10.7. Polar Coordinates. Introduction. What you should learn. Why you should learn it. Example 1. Plotting Points on the Polar Coordinate System


 Horace Ward
 2 years ago
 Views:
Transcription
1 _7.qxd /8/5 9: AM Page 779 Section.7 Polar Coordinates Polar Coordinates What ou should learn Plot points on the polar coordinate sstem. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and vice versa. Wh ou should learn it Polar coordinates offer a different mathematical perspective on graphing. For instance, in Exercises 8 on page 78, ou are asked to find multiple representations of polar coordinates. Introduction So far, ou have been representing graphs of equations as collections of points x, on the rectangular coordinate sstem, where x and represent the directed distances from the coordinate axes to the point x,. In this section, ou will stud a different sstem called the polar coordinate sstem. To form the polar coordinate sstem in the plane, fix a point O, called the pole (or origin), and construct from O an initial ra called the polar axis, as shown in Figure.57. Then each point P in the plane can be assigned polar coordinates r, as follows.. r directed distance from O to P. directed angle, counterclockwise from polar axis to segment OP O r = directed distance θ FIGURE.57 P= ( r, θ) = directed angle Polar axis Example Plotting Points on the Polar Coordinate Sstem a. The point r,, lies two units from the pole on the terminal side of the angle as shown in Figure.58. b. The point r,, lies three units from the pole on the terminal side of the angle as shown in Figure.59. c. The point r,, coincides with the point,, as shown in Figure..,, θ =, FIGURE.58 θ = FIGURE.59, Now tr Exercise. FIGURE., θ =
2 _7.qxd /8/5 9: AM Page Chapter Topics in Analtic Geometr, θ =, = (, 5 ) =, 7 =, =... FIGURE. Pole Exploration Most graphing calculators have a polar graphing mode. If ours does, graph the equation r. (Use a setting in which x and.) You should obtain a circle of radius. a. Use the trace feature to cursor around the circle. Can ou locate the point, 5? b. Can ou find other polar representations of the point, 5? If so, explain how ou did it. (Origin) FIGURE. θ r x (r, θ) (x, ) x Polar axis (xaxis) In rectangular coordinates, each point x, has a unique representation. This is not true for polar coordinates. For instance, the coordinates r, and r, represent the same point, as illustrated in Example. Another wa to obtain multiple representations of a point is to use negative values for r. Because r is a directed distance, the coordinates r, and r, represent the same point. In general, the point r, can be represented as r, r, ± n or r, r, ± n where n is an integer. Moreover, the pole is represented b,, where is an angle. Example Multiple Representations of Points Plot the point, and find three additional polar representations of this point, using < <. The point is shown in Figure.. Three other representations are as follows., 5,, 7,,, Now tr Exercise. Coordinate Conversion Add to. Replace r b r; subtract Replace r b r; add to. from. To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive xaxis and the pole with the origin, as shown in Figure.. Because x, lies on a circle of radius r, it follows that r x. Moreover, for r >, the definitions of the trigonometric functions impl that tan x, cos x r, and sin r. If r <, ou can show that the same relationships hold. Coordinate Conversion The polar coordinates r, are related to the rectangular coordinates x, as follows. PolartoRectangular RectangulartoPolar x r cos r sin tan x r x
3 _7.qxd /8/5 9: AM Page 78 Example Section.7 Polar Coordinates 78 PolartoRectangular Conversion (, r θ) = (, ) (, x ) = (, ) FIGURE. Activities. Find three additional polar representations of the point (, r θ) =, (, x ) =,,., Answer:,, and, 5,. Convert the point, from rectangular to polar form. Answer: 5,. or 5,.779 x Convert each point to rectangular coordinates. a., b., a. For the point r,,, ou have the following. x r cos cos r sin sin The rectangular coordinates are x,,. (See Figure..) b. For the point r,, ou have the following. x cos sin, The rectangular coordinates are Now tr Exercise. x,,. Example RectangulartoPolar Conversion (x, ) = (, ) (r, θ) =, FIGURE. (x, ) = (, ) FIGURE.5 (r, θ) =, Convert each point to polar coordinates. a., b., a. For the secondquadrant point x,,, ou have tan x Because lies in the same quadrant as x,, use positive r. r x So, one set of polar coordinates is r,,, as shown in Figure.. b. Because the point x,, lies on the positive axis, choose. and r. This implies that one set of polar coordinates is r,,, as shown in Figure.5. Now tr Exercise 9.
