2 Unit Bridging Course Day 10
|
|
- Gilbert Sutton
- 5 years ago
- Views:
Transcription
1 1 / 31 Unit Bridging Course Day 10 Circular Functions III The cosine function, identities and derivatives Clinton Boys
2 / 31 The cosine function The cosine function, abbreviated to cos, is very similar to the sine function. In fact, the cos function is exactly the same, except shifted π/ units to the left.
3 3 / 31 The cosine function The cosine function, abbreviated to cos, is very similar to the sine function. In fact, the cos function is exactly the same, except shifted π/ units to the left.
4 4 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.
5 5 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.
6 6 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.
7 7 / 31 Graph of y = cos(x) Below is the graph of y = cos(x) between x = 4π and x = 4π. 1 4π 7π 3π 5π π 3π π π 1 π π 3π π 5π 3π 7π 4π The graph continues forever in both directions. Notice the similarities between cos and sin, as well as the differences.
8 8 / 31 Properties of cosine cos shares the following properties with sin: (i) 1 cos x 1 for all x. (ii) cos(x + π) = cos x for all x, i.e. cos x is periodic with period π, just like sin x.
9 9 / 31 Properties of cosine Unlike sin, however, cos is not odd: (iii) cos( x) = cos(x). 1 π π 1 y = cos x is symmetric about the y-axis we say it is an even function.
10 10 / 31 Sketching cosine curves Practice questions See if you can sketch the following cosine curves, using the same ideas we used to sketch sine curves. (i) y = cos x (ii) y = cos(x) (iii) y = 3 cos(x).
11 11 / 31 Sketching cosine curves Answers (i) y = cos x π π
12 1 / 31 Sketching cosine curves Answers (ii) y = cos(x) 1 1 π 4 π
13 13 / 31 Sketching cosine curves Answers (iii) y = 3 cos(x) 3 π 4 π 3
14 14 / 31 Identities involving circular functions Together, sin and cos are called the circular functions. There are many important identities involving circular functions which you should remember. (i) sin x + cos x = 1 (where sin x = (sin x) ) (ii) sin(x + y) = sin x cos y + cos x sin y (iii) cos(x + y) = cos x cos y sin x sin y (ii) and (iii) are known as double angle formulas. You can find plenty more such identities, for example on Wikipedia.
15 15 / 31 Identities involving circular functions Together, sin and cos are called the circular functions. There are many important identities involving circular functions which you should remember. (i) sin x + cos x = 1 (where sin x = (sin x) ) (ii) sin(x + y) = sin x cos y + cos x sin y (iii) cos(x + y) = cos x cos y sin x sin y (ii) and (iii) are known as double angle formulas. You can find plenty more such identities, for example on Wikipedia.
16 16 / 31 Identities involving circular functions Together, sin and cos are called the circular functions. There are many important identities involving circular functions which you should remember. (i) sin x + cos x = 1 (where sin x = (sin x) ) (ii) sin(x + y) = sin x cos y + cos x sin y (iii) cos(x + y) = cos x cos y sin x sin y (ii) and (iii) are known as double angle formulas. You can find plenty more such identities, for example on Wikipedia.
17 17 / 31 Derivatives of circular functions The circular functions, sin and cos, have particularly simple derivatives. Derivatives of the circular functions d d (sin x) = cos x Notice the derivative of cos is negative sin. (cos x) = sin x.
18 18 / 31 Derivatives of circular functions The circular functions, sin and cos, have particularly simple derivatives. Derivatives of the circular functions d d (sin x) = cos x Notice the derivative of cos is negative sin. (cos x) = sin x.
19 19 / 31 Derivatives of circular functions The circular functions, sin and cos, have particularly simple derivatives. Derivatives of the circular functions d d (sin x) = cos x Notice the derivative of cos is negative sin. (cos x) = sin x.
20 0 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule.
21 1 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule.
22 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule. df = 3 cos(x) d (x)
23 3 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule. df = 3 cos(x) d (x) = 3 cos(x)
24 4 / 31 Derivatives of circular functions Example Find the derivative of the function f (x) = 3 sin(x). We need to use the chain rule. df = 3 cos(x) d (x) = 3 cos(x) = 6 cos(x).
