y= sin( x) y= cos( x)
|
|
- Miles Hodges
- 6 years ago
- Views:
Transcription
1 . The graphs of sin(x) and cos(x). Now I am going to define the two basic trig functions: sin(x) and cos(x). Study the diagram at the right. The circle has radius. The arm OP starts at the positive horizontal axis and rotates in a counter-clockwise direction. When it has turned through an angle of x radians, we define: sin(x) the distance from P to the horizontal axis cos(x) the distance from P to the vertical axis Now is a good moment to stop and get the class to try to draw what the graphs of sin(x) and cos(x) ought to look like. These graphs are displayed below, for x, along with their generator. Study these diagrams carefully. Note the periodicity it s clear that when the arm has turned through a complete revolution ( radians) and keeps on turning, the values of sin and cos repeat. Note also the relationship between the two graphs. Can you formulate that algebraically? O cos( x) x sin( x ) P / sin( ) 6 6 y sin( x) / y cos( x) cos( ) Note that in the upper graph, the circle is exactly the one from the original definition at the top. But the circle in the lower graph is turned from that through 9º (or /). Why has that happened? Well, function graphs are conventionally plotted with f(x) on the vertical axis. The original definition of sin(x) gives it as the length of a vertical segment, so it projects directly onto the graph. But cos(x) is defined as the length of a horizontal segment, and to generate the cos graph I had to rotate the circle so that the x-axis was vertical. sin&cos 8/6/7
2 In summary, here are more complete versions of the graphs, plotted together, emphasizing the relationship between them, that the cos graph is obtained by translating the sin graph an amount / to the left. y cosx y sinx There are lots of simple arithmetic relationships involving sin and cos, and I will record some of these here. But I warn you not to overburden your memory with these. When you need to use them, look at the circle definition or look at the graph (or look at them both!) Observe that sin is an odd function (graph symmetric through the origin) and cos is an even function (graph symmetric about the y-axis). Algebraically, this tells us that: sin( x) sin(x) cos( x) cos(x) The internal symmetries of each of the graphs give us: x cos( x) x sin(x) sin( x) sin(+x) sin( x). cos(x) cos( x) cos(+x) cos( x). O sin( x ) This can be seen both from the symmetry of the graphs and from the circle diagram at the right. +x x There are two important ways of getting a relationship between sin and cos. The first is through translation. The fact that if we move the singraph / units left we get the cos-graph can be written: cos(x) sin( +x) and since cos(x) cos( x), this can be also written: cos(x) sin( x). The second relationship between sin and cos comes from the Pythagorean Theorem applied to the bold triangle at the right: sin (x) + cos (x). O x sin( x ) cos( x) sin&cos 8/6/7
3 What should sin(x+y) be? Should it be the sum of sin(x) and sin(y)? The answer is emphatically no! If it were, we would have a great proof that : sin() sin( ) + sin( ) +!!! In fact, here are the formulas for the sin and cos of the sum and difference of angles. sin(x+y) sin(x)cos(y) + cos(x)sin(y) sin(x y) sin(x)cos(y) cos(x)sin(y) cos(x+y) cos(x)cos(y) sin(x)sin(y) cos(x y) cos(x)cos(y) + sin(x)sin(y) There are a number of ways to establish these formulas, but the most elegant I ve seen I pulled off the web at Some time ago a student submitted an assignment with the assertion sin( 4 ) ½. When questioned, he replied that since sin( ), should not sin( 4 ) be half that? In fact, he was implicitly assuming that sin(x) is a linear function of the form kx. But it isn t. That same error leads popular misstatements such as: sin(x) sin(x) sin(x+y) sin(x) + sin(y) neither of which are true. See below. It uses the simple fact that if you take the basic sin-cos triangle and scale it by a factor of k, you will have ksinx and kcosx appearing as the lengths of the legs of the triangle. Let s try to establish the sin(x+y) formula. For this we would want to organize things so that sin(x) and cos(x) appeared in the role of k in a couple of angle y triangles. Well here s the diagram. x y sinx cosx y sinx cosy cosx siny sin( x+y) sin(x+y) sinx cosy + cosx siny This is a lovely proof without words. However, for those who need a kick start, I ll add a few words. There are three right-angled triangles. The big one, with angle x, has hypotenuse and sides cosx and sinx. The two smaller ones have angle y and would have side lengths cosy and siny, except they are scaled down, one by the factor cosx and the other by the factor sinx. Thus, instead of side length cosy and siny, these triangles have side length cosx siny and sinx cosy. If we now step back and imagine the triangle with angle x+y (which is not drawn) the length of the vertical side is the sum of these. And that gives us the sin(x+y) formula. sin&cos 8/6/7 3
