities

Verifying Trigonometric Identities Verifying Trigonometric Identities What you should learn Verify trigonometric identities. Why you should learn it You can use trigonometric identities to rewrite trigonometric equations that model real-life

More information

Math 1330 Final Exam Review Covers all material covered in class this semester.

Math 1330 Final Exam Review Covers all material covered in class this semester. Math 1330 Final Exam Review Covers all material covered in class this semester. 1. Give an equation that could represent each graph. A. Recall: For other types of polynomials: End Behavior An even-degree

More information

Chapter 5.6: The Other Trig Functions

Chapter 5.6: The Other Trig Functions Chapter 5.6: The Other Trig Functions The other four trig functions, tangent, cotangent, cosecant, and secant are not sinusoids, although they are still periodic functions. Each of the graphs of these

More information

Name: Teacher: Pd: Algebra 2/Trig: Trigonometric Graphs (SHORT VERSION)

Name: Teacher: Pd: Algebra 2/Trig: Trigonometric Graphs (SHORT VERSION) Algebra 2/Trig: Trigonometric Graphs (SHORT VERSION) In this unit, we will Learn the properties of sine and cosine curves: amplitude, frequency, period, and midline. Determine what the parameters a, b,

More information

Unit T Student Success Sheet (SSS) Graphing Trig Functions (sections )

Unit T Student Success Sheet (SSS) Graphing Trig Functions (sections ) Unit T Student Success Sheet (SSS) Graphing Trig Functions (sections 4.5-4.7) Standards: Trig 4.0, 5.0,6.0 Segerstrom High School -- Math Analysis Honors Name: Period: Thinkbinder Study Group: www.bit.ly/chatunitt

More information

( ) = 1 4. (Section 4.6: Graphs of Other Trig Functions) Example. Use the Frame Method to graph one cycle of the graph of

( ) = 1 4. (Section 4.6: Graphs of Other Trig Functions) Example. Use the Frame Method to graph one cycle of the graph of (Section 4.6: Graphs of Other Trig Functions) 4.63 Example Use the Frame Method to graph one cycle of the graph of y = 2 tan 2 5 x 3. (There are infinitely many possible cycles.) Solution Fortunately,

More information

SECTION 6-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions

SECTION 6-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions 6-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions 9 duce a scatter plot in the viewing window. Choose 8 for the viewing window. (B) It appears that a sine curve of the form k

More information

Chapter 4 Using Fundamental Identities Section USING FUNDAMENTAL IDENTITIES. Fundamental Trigonometric Identities. Reciprocal Identities

Chapter 4 Using Fundamental Identities Section USING FUNDAMENTAL IDENTITIES. Fundamental Trigonometric Identities. Reciprocal Identities Chapter 4 Using Fundamental Identities Section 4.1 4.1 USING FUNDAMENTAL IDENTITIES Fundamental Trigonometric Identities Reciprocal Identities csc x sec x cot x Quotient Identities tan x cot x Pythagorean

More information

x,,, (All real numbers except where there are

x,,, (All real numbers except where there are Section 5.3 Graphs of other Trigonometric Functions Tangent and Cotangent Functions sin( x) Tangent function: f( x) tan( x) ; cos( x) 3 5 Vertical asymptotes: when cos( x ) 0, that is x,,, Domain: 3 5

More information

Parametric Equations of Line Segments: what is the slope? what is the y-intercept? how do we find the parametric eqtn of a given line segment?

Parametric Equations of Line Segments: what is the slope? what is the y-intercept? how do we find the parametric eqtn of a given line segment? Shears Math 122/126 Parametric Equations Lecture Notes Use David Little's program for the following: Parametric Equations in General: look at default in this program, also spiro graph Parametric Equations

More information

Contents 10. Graphs of Trigonometric Functions

Contents 10. Graphs of Trigonometric Functions Contents 10. Graphs of Trigonometric Functions 2 10.2 Sine and Cosine Curves: Horizontal and Vertical Displacement...... 2 Example 10.15............................... 2 10.3 Composite Sine and Cosine

More information

Math 144 Activity #3 Coterminal Angles and Reference Angles

Math 144 Activity #3 Coterminal Angles and Reference Angles 144 p 1 Math 144 Activity #3 Coterminal Angles and Reference Angles For this activity we will be referring to the unit circle. Using the unit circle below, explain how you can find the sine of any given

More information

AP Calculus Summer Review Packet

AP Calculus Summer Review Packet AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** Email anytime with questions: danna.seigle@henry.k1.ga.us Complex Fractions