25 5 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then
26 6 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then
27 7 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then dy = u dv + v du
28 8 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then dy = u dv + v du = sin x ( sin x) + cos x (cos x)
29 9 / 31 Derivatives of circular functions Example Find dy if y = sin x cos x. We need to use the product rule. Let u = sin x and v = cos x. Then dy = u dv + v du = sin x ( sin x) + cos x (cos x) = sin x + cos x.
30 30 / 31 Derivatives of circular functions Practice questions Find the derivatives of the following functions: (i) f (x) = sin x (ii) f (x) = x cos x (iii) f (x) = sin(x ) (iv) f (x) = sin x (usually written tan x). cos x
31 31 / 31 Derivatives of circular functions Answers to practice questions df (i) df (ii) = sin x cos x = x sin x + cos x (iii) df = x cos(x ) (iv) df = cos x+sin x cos x = 1 cos x.
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 informationSection 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 informationMath 144 Activity #4 Connecting the unit circle to the graphs of the trig functions
144 p 1 Math 144 Activity #4 Connecting the unit circle to the graphs of the trig functions Graphing the sine function We are going to begin this activity with graphing the sine function ( y = sin x).
More informationReview 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 informationMid-Chapter Quiz: Lessons 9-1 through 9-3
Graph each point on a polar grid. 1. A( 2, 45 ) 3. Because = 45, locate the terminal side of a 45 angle with the polar axis as its initial side. Because r = 2, plot a point 2 units from the pole in the
More information7.2 Trigonometric Integrals
7. Trigonometric Integrals The three identities sin x + cos x, cos x (cos x + ) and sin x ( cos x) can be used to integrate expressions involving powers of Sine and Cosine. The basic idea is to use an
More information2/3 Unit Math Homework for Year 12
Yimin Math Centre 2/3 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 12 Trigonometry 2 1 12.1 The Derivative of Trigonometric Functions....................... 1 12.2
More informationUnit 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 informationThis 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 informationHigher. The Wave Equation. The Wave Equation 146
Higher Mathematics UNIT OUTCOME 4 The Wave Equation Contents The Wave Equation 146 1 Expressing pcosx + qsinx in the form kcos(x a 146 Expressing pcosx + qsinx in other forms 147 Multiple Angles 148 4
More informationModule 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 informationMathematics 134 Calculus 2 With Fundamentals Exam 2 Answers/Solutions for Sample Questions March 2, 2018
Sample Exam Questions Mathematics 1 Calculus 2 With Fundamentals Exam 2 Answers/Solutions for Sample Questions March 2, 218 Disclaimer: The actual exam questions may be organized differently and ask questions
More informationGraphing Trig Functions - Sine & Cosine
Graphing Trig Functions - Sine & Cosine Up to this point, we have learned how the trigonometric ratios have been defined in right triangles using SOHCAHTOA as a memory aid. We then used that information
More informationSUM AND DIFFERENCES. Section 5.3 Precalculus PreAP/Dual, Revised 2017
SUM AND DIFFERENCES Section 5. Precalculus PreAP/Dual, Revised 2017 Viet.dang@humbleisd.net 8/1/2018 12:41 AM 5.4: Sum and Differences of Trig Functions 1 IDENTITY Question 1: What is Cosine 45? Question
More informationBasic Graphs of the Sine and Cosine Functions
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
More informationTranslation 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 informationSection 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 informationVerifying 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 informationSection 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 informationDownloaded 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 informationTrigonometric Integrals
Most trigonometric integrals can be solved by using trigonometric identities or by following a strategy based on the form of the integrand. There are some that are not so easy! Basic Trig Identities and
More informationScience One Math. Jan
Science One Math Jan 24 2018 Announcements Midterm : February 13 th, 2018 Details on topics and list of practice problems to appear on course webpage soon. What integration techniques have we learned so
More informationMath 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 informationYou are not expected to transform y = tan(x) or solve problems that involve the tangent function.
In this unit, we will develop the graphs for y = sin(x), y = cos(x), and later y = tan(x), and identify the characteristic features of each. Transformations of y = sin(x) and y = cos(x) are performed and
More informationUnit #11 : Integration by Parts, Average of a Function. Goals: Learning integration by parts. Computing the average value of a function.