4 Double and half angle. Setting xy in the sum formulae, we get: sin(x) sin(x)cos(x) cos(x) cos x sin x. cos x sin x The last two formulae for cos(x) give us the half-angle formulas (replace x by x/). x + cos x cos x cos x sin The cos(x) formula has three forms, the nd and 3 rd obtained from the st with a substitution from sin + cos. The last two forms are useful because they can be solved for cosx and sinx in terms of cos(x), and that gives us the half-angle formulae. Special angles Recall that the special and triangles give us simple exact expressions for the sin and cos of the angles. If we write the angles in radians, we get the same formulae with a different look. /6 /6 We summarize the results in the following table. 3 sin sin /6 / sin /3 3 / sin /4 / sin / cos cos /6 3 / cos /3 / cos /4 / cos / These angles are unusual in that we get exact expressions for sin and cos. Are there other angles for which this is also true? Of course there are the corresponding angles in other quadrants such as /3 and /4, but what about angles other than these? Well the sum and difference formulae for sin and cos will give us expressions for any angle that can be built out of special ones by adding or subtracting. And they in turn give rise to the double- and half-angle formulae. These all give us ways of generating expressions for the sin and cos of new angles. /3 /3 /4 /4 sin&cos 8/6/7 4
5 Example. Here are two different ways to write 5 in terms of the special angles 5º 6º 45º. In radians: 5 3 / Use both the sin difference formula and the half-angle formula to get an exact expression for sin(/). Compare your answers. Solution: The sin difference formula gives us: sin sin( ) sin cos cos sin The half-angle formula gives us: sin 3 cos( / 6) 3 4 ( 3 / ) 3 That s an answer of sorts. But it s a bit more complicated than it ought to be, and besides it s not at all clear that the two answers are the same! We certainly have to do something about that. There are a number of things we might do at this point. Now this is interesting. The two standard ways we might do this calculation give us two different answers and it s not at all clear they are the same. There s an interesting new technique comes out of this: how to take the square root of something like a + b 3. ) Show directly that the two answers are the same. The simplest way to do that is to square both answers and compare. ) Suppose we didn t have the first answer and ask how we might simplify the second answer. That seems like an interesting (and possibly useful) question. How do you take the square root of something of the form: a + b 3? See problem. sin&cos 8/6/7 5
6 Problems.. Calculate the following. Give exact answers. Work with a picture, either the cos and sin graphs, or the circle diagram (or both!) (a) sin(/3) (b) cos(3/4) (c) sin(7/4) (d) sin( 5/4) (e) cos(4/3) (f) cos( 5/6) (g) sin(/6) (h) cos(4/3) (i) sin( 9/6) (j) sin(/4). Find all values of x in the interval [, 4] which satisfy the equation sinx sin 9. Display all your solutions on a copy of the graph y sinx. 3. Use the internal symmetry of the sin and cos graphs to simplify the following: (a) sin(x + ) (b) cos(x /) (c) sin(x + 3/) (d) sin(5/ x) (e) cos(4/3 x) 4. Use the addition laws for sin and cos to calculate the following. Simplify as much as possible. (a) sin(x + /4) (b) cos(x /3) (c) sin(x 5/4) (d) sin(5/6 x) (e) cos(3/4 x) 5.(a) Use the sin sum formula to calculate sin(5/). (b) Use the half-angle formula to calculate sin(/8). Simplify. (c) Noting that /6 /8 /4, calculate cos(/4). 6. Investigate the solutions of the equation sin(x) cos(x/) ( x 4) using two approaches: (a) On the same set of axes, draw rough graphs of sin(x) and cos(x/) and approximately locate the intersections. (b) Use the half-angle formula for cos(x/) and work algebraically with the resulting equation to get more exact decimal approximations for the solutions. Use the cos button on your calculator. 7. How many solutions are there to the equation: x 9 sinx? Illustrate your answer on a graph of sinx, and find exact values for as many solutions as you can. 8. Use the formula sin (x) + cos (x) to find all solutions of the equation sinx + 3cosx in the interval x. sin&cos 8/6/7 6