More information

Algebra II Trigonometric Functions

Algebra II Trigonometric Functions Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc

More information

Unit 13: Periodic Functions and Trig

Unit 13: Periodic Functions and Trig Date Period Unit 13: Periodic Functions and Trig Day Topic 0 Special Right Triangles and Periodic Function 1 Special Right Triangles Standard Position Coterminal Angles 2 Unit Circle Cosine & Sine (x,

More information

Chapter 4: Trigonometry

Chapter 4: Trigonometry Chapter 4: Trigonometry Section 4-1: Radian and Degree Measure INTRODUCTION An angle is determined by rotating a ray about its endpoint. The starting position of the ray is the of the angle, and the position

More information

A trigonometric ratio is a,

A trigonometric ratio is a, ALGEBRA II Chapter 13 Notes The word trigonometry is derived from the ancient Greek language and means measurement of triangles. Section 13.1 Right-Triangle Trigonometry Objectives: 1. Find the trigonometric

More information

Pre-calculus Chapter 4 Part 1 NAME: P.

Pre-calculus Chapter 4 Part 1 NAME: P. Pre-calculus NAME: P. Date Day Lesson Assigned Due 2/12 Tuesday 4.3 Pg. 284: Vocab: 1-3. Ex: 1, 2, 7-13, 27-32, 43, 44, 47 a-c, 57, 58, 63-66 (degrees only), 69, 72, 74, 75, 78, 79, 81, 82, 86, 90, 94,

More information

to and go find the only place where the tangent of that

to and go find the only place where the tangent of that Study Guide for PART II of the Spring 14 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

More information

Review of Trigonometry

Review of Trigonometry Worksheet 8 Properties of Trigonometric Functions Section Review of Trigonometry This section reviews some of the material covered in Worksheets 8, and The reader should be familiar with the trig ratios,

More information

The Sine and Cosine Functions

The Sine and Cosine Functions Concepts: Graphs of Tangent, Cotangent, Secant, and Cosecant. We obtain the graphs of the other trig functions by thinking about how they relate to the sin x and cos x. The Sine and Cosine Functions Page

More information

Algebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions

Algebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions 2015-12-17 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Co-terminal angles Arc

More information

Verifying Trigonometric Identities

Verifying Trigonometric Identities 40 Chapter Analytic Trigonometry. f x sec x Sketch the graph of y cos x Amplitude: Period: One cycle: first. The x-intercepts of y correspond to the vertical asymptotes of f x. cos x sec x 4 x, x 4 4,...

More information

Chapter 5. An Introduction to Trigonometric Functions 1-1

Chapter 5. An Introduction to Trigonometric Functions 1-1 Chapter 5 An Introduction to Trigonometric Functions Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1-1 5.1 A half line or all points extended from a single

More information

Section Graphs of the Sine and Cosine Functions

Section Graphs of the Sine and Cosine Functions Section 5. - Graphs of the Sine and Cosine Functions In this section, we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations.

More information

5.1 Angles & Their Measures. Measurement of angle is amount of rotation from initial side to terminal side. radians = 60 degrees

5.1 Angles & Their Measures. Measurement of angle is amount of rotation from initial side to terminal side. radians = 60 degrees .1 Angles & Their Measures An angle is determined by rotating array at its endpoint. Starting side is initial ending side is terminal Endpoint of ray is the vertex of angle. Origin = vertex Standard Position:

More information

Trigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are:

Trigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are: TRIGONOMETRY TRIGONOMETRIC RATIOS If one of the angles of a triangle is 90º (a right angle), the triangle is called a right angled triangle. We indicate the 90º (right) angle by placing a box in its corner.)

More information

5.2. The Sine Function and the Cosine Function. Investigate A

5.2. The Sine Function and the Cosine Function. Investigate A 5.2 The Sine Function and the Cosine Function What do an oceanographer, a stock analyst, an audio engineer, and a musician playing electronic instruments have in common? They all deal with periodic patterns.

More information

Walt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC

Walt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be

More information

A lg e b ra II. Trig o n o m e tric F u n c tio

A lg e b ra II. Trig o n o m e tric F u n c tio 1 A lg e b ra II Trig o n o m e tric F u n c tio 2015-12-17 www.njctl.org 2 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector

More information

Precalculus Solutions Review for Test 6 LMCA Section

Precalculus Solutions Review for Test 6 LMCA Section Precalculus Solutions Review for Test 6 LMCA Section 4.5-4.8 Memorize all of the formulas and identities. Here are some of the formulas for chapter 5. BasicTrig Functions opp y hyp r sin csc hyp r opp

More information

Check In before class starts:

Check In before class starts: Name: Date: Lesson 5-3: Graphing Trigonometric Functions Learning Goal: How do I use the critical values of the Sine and Cosine curve to graph vertical shift and vertical stretch? Check In before class

More information

Sum and Difference Identities. Cosine Sum and Difference Identities: cos A B. does NOT equal cos A. Cosine of a Sum or Difference. cos B.

