MidChapter Quiz: Lessons 91 through 93


 Myrtle Gordon
 6 years ago
 Views:
Transcription
1 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 opposite direction of the terminal side of the angle. Because =, locate the terminal side of a  angle with the polar axis as its initial side. Because r = 1.5, plot a point 1.5 units from the pole in the opposite direction of the terminal side of the angle. 2. D(1, 315 ) Because = 315, locate the terminal side of a angle with the polar axis as its initial side. Because r = 1, plot a point 1 unit from the pole along the terminal side of the angle. esolutions Manual  Powered by Cognero Page 1
2 4. 6. Because =, locate the terminal side of a The solutions of = are ordered pairs of the  angle with the polar axis as its initial side. form, where r is any real number. The graph consists of all points on the line that make an angle of with the positive polar axis. Because r = 3, plot a point 3 units from the pole along the terminal side of the angle. 7. = 60 Graph each polar equation. 5. r = 3 The solutions of r = 3 are ordered pairs of the form (3, ), where is any real number. The graph consists of all points that are 3 units from the pole, so the graph is a circle centered at the origin with radius 3. The solutions of = 60 are ordered pairs of the form (r, 60 ), where r is any real number. The graph consists of all points on the line that make an angle of 60 with the positive polar axis. esolutions Manual  Powered by Cognero Page 2
3 8. r = 1.5 The solutions of r = 1.5 are ordered pairs of the form ( 1.5, ), where is any real number. The graph consists of all points that are 1.5 units from the pole, so the graph is a circle centered at the origin with radius or 72. The length of each blade is 11.5 feet. So, r = 11.5 for each blade. The angle blade A makes with the polar axis is 3, therefore the tip of blade A can be represented by the polar coordinates (11.5, 3 ). The angle that blade B makes with the polar axis is or 75. Thus, the tip of blade B can be represented by the polar coordinates (11.5, 75 ). By adding 72 to 75, the polar coordinates for blade C can be found as (11.5, 147 ). By adding 72 to 147, the polar coordinates for blade D can be found as (11.5, 219 ). By adding 72 to 219, the polar coordinates for blade E can be found as (11.5, 291 ). 9. HELICOPTERS A helicopter rotor consists of five equally spaced blades. Each blade is 11.5 feet long. b. Blade B has the polar coordinates (11.5, 75 ) and blade C has the polar coordinates (11.5, 147 ). Use the Polar Distance Formula to find the distance between them. The distance between the tips of the two blades is approximately 13.5 feet. a. If the angle blade A makes with the polar axis is 3, write an ordered pair to represent the tip of each blade on a polar grid. Assume that the rotor is centered at the pole. b. What is the distance d between the tips of the helicopter blades to the nearest tenth of a foot? a. Sample answer: Sketch a diagram of the situation. Since the blades of the rotor create 5 angles, the angle between each pair of adjacent blades is 360 esolutions Manual  Powered by Cognero Page 3
4 Graph each equation. 10. r = sec Make a table of values to find the rvalues corresponding to various values of on the interval [0, 2π]. Round each rvalue to the nearest tenth. r = π π 0.3 sec 11. r = cos Because the polar equation is a function of the cosine function, it is symmetric with respect to the polar axis. Therefore, make a table and calculate the values of r on [0, π]. r = π 0.3 cos Use these points and polar axis symmetry to graph the function. Graph the ordered pairs (r, ) and connect them with a line. esolutions Manual  Powered by Cognero Page 4
5 12. r = 3 csc Make a table of values to find the rvalues corresponding to various values of on the interval [0, 2π]. Round each rvalue to the nearest tenth. r = 3 csc 0 π r = 4 sin Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = 4 sin Use these points and symmetry with respect to the line = to graph the function. Graph the ordered pairs (r, ) and connect them with a line. 14. STAINED GLASS A rose window is a circular window seen in gothic architecture. The pattern of the window radiates from the center. The window shown can be approximated by the equation r = 3 sin 6. Use symmetry, zeros, and maximum rvalues of the function to graph the function. Refer to the image on Page 560. esolutions Manual  Powered by Cognero Page 5
6 Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Sketch the graph of the rectangular function y = 3 sin 6x on the interval. From the graph, you Use these and a few additional points to sketch the graph of the function. can see that = 3 when and y = 0 when Interpreting these results in terms of the polar equation r = 3 sin 6, we can say that has a maximum value of 3 when and r = 0 when Since the function is symmetric with respect to the line =, make a table and calculate the values of r on. r = 3 sin esolutions Manual  Powered by Cognero Page 6
7 Identify and graph each classic curve. 15. r = sin The equation is of the form r = a sin, so its graph is a circle. Because the polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = sin r = + 3, 0 The equation is of the form r = a + b, so its graph is a spiral of Archimedes. Use points on the interval [0, 2π] to sketch the graph of the function. r = π π 5.1 Use these points and symmetry with respect to the line = to graph the function. esolutions Manual  Powered by Cognero Page 7
8 17. r = cos The equation is of the form r = a + b cos, so its graph is a limacon. Since a < b, the graph with have an inner loop. Because this polar equation is a function of the cosine function, it is symmetric with respect to the polar axis. Therefore, make a table and calculate the values of r on. r = cos 0 3 π Use these points and polar axis symmetry to graph the function. 18. r = 5 sin 3 The equation is of the form r = a sin n, so its graph is a rose. Because this polar equation is a function of the sine function, it is symmetric with respect to the line =. Therefore, make a table and calculate the values of r on. r = 5 sin Use these points and symmetry with respect to the line = to graph the function. esolutions Manual  Powered by Cognero Page 8