4 _7.qxd /8/5 9: AM Page Chapter Topics in Analtic Geometr Multiple representations of points and equations in the polar sstem can cause confusion. You ma want to discuss several examples. Activities. Convert the polar equation r cos to rectangular form. Answer: x x. Convert the rectangular equation x to polar form. Answer: r sec Equation Conversion B comparing Examples and, ou can see that point conversion from the polar to the rectangular sstem is straightforward, whereas point conversion from the rectangular to the polar sstem is more involved. For equations, the opposite is true. To convert a rectangular equation to polar form, ou simpl replace x b r cos and b r sin. For instance, the rectangular equation x can be written in polar form as follows. x r sin r cos r sec tan Simplest form On the other hand, converting a polar equation to rectangular form requires considerable ingenuit. Example 5 demonstrates several polartorectangular conversions that enable ou to sketch the graphs of some polar equations. Example 5 Converting Polar Equations to Rectangular Form FIGURE. Describe the graph of each polar equation and find the corresponding rectangular equation. a. r b. c. r sec a. The graph of the polar equation r consists of all points that are two units from the pole. In other words, this graph is a circle centered at the origin with a radius of, as shown in Figure.. You can confirm this b converting to rectangular form, using the relationship r x. r r x FIGURE.7 b. The graph of the polar equation consists of all points on the line that makes an angle of with the positive polar axis, as shown in Figure.7. To convert to rectangular form, make use of the relationship tan x. tan x c. The graph of the polar equation r sec is not evident b simple inspection, so convert to rectangular form b using the relationship r cos x. r sec r cos x FIGURE.8 Now ou see that the graph is a vertical line, as shown in Figure.8. Now tr Exercise 5.
5 _7.qxd /8/5 :9 AM Page 78 Section.7 Polar Coordinates 78.7 Exercises VOCABULARY CHECK: Fill in the blanks.. The origin of the polar coordinate sstem is called the.. For the point r,, r is the from O to P and is the counterclockwise from the polar axis to the line segment OP.. To plot the point r,, use the coordinate sstem.. The polar coordinates r, are related to the rectangular coordinates x, as follows: x tan r PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Exercises 8, plot the point given in polar coordinates and find two additional polar representations of the point, using < <..,.. 5, 7., 5.,..,.57 7., ,. In Exercises 9, a point in polar coordinates is given. Convert the point to rectangular coordinates. 9.,. (, r θ) =, 5.,., (r, θ) =, 5,, (, r θ) =, (r, θ) = (, ) 7.,., 5..5,.. 8.5,.5 In Exercises 7, a point in rectangular coordinates is given. Convert the point to polar coordinates. 7., 8., 9.,., 5.,.,.,., 5., 9. 5, In Exercises 7, use a graphing utilit to find one set of polar coordinates for the point given in rectangular coordinates. 7., 8. 5, 9.,.,,.. 5, 7, In Exercises 8, convert the rectangular equation to polar form. Assume a >.. x 9. x 5.. x 7. x 8. x a 9. x. x 5. x. x. 8x. x 9x 5. x a. x 9a 7. x ax 8. x a
6 _7.qxd /8/5 9: AM Page Chapter Topics in Analtic Geometr In Exercises 9,convert the polar equation to rectangular form. 9. r sin 5. r cos r 5. r 55. r csc 5. r sec 57. r cos 58. r sin 59. r sin. r cos. r. r sin cos. r. r sin cos sin In Exercises 5 7, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. 5. r. r r sec 7. r csc Snthesis 5 True or False? In Exercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. If n for some integer n, then r, and r, represent the same point on the polar coordinate sstem. 7. If r r, then r, and r, represent the same point on the polar coordinate sstem. 7. Convert the polar equation r h cos k sin to rectangular form and verif that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle. 7. Convert the polar equation r cos sin to rectangular form and identif the graph. 75. Think About It (a) Show that the distance between the points r, and r, is r r r r cos. (b) Describe the positions of the points relative to each other for. Simplif the Distance Formula for this case. Is the simplification what ou expected? Explain. (c) Simplif the Distance Formula for 9. Is the simplification what ou expected? Explain. (d) Choose two points on the polar coordinate sstem and find the distance between them. Then choose different polar representations of the same two points and appl the Distance Formula again. Discuss the result. 