Sum and Difference Identities. Cosine Sum and Difference Identities: cos A B. does NOT equal cos A. Cosine of a Sum or Difference. cos B. 7.3 Sum and Difference Identities 7-1 Cosine Sum and Difference Identities: cos A B Cosine of a Sum or Difference cos cos does NOT equal cos A cos B. AB AB EXAMPLE 1 Finding Eact Cosine Function Values

More information

4.8. Solving Problems with Trigonometry. Copyright 2011 Pearson, Inc.

4.8. Solving Problems with Trigonometry. Copyright 2011 Pearson, Inc. 4.8 Solving Problems with Trigonometry Copyright 2011 Pearson, Inc. What you ll learn about More Right Triangle Problems Simple Harmonic Motion and why These problems illustrate some of the better- known

More information

Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231

Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231 1 Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231 What is Trigonometry? 2 It is defined as the study of triangles and the relationships between their sides and the angles between these sides.

More information

MATHEMATICS 105 Plane Trigonometry

MATHEMATICS 105 Plane Trigonometry Chapter I THE TRIGONOMETRIC FUNCTIONS MATHEMATICS 105 Plane Trigonometry INTRODUCTION The word trigonometry literally means triangle measurement. It is concerned with the measurement of the parts, sides,

More information

AP Calculus Summer Review Packet School Year. Name

AP Calculus Summer Review Packet School Year. Name AP Calculus Summer Review Packet 016-017 School Year Name Objectives for AP/CP Calculus Summer Packet 016-017 I. Solving Equations & Inequalities (Problems # 1-6) Using the properties of equality Solving

More information

8B.2: Graphs of Cosecant and Secant

8B.2: Graphs of Cosecant and Secant Opp. Name: Date: Period: 8B.: Graphs of Cosecant and Secant Or final two trigonometric functions to graph are cosecant and secant. Remember that So, we predict that there is a close relationship between

More information

Common Core Standards Addressed in this Resource

Common Core Standards Addressed in this Resource Common Core Standards Addressed in this Resource N-CN.4 - Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular

More information

CCNY Math Review Chapters 5 and 6: Trigonometric functions and graphs

CCNY Math Review Chapters 5 and 6: Trigonometric functions and graphs Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions CCNY Math Review Chapters 5 and 6: Trigonometric functions and

More information

1. Let be a point on the terminal side of θ. Find the 6 trig functions of θ. (Answers need not be rationalized). b. P 1,3. ( ) c. P 10, 6.

1. Let be a point on the terminal side of θ. Find the 6 trig functions of θ. (Answers need not be rationalized). b. P 1,3. ( ) c. P 10, 6. Q. Right Angle Trigonometry Trigonometry is an integral part of AP calculus. Students must know the basic trig function definitions in terms of opposite, adjacent and hypotenuse as well as the definitions

More information

PRECALCULUS MATH Trigonometry 9-12

PRECALCULUS MATH Trigonometry 9-12 1. Find angle measurements in degrees and radians based on the unit circle. 1. Students understand the notion of angle and how to measure it, both in degrees and radians. They can convert between degrees

More information

Unit Circle. Project Response Sheet

Unit Circle. Project Response Sheet NAME: PROJECT ACTIVITY: Trigonometry TOPIC Unit Circle GOALS MATERIALS Explore Degree and Radian Measure Explore x- and y- coordinates on the Unit Circle Investigate Odd and Even functions Investigate

More information

8.6 Other Trigonometric Functions

8.6 Other Trigonometric Functions 8.6 Other Trigonometric Functions I have already discussed all the trigonometric functions and their relationship to the sine and cosine functions and the x and y coordinates on the unit circle, but let

More information

5.5 Multiple-Angle and Product-to-Sum Formulas

5.5 Multiple-Angle and Product-to-Sum Formulas Section 5.5 Multiple-Angle and Product-to-Sum Formulas 87 5.5 Multiple-Angle and Product-to-Sum Formulas Multiple-Angle Formulas In this section, you will study four additional categories of trigonometric