Unit #11 : Integration by Parts, Average of a Function Goals: Learning integration by parts. Computing the average value of a function. Integration Method - By Parts - 1 Integration by Parts So far in
More informationWalt 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 information4.2 Graphing Inverse Trigonometric Functions
4.2 Graphing Inverse Trigonometric Functions Learning Objectives Understand the meaning of restricted domain as it applies to the inverses of the six trigonometric functions. Apply the domain, range and
More informationSection 5.4: Modeling with Circular Functions
Section 5.4: Modeling with Circular Functions Circular Motion Example A ferris wheel with radius 25 feet is rotating at a rate of 3 revolutions per minute, When t = 0, a chair starts at its lowest point
More informationPolar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 28 / 46
Polar Coordinates Polar Coordinates: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ)
More informationUse the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before.
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationChapter 7. Exercise 7A. dy dx = 30x(x2 3) 2 = 15(2x(x 2 3) 2 ) ( (x 2 3) 3 ) y = 15
Chapter 7 Exercise 7A. I will use the intelligent guess method for this question, but my preference is for the rearranging method, so I will use that for most of the questions where one of these approaches
More informationChapter 5.4: Sinusoids
Chapter 5.4: Sinusoids If we take our circular functions and unwrap them, we can begin to look at the graphs of each trig function s ratios as a function of the angle in radians. We will begin by looking
More informationScience One Math. Jan
Science One Math Jan 21 2019 Today s Goal: Compute Trigonometric Integrals Apply the technique of substitution to compute trigonometric integrals of the form sin % x cos * x dx for both m > 1 and n > 1
More informationMath 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 informationPre Calculus Worksheet: Fundamental Identities Day 1
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationTImath.com Algebra 2. Proof of Identity
TImath.com Algebra Proof of Identity ID: 9846 Time required 45 minutes Activity Overview Students use graphs to verify the reciprocal identities. They then use the handheld s manual graph manipulation
More informationPolar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 27 / 45
: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations
More informationLesson 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 informationMHF4U. Advanced Functions Grade 12 University Mitchell District High School. Unit 5 Trig Functions & Equations 5 Video Lessons
MHF4U Advanced Functions Grade 12 University Mitchell District High School Unit 5 Trig Functions & Equations 5 Video Lessons Allow no more than 12 class days for this unit! This includes time for review
More informationSOME PROPERTIES OF TRIGONOMETRIC FUNCTIONS. 5! x7 7! + = 6! + = 4! x6
SOME PROPERTIES OF TRIGONOMETRIC FUNCTIONS PO-LAM YUNG We defined earlier the sine cosine by the following series: sin x = x x3 3! + x5 5! x7 7! + = k=0 cos x = 1 x! + x4 4! x6 6! + = k=0 ( 1) k x k+1
More information8-1 Simple Trigonometric Equations. Objective: To solve simple Trigonometric Equations and apply them
Warm Up Use your knowledge of UC to find at least one value for q. 1) sin θ = 1 2 2) cos θ = 3 2 3) tan θ = 1 State as many angles as you can that are referenced by each: 1) 30 2) π 3 3) 0.65 radians Useful
More information4.7a Trig Inverses.notebook September 18, 2014
WARM UP 9 18 14 Recall from Algebra 2 (or possibly see for the first time...): In order for a function to have an inverse that is also a function, it must be one to one, which means it must pass the horizontal
More informationAP 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 information1. 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 informationTrigonometric 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 information2.3 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 informationReminder: y =f(x) mean that a function f uses a variable (an ingredient) x to make the result y.