7. Exploration (a) Set the window format of our graphing utilit on rectangular coordinates and locate the cursor at an position off the coordinate axes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor verticall. (b) Set the window format of our graphing utilit on polar coordinates and locate the cursor at an position off the coordinate axes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor verticall. (c) Explain wh the results of parts (a) and (b) are not the same. Skills Review In Exercises 77 8, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 77. log x z 78. log x 79. ln xx 8. ln 5x x In Exercises 8 8, condense the expression to the logarithm of a single quantit. 8. log 7 x log 7 8. log 5 a 8 log 5 x 8. ln x lnx 8. ln ln lnx In Exercises 85 9, use Cramer s Rule to solve the sstem of equations x 7 x a b c a b c a b 9c x z x z 5x z 9. x 5 x 5u u 8u x x x 7v v v 5 x x x 9w w w x x In Exercises 9 9, use a determinant to determine whether the points are collinear. 9.,,, 7,, 9.,,,,, 5 9.,,,,.5,.5 9.., 5,.5,,.5, 5 7 5
6.7. POLAR COORDINATES
6.7. POLAR COORDINATES What You Should Learn Plot points on the polar coordinate system. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and
More information9.5 Polar Coordinates. Copyright Cengage Learning. All rights reserved.
9.5 Polar Coordinates Copyright Cengage Learning. All rights reserved. Introduction Representation of graphs of equations as collections of points (x, y), where x and y represent the directed distances
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 informationTopics in Analytic Geometry Part II
Name Chapter 9 Topics in Analytic Geometry Part II Section 9.4 Parametric Equations Objective: In this lesson you learned how to evaluate sets of parametric equations for given values of the parameter
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 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 informationPolar Functions Polar coordinates
548 Chapter 1 Parametric, Vector, and Polar Functions 1. What ou ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed b Polar Curves A Small Polar Galler... and wh Polar
More informationInclination of a Line
0_00.qd 78 /8/05 Chapter 0 8:5 AM Page 78 Topics in Analtic Geometr 0. Lines What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and
More informationturn counterclockwise from the positive xaxis. However, we could equally well get to this point by a 3 4 turn clockwise, giving (r, θ) = (1, 3π 2
Math 133 Polar Coordinates Stewart 10.3/I,II Points in polar coordinates. The first and greatest achievement of modern mathematics was Descartes description of geometric objects b numbers, using a sstem
More informationGetting a New Perspective
Section 6.3 Polar Coordinates Getting a New Perspective We have worked etensively in the Cartesian coordinate system, plotting points, graphing equations, and using the properties of the Cartesian plane
More informationMath 231E, Lecture 34. Polar Coordinates and Polar Parametric Equations
Math 231E, Lecture 34. Polar Coordinates and Polar Parametric Equations 1 Definition of polar coordinates Let us first recall the definition of Cartesian coordinates: to each point in the plane we can
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 information5.2 Verifying Trigonometric Identities
360 Chapter 5 Analytic Trigonometry 5. Verifying Trigonometric Identities Introduction In this section, you will study techniques for verifying trigonometric identities. In the next section, you will study
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 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 informationPreCalc Unit 14: Polar Assignment Sheet April 27 th to May 7 th 2015
PreCalc Unit 14: Polar Assignment Sheet April 27 th to May 7 th 2015 Date Objective/ Topic Assignment Did it Monday Polar Discovery Activity pp. 45 April 27 th Tuesday April 28 th Converting between
More informationSum 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 71 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 informationUnit 4 Trigonometry. Study Notes 1 Right Triangle Trigonometry (Section 8.1)
Unit 4 Trigonometr Stud Notes 1 Right Triangle Trigonometr (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric
More information5.3 Angles and Their Measure
5.3 Angles and Their Measure 1. Angles and their measure 1.1. Angles. An angle is formed b rotating a ra about its endpoint. The starting position of the ra is called the initial side and the final position
More informationExample 1: Give the coordinates of the points on the graph.
Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,
More informationPolar Coordinates. OpenStax. 1 Dening Polar Coordinates
OpenStaxCNX module: m53852 1 Polar Coordinates OpenStax This work is produced by OpenStaxCNX and licensed under the Creative Commons AttributionNonCommercialShareAlike License 4.0 Abstract Locate points
More information5.4 Sum and Difference Formulas
380 Capter 5 Analtic Trigonometr 5. Sum and Difference Formulas Using Sum and Difference Formulas In tis section and te following section, ou will stud te uses of several trigonometric identities and formulas.
More informationMidChapter Quiz: Lessons 91 through 93
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 information6.1 Polar Coordinates
6.1 Polar Coordinates Introduction This chapter introduces and explores the polar coordinate system, which is based on a radius and theta. Students will learn how to plot points and basic graphs in this
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 xaxis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations
More informationMultiple Angle and ProducttoSum Formulas. MultipleAngle Formulas. DoubleAngle 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 MultipleAngle and ProducttoSum Formulas 407 5.5 Multiple Angle and ProducttoSum Formulas What you should learn Use multipleangle formulas to rewrite
More information7.5. Systems of Inequalities. The Graph of an Inequality. What you should learn. Why you should learn it
0_0705.qd /5/05 9:5 AM Page 5 Section 7.5 7.5 Sstems of Inequalities 5 Sstems of Inequalities What ou should learn Sketch the graphs of inequalities in two variables. Solve sstems of inequalities. Use
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 informationJim Lambers MAT 169 Fall Semester Lecture 33 Notes
Jim Lambers MAT 169 Fall Semester 200910 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 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 information4.6 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 informationReview Notes for the Calculus I/Precalculus Placement Test
Review Notes for the Calculus I/Precalculus Placement Test Part 9 . Degree and radian angle measures a. Relationship between degrees and radians degree 80 radian radian 80 degree Example Convert each
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 informationModule 2, Section 2 Graphs of Trigonometric Functions
Principles of Mathematics Section, Introduction 5 Module, Section Graphs of Trigonometric Functions Introduction You have studied trigonometric ratios since Grade 9 Mathematics. In this module ou will
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 information5/27/12. Objectives. Plane Curves and Parametric Equations. Sketch the graph of a curve given by a set of parametric equations.
Objectives Sketch the graph of a curve given by a set of parametric equations. Eliminate the parameter in a set of parametric equations. Find a set of parametric equations to represent a curve. Understand
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 informationChapter 9 Topics in Analytic Geometry
Chapter 9 Topics in Analytic Geometry What You ll Learn: 9.1 Introduction to Conics: Parabolas 9.2 Ellipses 9.3 Hyperbolas 9.5 Parametric Equations 9.6 Polar Coordinates 9.7 Graphs of Polar Equations 9.1
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 information63. Transformations of Square Root Functions. Key Concept Square Root Function Family VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING
3 Transformations of Square Root Functions TEKS FOCUS TEKS ()(C) Determine the effect on the graph of f() = when f() is replaced b af(), f() + d, f(b), and f(  c) for specific positive and negative values
More informationTable of Contents. Unit 5: Trigonometric Functions. Answer Key...AK1. Introduction... v
These materials ma not be reproduced for an purpose. The reproduction of an part for an entire school or school sstem is strictl prohibited. No part of this publication ma be transmitted, stored, or recorded
More informationUnit 6 Introduction to Trigonometry The Unit Circle (Unit 6.3)
Unit Introduction to Trigonometr The Unit Circle Unit.) William Bill) Finch Mathematics Department Denton High School Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic
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 RightTriangle Trigonometry Objectives: 1. Find the trigonometric
More information5.5 MultipleAngle and ProducttoSum Formulas
Section 5.5 MultipleAngle and ProducttoSum Formulas 87 5.5 MultipleAngle and ProducttoSum Formulas MultipleAngle Formulas In this section, you will study four additional categories of trigonometric
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 8.2 Trigonometric (Polar) Form of Complex Numbers The Complex Plane and Vector Representation
More information13.2. General Angles and Radian Measure. What you should learn
Page 1 of 1. General Angles and Radian Measure What ou should learn GOAL 1 Measure angles in standard position using degree measure and radian measure. GOAL Calculate arc lengths and areas of sectors,
More informationGraphs of Equations. MATH 160, Precalculus. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Graphs of Equations
Graphs of Equations MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: sketch the graphs of equations, find the x and yintercepts
More informationName Class Date. subtract 3 from each side. w 5z z 5 2 w p  9 = = 15 + k = 10m. 10. n =
Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations
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 xaxis to OP. Polar coordinate: (r, θ)
More informationTrigonometry Curriculum Guide Scranton School District Scranton, PA
Trigonometry Scranton School District Scranton, PA Trigonometry Prerequisite: Algebra II, Geometry, Algebra I Intended Audience: This course is designed for the student who has successfully completed Algebra
More informationReteaching Golden Ratio
Name Date Class Golden Ratio INV 11 You have investigated fractals. Now ou will investigate the golden ratio. The Golden Ratio in Line Segments The golden ratio is the irrational number 1 5. c On the line
More informationRational Numbers: Graphing: The Coordinate Plane
Rational Numbers: Graphing: The Coordinate Plane A special kind of plane used in mathematics is the coordinate plane, sometimes called the Cartesian plane after its inventor, René Descartes. It is one
More informationUsing Polar Coordinates. Graphing and converting polar and rectangular coordinates
Using Polar Coordinates Graphing and converting polar and rectangular coordinates Butterflies are among the most celebrated of all insects. It s hard not to notice their beautiful colors and graceful flight.
More informationChapter 4: Trigonometry
Chapter 4: Trigonometry Section 41: 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 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 informationCh. 7.4, 7.6, 7.7: Complex Numbers, Polar Coordinates, ParametricFall equations / 17
Ch. 7.4, 7.6, 7.7: Complex Numbers, Polar Coordinates, Parametric equations Johns Hopkins University Fall 2014 Ch. 7.4, 7.6, 7.7: Complex Numbers, Polar Coordinates, ParametricFall equations 2014 1 / 17
More informationNumber Algebra. Problem solving Statistics Investigations
Place Value Addition, Subtraction, Multiplication and Division Fractions Position and Direction Decimals Percentages Algebra Converting units Perimeter, Area and Volume Ratio Properties of Shapes Problem
More informationChapter 1. Limits and Continuity. 1.1 Limits
Chapter Limits and Continuit. Limits The its is the fundamental notion of calculus. This underling concept is the thread that binds together virtuall all of the calculus ou are about to stud. In this section,
More informationLESSON 3.1 INTRODUCTION TO GRAPHING
LESSON 3.1 INTRODUCTION TO GRAPHING LESSON 3.1 INTRODUCTION TO GRAPHING 137 OVERVIEW Here s what ou ll learn in this lesson: Plotting Points a. The plane b. The ais and ais c. The origin d. Ordered
More informationθ as rectangular coordinates)
Section 11.1 Polar coordinates 11.1 1 Learning outcomes After completing this section, you will inshaallah be able to 1. know what are polar coordinates. see the relation between rectangular and polar
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 informationMAC Learning Objectives. Module 12 Polar and Parametric Equations. Polar and Parametric Equations. There are two major topics in this module:
MAC 4 Module 2 Polar and Parametric Equations Learning Objectives Upon completing this module, you should be able to:. Use the polar coordinate system. 2. Graph polar equations. 3. Solve polar equations.