9 19. MULTIPLE CHOICE Identify the polar graph of y 2 = x. 20. Find the rectangular coordinates for each point with the given polar coordinates. For, r = 4 and =. Write the rectangular equation y 2 = x in polar The rectangular coordinates of. are form. 21. For, r = 2 and =. Graph r = cos csc 2 using a graphing calculator. Let = and solve for r. The rectangular coordinates of. are The point corresponds to graph B. The correct answer is B. esolutions Manual  Powered by Cognero Page 9
10 22. ( 1, 210 ) For ( 1, 210 ), r = 1 and = 210. Find two pairs of polar coordinates for each point with the given rectangular coordinates if 0 2π. Round to the nearest hundredth. 24. ( 3, 5) For ( 3, 5), x = 3 and y = 5. Since x < 0, use to find. The rectangular coordinates of ( 1, 210 ) are. 23. (3, 30 ) For (3, 30 ), r = 3 and = 30. One set of polar coordinates is (5.83, 2.11). Another representation that uses a negative rvalue is ( 5.83, π) or ( 5.83, 5.25). 25. (8, 1) The rectangular coordinates of (3, 30 ) are. For (8, 1), x = 8 and y = 1. Since x > 0, use to find. One set of polar coordinates is (8.06, 0.12). Another representation that uses a negative rvalue is ( 8.06, π) or ( 8.06, 3.27). esolutions Manual  Powered by Cognero Page 10
11 26. (7, 6) Write a rectangular equation for each graph. For (7, 6), x = 7 and y = 6. Since x > 0, use to find. 28. One set of polar coordinates is (9.22, 0.71). Since this set is not in the required domain, two more sets have to be found. A representation that uses a positive rvalue is (9.22, π) or (9.22, 5.57). A representation that uses a negative rvalue is ( 9.22, π) or ( 9.22, 2.43). 27. ( 4, 10) For ( 4, 10), x = 4 and y = 10. Since x < 0, use tan 1 + π to find. 29. One set of polar coordinates is (10.77, 4.33). Another representation that uses a negative rvalue is ( 10.77, 4.33 π) or ( 10.77, 1.19). esolutions Manual  Powered by Cognero Page 11
92 Graphs of Polar Equations
Graph each equation by plotting points. 3. r = cos Make a table of values to find the rvalues corresponding to various values of on the interval [, 2π]. Round each rvalue to the nearest tenth. r = θ
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 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 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 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 information6.7. 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 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 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 information75 Parametric Equations
3. Sketch the curve given by each pair of parametric equations over the given interval. Make a table of values for 6 t 6. t x y 6 19 28 5 16.5 17 4 14 8 3 11.5 1 2 9 4 1 6.5 7 0 4 8 1 1.5 7 2 1 4 3 3.5
More information, minor axis of length 12. , asymptotes y 2x. 16y
Math 4 Midterm 1 Review CONICS [1] Find the equations of the following conics. If the equation corresponds to a circle find its center & radius. If the equation corresponds to a parabola find its focus
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 informationObjectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using
Ch 13  RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric
More informationStudy Guide and Review
Choose the term that best matches the statement or phrase. a square of a whole number A perfect square is a square of a whole number. a triangle with no congruent sides A scalene triangle has no congruent
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 informationMATH115. Polar Coordinate System and Polar Graphs. Paolo Lorenzo Bautista. June 14, De La Salle University
MATH115 Polar Coordinate System and Paolo Lorenzo Bautista De La Salle University June 14, 2014 PLBautista (DLSU) MATH115 June 14, 2014 1 / 30 Polar Coordinates and PLBautista (DLSU) MATH115 June 14, 2014
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. 1)
More information46 Inverse Trigonometric Functions
Find the exact value of each expression, if it exists. 29. The inverse property applies, because lies on the interval [ 1, 1]. Therefore, =. 31. The inverse property applies, because lies on the interval
More information2 Unit Bridging Course Day 10
1 / 31 Unit Bridging Course Day 10 Circular Functions III The cosine function, identities and derivatives Clinton Boys / 31 The cosine function The cosine function, abbreviated to cos, is very similar
More 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 informationSM 2. Date: Section: Objective: The Pythagorean Theorem: In a triangle, or
SM 2 Date: Section: Objective: The Pythagorean Theorem: In a triangle, or. It doesn t matter which leg is a and which leg is b. The hypotenuse is the side across from the right angle. To find the length
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 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 informationSecondary Math 3 Honors. 74 Inverse Trigonometric Functions
Secondary Math 3 Honors 74 Inverse Trigonometric Functions Warm Up Fill in the Unit What You Will Learn How to restrict the domain of trigonometric functions so that the inverse can be constructed. How
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 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 informationTriangle Trigonometry
Honors Finite/Brief: Trigonometry review notes packet Triangle Trigonometry Right Triangles All triangles (including nonright triangles) Law of Sines: a b c sin A sin B sin C Law of Cosines: a b c bccos
More information1) The domain of y = sin1x is The range of y = sin1x is. 2) The domain of y = cos1x is The range of y = cos1x is
MAT 204 NAME TEST 4 REVIEW ASSIGNMENT Sections 8.1, 8.38.5, 9.29.3, 10.1 For # 13, fill in the blank with the appropriate interval. 1) The domain of y = sin1x is The range of y = sin1x is 2) The domain