Functions (.3) Reminder: y =f(x) mean that a function f uses a variable (an ingredient) x to make the result y.. Transformation of functions 3 We know many elementary functions like f ( x) = x + x + 3x+,
More informationMATH 19520/51 Class 15
MATH 19520/51 Class 15 Minh-Tam Trinh University of Chicago 2017-11-01 1 Change of variables in two dimensions. 2 Double integrals via change of variables. Change of Variables Slogan: An n-variable substitution
More informationGraphing 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 informationPARAMETRIC EQUATIONS AND POLAR COORDINATES
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. PARAMETRIC EQUATIONS
More informationGraphs 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 informationName: 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 informationSec 4.1 Trigonometric Identities Basic Identities. Name: Reciprocal Identities:
Sec 4. Trigonometric Identities Basic Identities Name: Reciprocal Identities: Quotient Identities: sin csc cos sec csc sin sec cos sin tan cos cos cot sin tan cot cot tan Using the Reciprocal and Quotient
More informationMath 2412 Activity 4(Due with Final Exam)
Math Activity (Due with Final Exam) Use properties of similar triangles to find the values of x and y x y 7 7 x 5 x y 7 For the angle in standard position with the point 5, on its terminal side, find the
More informationNotice 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 informationTrigonometric Graphs Dr. Laura J. Pyzdrowski
1 Names: About this Laboratory In this laboratory, we will examine trigonometric functions and their graphs. Upon completion of the lab, you should be able to quickly sketch such functions and determine
More informationSection 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 informationPresented, and Compiled, By. Bryan Grant. Jessie Ross
P a g e 1 Presented, and Compiled, By Bryan Grant Jessie Ross August 3 rd, 2016 P a g e 2 Day 1 Discovering Polar Graphs Days 1 & 2 Adapted from Nancy Stephenson - Clements High School, Sugar Land, Texas
More informationCLEP Pre-Calculus. Section 1: Time 30 Minutes 50 Questions. 1. According to the tables for f(x) and g(x) below, what is the value of [f + g]( 1)?
CLEP Pre-Calculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. An online graphing calculator (non-cas) is allowed to be used for this section..
More informationEducation Resources. This section is designed to provide examples which develop routine skills necessary for completion of this section.
Education Resources Trigonometry Higher Mathematics Supplementary Resources Section A This section is designed to provide examples which develop routine skills necessary for completion of this section.
More informationJune 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 informationFirst of all, we need to know what it means for a parameterize curve to be differentiable. FACT:
CALCULUS WITH PARAMETERIZED CURVES In calculus I we learned how to differentiate and integrate functions. In the chapter covering the applications of the integral, we learned how to find the length of
More informationMath 144 Activity #2 Right Triangle Trig and the Unit Circle
1 p 1 Right Triangle Trigonometry Math 1 Activity #2 Right Triangle Trig and the Unit Circle We use right triangles to study trigonometry. In right triangles, we have found many relationships between the
More information1. 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 informationA 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 informationNational 5 Portfolio Relationships 1.5 Trig equations and Graphs
National 5 Portfolio Relationships 1.5 Trig equations and Graphs N5 Section A - Revision This section will help you revise previous learning which is required in this topic. R1 I can use Trigonometry in
More informationTrigonometric Functions. Copyright Cengage Learning. All rights reserved.
4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Evaluate and graph
More informationPacket Unit 5 Trigonometry Honors Math 2 17
Packet Unit 5 Trigonometry Honors Math 2 17 Homework Day 12 Part 1 Cumulative Review of this unit Show ALL work for the following problems! Use separate paper, if needed. 1) If AC = 34, AB = 16, find sin
More informationUsing Fundamental Identities. Fundamental Trigonometric Identities. Reciprocal Identities. sin u 1 csc u. sec u. sin u Quotient Identities
3330_050.qxd /5/05 9:5 AM Page 374 374 Chapter 5 Analytic Trigonometry 5. Using Fundamental Identities What you should learn Recognize and write the fundamental trigonometric identities. Use the fundamental
More informationUnit 3, Lesson 1.3 Special Angles in the Unit Circle
Unit, Lesson Special Angles in the Unit Circle Special angles exist within the unit circle For these special angles, it is possible to calculate the exact coordinates for the point where the terminal side
More informationChapter 7: Analytic Trigonometry
Chapter 7: Analytic Trigonometry 7. Trigonometric Identities Below are the basic trig identities discussed in previous chapters. Reciprocal csc(x) sec(x) cot(x) sin(x) cos(x) tan(x) Quotient sin(x) cos(x)
More informationAngle Measure 1. Use the relationship π rad = 180 to express the following angle measures in radian measure. a) 180 b) 135 c) 270 d) 258
Chapter 4 Prerequisite Skills BLM 4-1.. Angle Measure 1. Use the relationship π rad = 180 to express the following angle measures in radian measure. a) 180 b) 135 c) 70 d) 58. Use the relationship 1 =!