7 9. Refer to Example. (a) Show that an expression of the form a + b 3 has a square root that is of the same form (if it is positive). (b) Calculate the square root of 3. Use this to show that the two answers we obtained in Example are equal..(a) The sin graph is drawn below. Sketch on this set of axes what you think the graph of sin (x) ought to look like. Justify the principle features of your graph. Use one of the formulae of this section to obtain a very revealing alternative formula for sin (x). [Don t use your graphing calculator for this problem!] (b) Now use the translation property that cos(x) sin( +x) to draw, on the same set of axes, the graph of cos (x). Finally make a sketch of the curve that lies exactly halfway between your two graphs. Use the Pythagorean relation: sin (x) + cos (x) to explain an unexpected property of this graph.. Study the graphs of sinx and cosx drawn below: y cosx y sinx (a) Based on this diagram, sketch the graph of the average of the two functions: sin x + cos x f(x). (b) Argue that the graph of f(x) is periodic with period and bilateral symmetric about the line x /4. There are different ways to handle the bilateral symmetry, some quite elegant. (c) Now find a simplified expression for f(x) which makes it much clearer what the graph of f(x) really is. [Hint: try to adapt the sin-sum formula.] sin&cos 8/6/7 7
The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.
We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section
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 informationUnit 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 informationThis unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA.
Angular Rotations This unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA. sin x = opposite hypotenuse cosx =
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 informationEach point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
More informationMath 144 Activity #7 Trigonometric Identities
44 p Math 44 Activity #7 Trigonometric Identities What is a trigonometric identity? Trigonometric identities are equalities that involve trigonometric functions that are true for every single value of
More informationSection 4.1: Introduction to Trigonometry
Section 4.1: Introduction to Trigonometry Review of Triangles Recall that the sum of all angles in any triangle is 180. Let s look at what this means for a right triangle: A right angle is an angle which
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
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 information1
Zeros&asymptotes Example 1 In an early version of this activity I began with a sequence of simple examples (parabolas and cubics) working gradually up to the main idea. But now I think the best strategy
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 informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationMEI Desmos Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x² 4x + 1 2. Add a line, e.g. y = x 3 3. Select the points of intersection of the line and the curve. What
More informationDefinitions 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 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 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 informationTrigonometry and the Unit Circle. Chapter 4
Trigonometry and the Unit Circle Chapter 4 Topics Demonstrate an understanding of angles in standard position, expressed in degrees and radians. Develop and apply the equation of the unit circle. Solve
More information4.1: 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 informationPolar Coordinates. 2, π and ( )
Polar Coordinates Up to this point we ve dealt exclusively with the Cartesian (or Rectangular, or x-y) coordinate system. However, as we will see, this is not always the easiest coordinate system to work
More informationSection 10.1 Polar Coordinates
Section 10.1 Polar Coordinates Up until now, we have always graphed using the rectangular coordinate system (also called the Cartesian coordinate system). In this section we will learn about another system,
More informationTrigonometry Review Version 0.1 (September 6, 2004)
Trigonometry Review Version 0. (September, 00 Martin Jackson, University of Puget Sound The purpose of these notes is to provide a brief review of trigonometry for students who are taking calculus. The
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 informationTo graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
More informationLinear algebra deals with matrixes: two-dimensional arrays of values. Here s a matrix: [ x + 5y + 7z 9x + 3y + 11z
Basic Linear Algebra Linear algebra deals with matrixes: two-dimensional arrays of values. Here s a matrix: [ 1 5 ] 7 9 3 11 Often matrices are used to describe in a simpler way a series of linear equations.