More information

Downloaded from

Downloaded from Top Concepts Class XI: Maths Ch : Trigonometric Function Chapter Notes. An angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Trigonometric Functions of Any Angle MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of any angle,

More information

sin30 = sin60 = cos30 = cos60 = tan30 = tan60 =

sin30 = sin60 = cos30 = cos60 = tan30 = tan60 = Precalculus Notes Trig-Day 1 x Right Triangle 5 How do we find the hypotenuse? 1 sinθ = cosθ = tanθ = Reciprocals: Hint: Every function pair has a co in it. sinθ = cscθ = sinθ = cscθ = cosθ = secθ = cosθ

More information

Graphing Trigonometric Functions: Day 1

Graphing Trigonometric Functions: Day 1 Graphing Trigonometric Functions: Day 1 Pre-Calculus 1. Graph the six parent trigonometric functions.. Apply scale changes to the six parent trigonometric functions. Complete the worksheet Exploration:

More information

Section 7.6 Graphs of the Sine and Cosine Functions

Section 7.6 Graphs of the Sine and Cosine Functions Section 7.6 Graphs of the Sine and Cosine Functions We are going to learn how to graph the sine and cosine functions on the xy-plane. Just like with any other function, it is easy to do by plotting points.

More information

Algebra II. Chapter 13 Notes Sections 13.1 & 13.2

Algebra II. Chapter 13 Notes Sections 13.1 & 13.2 Algebra II Chapter 13 Notes Sections 13.1 & 13.2 Name Algebra II 13.1 Right Triangle Trigonometry Day One Today I am using SOHCAHTOA and special right triangle to solve trig problems. I am successful

More information

Math.1330 Section 5.2 Graphs of the Sine and Cosine Functions

Math.1330 Section 5.2 Graphs of the Sine and Cosine Functions Math.10 Section 5. Graphs of the Sine and Cosine Functions In this section, we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations.

More information

Definitions Associated w/ Angles Notation Visualization Angle Two rays with a common endpoint ABC

Definitions Associated w/ Angles Notation Visualization Angle Two rays with a common endpoint ABC Preface to Chapter 5 The following are some definitions that I think will help in the acquisition of the material in the first few chapters that we will be studying. I will not go over these in class and

More information

Vertical and Horizontal Translations

Vertical and Horizontal Translations SECTION 4.3 Vertical and Horizontal Translations Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the vertical translation of a sine or cosine function. Find the horizontal

More information

2.3 Circular Functions of Real Numbers

2.3 Circular Functions of Real Numbers www.ck12.org Chapter 2. Graphing Trigonometric Functions 2.3 Circular Functions of Real Numbers Learning Objectives Graph the six trigonometric ratios as functions on the Cartesian plane. Identify the

More information

Foundations of Math II

Foundations of Math II Foundations of Math II Unit 6b: Toolkit Functions Academics High School Mathematics 6.6 Warm Up: Review Graphing Linear, Exponential, and Quadratic Functions 2 6.6 Lesson Handout: Linear, Exponential,

More information

4.6 Graphs of Other Trigonometric Functions

4.6 Graphs of Other Trigonometric Functions .6 Graphs of Other Trigonometric Functions Section.6 Graphs of Other Trigonometric Functions 09 Graph of the Tangent Function Recall that the tangent function is odd. That is, tan tan. Consequentl, the

More information

Chapter 4/5 Part 1- Trigonometry in Radians

Chapter 4/5 Part 1- Trigonometry in Radians Chapter 4/5 Part - Trigonometry in Radians Lesson Package MHF4U Chapter 4/5 Part Outline Unit Goal: By the end of this unit, you will be able to demonstrate an understanding of meaning and application

More information

Unit 2 Intro to Angles and Trigonometry

Unit 2 Intro to Angles and Trigonometry HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 1 Unit 2 Intro to Angles and Trigonometry This is a BASIC CALCULATORS ONLY unit. (2) Definition of an Angle (3) Angle Measurements & Notation (4) Conversions of

More information

Math Handbook of Formulas, Processes and Tricks. Trigonometry

Math Handbook of Formulas, Processes and Tricks. Trigonometry Math Handbook of Formulas, Processes and Tricks (www.mathguy.us) Trigonometry Prepared by: Earl L. Whitney, FSA, MAAA Version 2.1 April 10, 2017 Copyright 2012 2017, Earl Whitney, Reno NV. All Rights Reserved