More informationUnit 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 informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13
MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus
More informationMAT137 Calculus! Lecture 31
MAT137 Calculus! Lecture 31 Today: Next: Integration Methods: Integration Methods: Trig. Functions (v. 9.10-9.12) Rational Functions Trig. Substitution (v. 9.13-9.15) (v. 9.16-9.17) Integration by Parts
More informationBil 104 Intiroduction To Scientific And Engineering Computing. Lecture 5. Playing with Data Modifiers and Math Functions Getting Controls
Readin from and Writint to Standart I/O BIL104E: Introduction to Scientific and Engineering Computing Lecture 5 Playing with Data Modifiers and Math Functions Getting Controls Pointers What Is a Pointer?
More informationMAT 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 informationTrigonometric Graphs. Graphs of Sine and Cosine
Trigonometric Graphs Page 1 4 Trigonometric Graphs Graphs of Sine and Cosine In Figure 13, we showed the graphs of = sin and = cos, for angles from 0 rad to rad. In reality these graphs extend indefinitely
More informationHW#50: Finish Evaluating Using Inverse Trig Functions (Packet p. 7) Solving Linear Equations (Packet p. 8) ALL
MATH 4R TRIGONOMETRY HOMEWORK NAME DATE HW#49: Inverse Trigonometric Functions (Packet pp. 5 6) ALL HW#50: Finish Evaluating Using Inverse Trig Functions (Packet p. 7) Solving Linear Equations (Packet
More informationMultiple 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 informationUnit 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 informationUnit 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 informationJim Lambers MAT 169 Fall Semester Lecture 33 Notes
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 33 Notes These notes correspond to Section 9.3 in the text. Polar Coordinates Throughout this course, we have denoted a point in the plane by an ordered
More information2.9 Linear Approximations and Differentials
2.9 Linear Approximations and Differentials 2.9.1 Linear Approximation Consider the following graph, Recall that this is the tangent line at x = a. We had the following definition, f (a) = lim x a f(x)
More informationBlue 21 Extend and Succeed Brain Growth Senior Phase. Trigonometry. Graphs and Equations
Blue 21 Extend and Succeed Brain Growth Senior Phase Trigonometry Graphs and Equations Trig Graphs O1 Trig ratios of angles of all sizes 1. Given the diagram above, find sin 130, cos 130 and tan 130 correct
More informationUnit 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 informationTopic 3 - Circular Trigonometry Workbook
Angles between 0 and 360 degrees 1. Set your GDC to degree mode. Topic 3 - Circular Trigonometry Workbook In the graph menu set the x-window from 0 to 90, and the y from -3 to 3. Draw the graph of y=sinx.
More information2 Second Derivatives. As we have seen, a function f (x, y) of two variables has four different partial derivatives: f xx. f yx. f x y.
2 Second Derivatives As we have seen, a function f (x, y) of two variables has four different partial derivatives: (x, y), (x, y), f yx (x, y), (x, y) It is convenient to gather all four of these into
More informationUnit 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 informationTrigonometry I. Exam 0
Trigonometry I Trigonometry Copyright I Standards 006, Test Barry Practice Mabillard. Exam 0 www.math0s.com 1. The minimum and the maximum of a trigonometric function are shown in the diagram. a) Write
More informationto 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 informationLESSON 1: Trigonometry Pre-test
LESSON 1: Trigonometry Pre-test Instructions. Answer each question to the best of your ability. If there is more than one answer, put both/all answers down. Try to answer each question, but if there is
More informationUse Parametric notation. Interpret the effect that T has on the graph as motion.
Learning Objectives Parametric Functions Lesson 3: Go Speed Racer! Level: Algebra 2 Time required: 90 minutes One of the main ideas of the previous lesson is that the control variable t does not appear
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
(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 information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 6: Trigonometric
More informationVertical 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 informationChapter 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 informationFunction f. Function f -1
Page 1 REVIEW (1.7) What is an inverse function? Do all functions have inverses? An inverse function, f -1, is a kind of undoing function. If the initial function, f, takes the element a to the element
More informationA Quick Review of Trigonometry
A Quick Review of Trigonometry As a starting point, we consider a ray with vertex located at the origin whose head is pointing in the direction of the positive real numbers. By rotating the given ray (initial
More information1. The Pythagorean Theorem
. The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be
More information