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 informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More information1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
More informationChapter 3A Rectangular Coordinate System
Fry Texas A&M University! Math 150! Spring 2015!!! Unit 4!!! 1 Chapter 3A Rectangular Coordinate System A is any set of ordered pairs of real numbers. The of the relation is the set of all first elements
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 informationComplex Numbers, Polar Equations, and Parametric Equations. Copyright 2017, 2013, 2009 Pearson Education, Inc.
8 Complex Numbers, Polar Equations, and Parametric Equations Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 8.5 Polar Equations and Graphs Polar Coordinate System Graphs of Polar Equations Conversion
More informationIn section 8.1, we began by introducing the sine function using a circle in the coordinate plane:
Chapter 8.: Degrees and Radians, Reference Angles In section 8.1, we began by introducing the sine function using a circle in the coordinate plane: y (3,3) θ x We now return to the coordinate plane, but
More informationThe Addition Formulas in Trigonometry. Scott Fallstrom Faculty Director, Math Learning Center
The Addition Formulas in Trigonometry Scott Fallstrom Faculty Director, Math Learning Center Why not the usual? In Mathematics, we know that the distributive property allows 7(x + 5) = 7x + 35 With derivatives,
More information+ b. From this we can derive the following equations:
A. GEOMETRY REVIEW Pythagorean Theorem (A. p. 58) Hypotenuse c Leg a 9º Leg b The Pythagorean Theorem is a statement about right triangles. A right triangle is one that contains a right angle, that is,
More informationMastery. PRECALCULUS Student Learning Targets
PRECALCULUS Student Learning Targets Big Idea: Sequences and Series 1. I can describe a sequence as a function where the domain is the set of natural numbers. Connections (Pictures, Vocabulary, Definitions,
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 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 informationTo graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6. ) is graphed below:
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
More informationAlgebra 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 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 informationCHAPTER 2 REVIEW COORDINATE GEOMETRY MATH Warm-Up: See Solved Homework questions. 2.2 Cartesian coordinate system
CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH6 2.1 Warm-Up: See Solved Homework questions 2.2 Cartesian coordinate system Coordinate axes: Two perpendicular lines that intersect at the origin O on each line.
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 informationLesson #64 First Degree Trigonometric Equations
Lesson #64 First Degree Trigonometric Equations A2.A.68 Solve trigonometric equations for all values of the variable from 0 to 360 How is the acronym ASTC used in trigonometry? If I wanted to put the reference
More informationFranklin Math Bowl 2008 Group Problem Solving Test Grade 6
Group Problem Solving Test Grade 6 1. The fraction 32 17 can be rewritten by division in the form 1 p + q 1 + r Find the values of p, q, and r. 2. Robert has 48 inches of heavy gauge wire. He decided to
More informationLesson 27: Angles in Standard Position
Lesson 27: Angles in Standard Position PreCalculus - Santowski PreCalculus - Santowski 1 QUIZ Draw the following angles in standard position 50 130 230 320 770-50 2 radians PreCalculus - Santowski 2 Fast
More informationSummer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.
Summer Review for Students Entering Pre-Calculus with Trigonometry 1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios
More informationPrecalculus: 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 informationWK # Given: f(x) = ax2 + bx + c
Alg2H Chapter 5 Review 1. Given: f(x) = ax2 + bx + c Date or y = ax2 + bx + c Related Formulas: y-intercept: ( 0, ) Equation of Axis of Symmetry: x = Vertex: (x,y) = (, ) Discriminant = x-intercepts: When
More informationSummer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.
1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios and Pythagorean Theorem 4. Multiplying and Dividing Rational Expressions
More informationMATHEMATICS 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 informationCCNY 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 information2. Periodic functions have a repeating pattern called a cycle. Some examples from real-life that have repeating patterns might include:
GRADE 2 APPLIED SINUSOIDAL FUNCTIONS CLASS NOTES Introduction. To date we have studied several functions : Function linear General Equation y = mx + b Graph; Diagram Usage; Occurence quadratic y =ax 2
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 informationAn interesting related problem is Buffon s Needle which was first proposed in the mid-1700 s.