More informationAlgebra I Notes Linear Functions & Inequalities Part I Unit 5 UNIT 5 LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES
UNIT LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES PREREQUISITE SKILLS: students must know how to graph points on the coordinate plane students must understand ratios, rates and unit rate VOCABULARY:
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x and yintercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More information13 Trigonometric Graphs
Trigonometric Graphs Concepts: Period The Graph of the sin, cos, tan, csc, sec, and cot Functions Appling Graph Transformations to the Graphs of the sin, cos, tan, csc, sec, and cot Functions Using Graphical
More information1.5 LIMITS. The Limit of a Function
60040_005.qd /5/05 :0 PM Page 49 SECTION.5 Limits 49.5 LIMITS Find its of functions graphicall and numericall. Use the properties of its to evaluate its of functions. Use different analtic techniques to
More informationTo sketch the graph we need to evaluate the parameter t within the given interval to create our x and y values.
Module 10 lesson 6 Parametric Equations. When modeling the path of an object, it is useful to use equations called Parametric equations. Instead of using one equation with two variables, we will use two
More informationTrigonometric Ratios and Functions
Algebra 2/Trig Unit 8 Notes Packet Name: Date: Period: # Trigonometric Ratios and Functions (1) Worksheet (Pythagorean Theorem and Special Right Triangles) (2) Worksheet (Special Right Triangles) (3) Page
More informationHORIZONTAL AND VERTICAL LINES
the graph of the equation........... AlgebraDate 4.2 Notes: Graphing Linear Equations In Lesson (pp 3.1210213) ou saw examples of linear equations in one variable. The solution of A an solution equation
More information16.6. Parametric Surfaces. Parametric Surfaces. Parametric Surfaces. Vector Calculus. Parametric Surfaces and Their Areas
16 Vector Calculus 16.6 and Their Areas Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and Their Areas Here we use vector functions to describe more general
More informationYear 6 Mathematics Overview
Year 6 Mathematics Overview Term Strand National Curriculum 2014 Objectives Focus Sequence Autumn 1 Number and Place Value read, write, order and compare numbers up to 10 000 000 and determine the value
More informationRational Numbers on the Coordinate Plane. 6.NS.C.6c
Rational Numbers on the Coordinate Plane 6.NS.C.6c Copy all slides into your composition notebook. Lesson 14 Ordered Pairs Objective: I can use ordered pairs to locate points on the coordinate plane. Guiding
More informationUNIT NUMBER 5.2. GEOMETRY 2 (The straight line) A.J.Hobson
JUST THE MATHS UNIT NUMBER 5.2 GEOMETRY 2 (The straight line) b A.J.Hobson 5.2.1 Preamble 5.2.2 Standard equations of a straight line 5.2. Perpendicular straight lines 5.2.4 Change of origin 5.2.5 Exercises
More informationTransformations. What are the roles of a, k, d, and c in polynomial functions of the form y a[k(x d)] n c, where n?
1. Transformations In the architectural design of a new hotel, a pattern is to be carved in the exterior crown moulding. What power function forms the basis of the pattern? What transformations are applied
More informationP.5 The Cartesian Plane
7_0P0.qp //07 8: AM Page 8 8 Chapter P Prerequisites P. The Cartesian Plane The Cartesian Plane Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real
More informationUnit 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 informationAppendix C: Review of Graphs, Equations, and Inequalities
Appendi C: Review of Graphs, Equations, and Inequalities C. What ou should learn Just as ou can represent real numbers b points on a real number line, ou can represent ordered pairs of real numbers b points
More information52 Verifying Trigonometric Identities
5 Verifying Trigonometric Identities Verify each identity. 1. (sec 1) cos = sin 3. sin sin 3 = sin cos 4 5. = cot 7. = cot 9. + tan = sec Page 1 5 Verifying Trigonometric Identities 7. = cot 9. + tan
More informationMultivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
More informationHigh School MATHEMATICS Trigonometry
High School MATHEMATICS Trigonometry Curriculum Curriculum Map USD 457 Math Framework Performance/OpenEnded Response Assessments o First Semester (Tasks 13) o Second Semester (Tasks 14) Available on
More informationPolar Coordinates
Polar Coordinates 772 Polar coordinates are an alternative to rectangular coordinates for referring to points in the plane. A point in the plane has polar coordinates r,θ). r is roughly) the distance
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 informationTrigonometry Review Day 1
Name Trigonometry Review Day 1 Algebra II Rotations and Angle Terminology II Terminal y I Positive angles rotate in a counterclockwise direction. Reference Ray Negative angles rotate in a clockwise direction.