More information

1 Trigonometry -Ideas and Applications

1 Trigonometry -Ideas and Applications 1 Trigonometry -Ideas and Applications 1.1 A second look at graphs The sine and cosine are basic entities of trigonometry, for the other four functions can be defined in terms of them. The graphs can be

More information

SNAP Centre Workshop. Introduction to Trigonometry

SNAP Centre Workshop. Introduction to Trigonometry SNAP Centre Workshop Introduction to Trigonometry 62 Right Triangle Review A right triangle is any triangle that contains a 90 degree angle. There are six pieces of information we can know about a given

More information

Multiple Choice Questions Circle the letter of the correct answer. 7 points each. is:

Multiple Choice Questions Circle the letter of the correct answer. 7 points each. is: This Math 114 final exam was administered in the Fall of 008. This is a sample final exam. The problems are not exhaustive. Be prepared for ALL CONCEPTS for the actual final exam. Multiple Choice Questions

More information

PRESCOTT UNIFIED SCHOOL DISTRICT District Instructional Guide Revised 6/3/15

PRESCOTT UNIFIED SCHOOL DISTRICT District Instructional Guide Revised 6/3/15 PRESCOTT UNIFIED SCHOOL DISTRICT District Instructional Guide Revised 6/3/15 Grade Level: PHS Subject: Precalculus Quarter/Semester: 1/1 Core Text: Precalculus with 1 st week Chapter P - Prerequisites

More information

MATH 181-Trigonometric Functions (10)

MATH 181-Trigonometric Functions (10) The Trigonometric Functions ***** I. Definitions MATH 8-Trigonometric Functions (0 A. Angle: It is generated by rotating a ray about its fixed endpoint from an initial position to a terminal position.

More information

4.1: Angles & Angle Measure

4.1: Angles & Angle Measure 4.1: Angles & Angle Measure In Trigonometry, we use degrees to measure angles in triangles. However, degree is not user friendly in many situations (just as % is not user friendly unless we change it into

More information

Multiple Angle and Product-to-Sum Formulas. Multiple-Angle Formulas. Double-Angle Formulas. sin 2u 2 sin u cos u. 2 tan u 1 tan 2 u. tan 2u.

Multiple Angle and Product-to-Sum Formulas. Multiple-Angle Formulas. Double-Angle Formulas. sin 2u 2 sin u cos u. 2 tan u 1 tan 2 u. tan 2u. 3330_0505.qxd 1/5/05 9:06 AM Page 407 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 407 5.5 Multiple Angle and Product-to-Sum Formulas What you should learn Use multiple-angle formulas to rewrite

More information

Lesson 5.2: Transformations of Sinusoidal Functions (Sine and Cosine)

Lesson 5.2: Transformations of Sinusoidal Functions (Sine and Cosine) Lesson 5.2: Transformations of Sinusoidal Functions (Sine and Cosine) Reflections Horizontal Translation (c) Vertical Translation (d) Remember: vertical stretch horizontal stretch 1 Part A: Reflections

More information

Intro Right Triangle Trig

Intro Right Triangle Trig Ch. Y Intro Right Triangle Trig In our work with similar polygons, we learned that, by definition, the angles of similar polygons were congruent and their sides were in proportion - which means their ratios

More information

Section 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.

Section 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis. 1 Section 7.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint

More information

2 Unit Bridging Course Day 10

2 Unit Bridging Course Day 10 1 / 31 Unit Bridging Course Day 10 Circular Functions III The cosine function, identities and derivatives Clinton Boys / 31 The cosine function The cosine function, abbreviated to cos, is very similar

More information

1. The circle below is referred to as a unit circle. Why is this the circle s name?

1. The circle below is referred to as a unit circle. Why is this the circle s name? Right Triangles and Coordinates on the Unit Circle Learning Task: 1. The circle below is referred to as a unit circle. Why is this the circle s name? Part I 2. Using a protractor, measure a 30 o angle

More information

Table of Contents Volume I

Table of Contents Volume I Precalculus Concepts Through Functions A Unit Circle Approach to Trigonometry 3rd Edition Sullivan SOLUTIONS MANUAL Full download at: https://testbankreal.com/download/precalculus-concepts-throughfunctions-a-unit-circle-approach-to-trigonometry-3rd-edition-sullivansolutions-manual/

More information

The Straight Line. m is undefined. Use. Show that mab

The Straight Line. m is undefined. Use. Show that mab The Straight Line What is the gradient of a horizontal line? What is the equation of a horizontal line? So the equation of the x-axis is? What is the gradient of a vertical line? What is the equation of

More information