Using Monte Carlo to Estimate π using Buffon s Needle Problem An interesting related problem is Buffon s Needle which was first proposed in the mid-1700 s. Here s the problem (in a simplified form). Suppose
More informationThe triangle
The Unit Circle The unit circle is without a doubt the most critical topic a student must understand in trigonometry. The unit circle is the foundation on which trigonometry is based. If someone were to
More informationCircles & Other Conics
SECONDARY MATH TWO An Integrated Approach MODULE 8 Circles & Other Conics The Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 2017 Original work 2013 in partnership with the
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 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 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 informationAP Calculus AB. Table of Contents. Slide 1 / 180. Slide 2 / 180. Slide 3 / 180. Review Unit
Slide 1 / 180 Slide 2 / 180 P alculus Review Unit 2015-10-20 www.njctl.org Table of ontents lick on the topic to go to that section Slide 3 / 180 Slopes Equations of Lines Functions Graphing Functions
More informationIn this chapter, we will investigate what have become the standard applications of the integral:
Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,
More informationExam 2 Review. 2. What the difference is between an equation and an expression?
Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? 2. What the difference is between an equation and an expression? 3. How to tell if an equation is linear? 4. How
More informationChapter 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 informationUnit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8)
Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8) Standards: Geom 19.0, Geom 20.0, Trig 7.0, Trig 8.0, Trig 12.0 Segerstrom High School -- Math Analysis Honors Name: Period:
More information2.0 Trigonometry Review Date: Pythagorean Theorem: where c is always the.
2.0 Trigonometry Review Date: Key Ideas: The three angles in a triangle sum to. Pythagorean Theorem: where c is always the. In trigonometry problems, all vertices (corners or angles) of the triangle are
More informationAQA GCSE Further Maths Topic Areas
AQA GCSE Further Maths Topic Areas This document covers all the specific areas of the AQA GCSE Further Maths course, your job is to review all the topic areas, answering the questions if you feel you need
More information12.4 Rotations. Learning Objectives. Review Queue. Defining Rotations Rotations
12.4. Rotations www.ck12.org 12.4 Rotations Learning Objectives Find the image of a figure in a rotation in a coordinate plane. Recognize that a rotation is an isometry. Review Queue 1. Reflect XY Z with
More informationFind the amplitude, period, and phase shift, and vertical translation of the following: 5. ( ) 6. ( )
1. Fill in the blanks in the following table using exact values. Reference Angle sin cos tan 11 6 225 2. Find the exact values of x that satisfy the given condition. a) cos x 1, 0 x 6 b) cos x 0, x 2 3.
More informationMath for Geometric Optics
Algebra skills Math for Geometric Optics general rules some common types of equations units problems with several variables (substitution) Geometry basics Trigonometry Pythagorean theorem definitions,
More informationNARROW CORRIDOR. Teacher s Guide Getting Started. Lay Chin Tan Singapore
Teacher s Guide Getting Started Lay Chin Tan Singapore Purpose In this two-day lesson, students are asked to determine whether large, long, and bulky objects fit around the corner of a narrow corridor.
More informationSAT Timed Section*: Math
SAT Timed Section*: Math *These practice questions are designed to be taken within the specified time period without interruption in order to simulate an actual SAT section as much as possible. Time --
More informationAppendix D Trigonometry
Math 151 c Lynch 1 of 8 Appendix D Trigonometry Definition. Angles can be measure in either degree or radians with one complete revolution 360 or 2 rad. Then Example 1. rad = 180 (a) Convert 3 4 into degrees.