More informationConics, Parametric Equations, and Polar Coordinates. Copyright Cengage Learning. All rights reserved.
10 Conics, Parametric Equations, and Polar Coordinates Copyright Cengage Learning. All rights reserved. 10.5 Area and Arc Length in Polar Coordinates Copyright Cengage Learning. All rights reserved. Objectives
More informationYear 6 Maths Long Term Plan
Week & Focus 1 Number and Place Value Unit 1 2 Subtraction Value Unit 1 3 Subtraction Unit 3 4 Subtraction Unit 5 5 Unit 2 6 Division Unit 4 7 Fractions Unit 2 Autumn Term Objectives read, write, order
More information20 Calculus and Structures
0 Calculus and Structures CHAPTER FUNCTIONS Calculus and Structures Copright LESSON FUNCTIONS. FUNCTIONS A function f is a relationship between an input and an output and a set of instructions as to how
More informationChapter 3. Exponential and Logarithmic Functions. Selected Applications
Chapter Eponential and Logarithmic Functions. Eponential Functions and Their Graphs. Logarithmic Functions and Their Graphs. Properties of Logarithms. Solving Eponential and Logarithmic Equations.5 Eponential
More informationNumber Mulitplication and Number and Place Value Addition and Subtraction Division
Number Mulitplication and Number and Place Value Addition and Subtraction Division read, write, order and compare numbers up to 10 000 000 and determine the value of each digit round any whole number to
More informationNew Swannington Primary School 2014 Year 6
Number Number and Place Value Number Addition and subtraction, Multiplication and division Number fractions inc decimals & % Ratio & Proportion Algebra read, write, order and compare numbers up to 0 000
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 01 Equations of Lines 0 Parallel and Perpendicular Lines 0 Intersecting Lines 0 Midpoints, Distance Formula, Segment Lengths 0 Equations of Circles 06 Problem
More informationTransformations of Functions. 1. Shifting, reflecting, and stretching graphs Symmetry of functions and equations
Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Horizontal Shifting/ Translation Horizontal Shifting/ Translation Shifting,
More informationYear 6.1 Number and Place Value 2 weeks Autumn 1 Read, write, order and compare numbers up to and determine the value of each digit.
Year 6.1 Number and Place Value 2 weeks Autumn 1 Read, write, order and compare numbers up to 10 000 000 and determine the value of each digit. Round any whole number to a required degree of accuracy.
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: We are very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will
More informationby Kevin M. Chevalier
Precalculus Review Handout.4 Trigonometric Functions: Identities, Graphs, and Equations, Part I by Kevin M. Chevalier Angles, Degree and Radian Measures An angle is composed of: an initial ray (side) 
More information81 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 information12.2 Techniques for Evaluating Limits
335_qd /4/5 :5 PM Page 863 Section Tecniques for Evaluating Limits 863 Tecniques for Evaluating Limits Wat ou sould learn Use te dividing out tecnique to evaluate its of functions Use te rationalizing
More informationTRIGONOMETRY. T.1 Angles and Degree Measure
403 TRIGONOMETRY Trigonometry is the branch of mathematics that studies the relations between the sides and angles of triangles. The word trigonometry comes from the Greek trigōnon (triangle) and metron
More informationACTIVITY: Frieze Patterns and Reflections. a. Is the frieze pattern a reflection of itself when folded horizontally? Explain.
. Reflections frieze pattern? How can ou use reflections to classif a Reflection When ou look at a mountain b a lake, ou can see the reflection, or mirror image, of the mountain in the lake. If ou fold
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 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 xyplane. Just like with any other function, it is easy to do by plotting points.
More information