More information1.6 Applying Trig Functions to Angles of Rotation
wwwck1org Chapter 1 Right Triangles and an Introduction to Trigonometry 16 Applying Trig Functions to Angles of Rotation Learning Objectives Find the values of the six trigonometric functions for angles
More informationMATH EXAM 1 - SPRING 2018 SOLUTION
MATH 140 - EXAM 1 - SPRING 018 SOLUTION 8 February 018 Instructor: Tom Cuchta Instructions: Show all work, clearly and in order, if you want to get full credit. If you claim something is true you must
More informationSection 1.1 Graphs Graphs
Section 1.1 Graphs 55 1.1 Graphs Much of algebra is concerned with solving equations. Many algebraic techniques have been developed to provide insights into various sorts of equations, and those techniques
More informationPre-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 informationMEI GeoGebra Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x 2 4x + 1 2. Add a line, e.g. y = x 3 3. Use the Intersect tool to find the points of intersection of
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS In this section, we assume that you have access to a graphing calculator or a computer with graphing software. FUNCTIONS AND MODELS 1.4 Graphing Calculators
More informationA Mathematica Tutorial
A Mathematica Tutorial -3-8 This is a brief introduction to Mathematica, the symbolic mathematics program. This tutorial is generic, in the sense that you can use it no matter what kind of computer you
More informationBasics of Computational Geometry
Basics of Computational Geometry Nadeem Mohsin October 12, 2013 1 Contents This handout covers the basic concepts of computational geometry. Rather than exhaustively covering all the algorithms, it deals
More informationIsometries. 1 Identifying Isometries
Isometries 1 Identifying Isometries 1. Modeling isometries as dynamic maps. 2. GeoGebra files: isoguess1.ggb, isoguess2.ggb, isoguess3.ggb, isoguess4.ggb. 3. Guessing isometries. 4. What can you construct
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 informationMEI Casio Tasks for A2 Core
Task 1: Functions The Modulus Function The modulus function, abs(x), is found using OPTN > NUMERIC > Abs 2. Add the graph y = x, Y1=Abs(x): iyqfl 3. Add the graph y = ax+b, Y2=Abs(Ax+B): iyqaff+agl 4.
More informationWhat is log a a equal to?
How would you differentiate a function like y = sin ax? What is log a a equal to? How do you prove three 3-D points are collinear? What is the general equation of a straight line passing through (a,b)
More informationIn this section, we will study the following topics:
6.1 Radian and Degree Measure In this section, we will study the following topics: Terminology used to describe angles Degree measure of an angle Radian measure of an angle Converting between radian and
More informationC. HECKMAN TEST 2A SOLUTIONS 170
C HECKMN TEST SOLUTIONS 170 (1) [15 points] The angle θ is in Quadrant IV and tan θ = Find the exact values of 5 sin θ, cos θ, tan θ, cot θ, sec θ, and csc θ Solution: point that the terminal side of the
More informationName Trigonometric Functions 4.2H
TE-31 Name Trigonometric Functions 4.H Ready, Set, Go! Ready Topic: Even and odd functions The graphs of even and odd functions make it easy to identify the type of function. Even functions have a line
More information1. 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 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 informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2017 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More informationMATHEMATICS FOR ENGINEERING TUTORIAL 5 COORDINATE SYSTEMS
MATHEMATICS FOR ENGINEERING TUTORIAL 5 COORDINATE SYSTEMS This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning by example.
More information5.5 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 informationInvestigating the Sine and Cosine Functions Part 1
Investigating the Sine and Cosine Functions Part 1 Name: Period: Date: Set-Up Press. Move down to 5: Cabri Jr and press. Press for the F1 menu and select New. Press for F5 and select Hide/Show > Axes.
More informationStandard Chapter/Unit Notes
. (C, F, T).,00 (C, F, S, T). (C, F, P, S, T).,,, (C, T) Workout. (C, F, P, S, T). (C, F, P, S, T). (C, F, G, P, T). (C, F, P, T). (C, E, G, P, T) 0. (C, F, P, S, T) Solution/Representation - Problem #0
More informationA 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 informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More informationB ABC is mapped into A'B'C'
h. 00 Transformations Sec. 1 Mappings & ongruence Mappings Moving a figure around a plane is called mapping. In the figure below, was moved (mapped) to a new position in the plane and the new triangle
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. NEW
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 informationUnit 1, Lesson 1: Moving in the Plane
Unit 1, Lesson 1: Moving in the Plane Let s describe ways figures can move in the plane. 1.1: Which One Doesn t Belong: Diagrams Which one doesn t belong? 1.2: Triangle Square Dance m.openup.org/1/8-1-1-2
More information