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 informationStudy Guide and Review  Chapter 10
State whether each sentence is true or false. If false, replace the underlined word, phrase, expression, or number to make a true sentence. 1. A triangle with sides having measures of 3, 4, and 6 is a
More informationStudy Guide and Review  Chapter 10
State whether each sentence is true or false. If false, replace the underlined word, phrase, expression, or number to make a true sentence. 1. A triangle with sides having measures of 3, 4, and 6 is a
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 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 informationPRECALCULUS MATH Trigonometry 912
1. Find angle measurements in degrees and radians based on the unit circle. 1. Students understand the notion of angle and how to measure it, both in degrees and radians. They can convert between degrees
More information: Find the values of the six trigonometric functions for θ. Special Right Triangles:
ALGEBRA 2 CHAPTER 13 NOTES Section 131 Right Triangle Trig Understand and use trigonometric relationships of acute angles in triangles. 12.F.TF.3 CC.9 Determine side lengths of right triangles by using
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 informationYoungstown State University Trigonometry Final Exam Review (Math 1511)
Youngstown State University Trigonometry Final Exam Review (Math 1511) 1. Convert each angle measure to decimal degree form. (Round your answers to thousandths place). a) 75 54 30" b) 145 18". Convert
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 informationStudy Guide and Review
State whether each sentence is or false. If false, replace the underlined term to make a sentence. 1. Euclidean geometry deals with a system of points, great circles (lines), and spheres (planes). false,
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 information52 Verifying Trigonometric Identities
Verify each identity 1 (sec 1) cos = sin sec (1 cos ) = tan 3 sin sin cos 3 = sin 4 csc cos cot = sin 4 5 = cot Page 1 4 5 = cot 6 tan θ csc tan = cot 7 = cot 8 + = csc Page 8 = csc + 9 + tan = sec 10
More informationPolar (BC Only) They are necessary to find the derivative of a polar curve in x and ycoordinates. The derivative
Polar (BC Only) Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole
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 informationMath 8 EXAM #5 Name: Any work or answers completed on this test form, other than the problems that require you to graph, will not be graded.
Math 8 EXAM #5 Name: Complete all problems in your blue book. Copy the problem into the bluebook then show all of the required work for that problem. Work problems out down the page, not across. Make only
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 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 informationMATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the triangle. 1) 1) 80 7 55 Solve the triangle. Round lengths to the nearest tenth
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 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 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 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 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 informationMAC Learning Objectives. Learning Objectives (Cont.) Module 2 Acute Angles and Right Triangles
MAC 1114 Module 2 Acute Angles and Right Triangles Learning Objectives Upon completing this module, you should be able to: 1. Express the trigonometric ratios in terms of the sides of the triangle given
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 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 informationParametric and Polar Curves
Chapter 2 Parametric and Polar Curves 2.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves Parametric Equations So far we ve described a curve by giving an equation that the coordinates
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 informationYou ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46
Math 1330 Section 6.2 Section 7.1: RightTriangle Applications In this section, we ll solve right triangles. In some problems you will be asked to find one or two specific pieces of information, but often
More informationAdvanced Math Final Exam Review Name: Bornoty May June Use the following schedule to complete the final exam review.
Advanced Math Final Exam Review Name: Bornoty May June 2013 Use the following schedule to complete the final exam review. Homework will e checked in every day. Late work will NOT e accepted. Homework answers
More informationParametric and Polar Curves
Chapter 2 Parametric and Polar Curves 2.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves Parametric Equations So far we ve described a curve by giving an equation that the coordinates
More informationParametric and Polar Curves
Chapter 2 Parametric and Polar Curves 2.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves Parametric Equations So far we ve described a curve by giving an equation that the coordinates
More informationName Trigonometric Functions 4.2H
TE31 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 informationCommon Core Standards Addressed in this Resource
Common Core Standards Addressed in this Resource NCN.4  Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular
More informationPART I You must complete this portion of the test without using a calculator. After you
Salt Lake Community College Math 1060 Final Exam A Fall Semester 2010 Name: Instructor: This Exam has three parts. Please read carefully the directions for each part. All problems are of equal point value.
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 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 informationsin30 = sin60 = cos30 = cos60 = tan30 = tan60 =
Precalculus Notes TrigDay 1 x Right Triangle 5 How do we find the hypotenuse? 1 sinθ = cosθ = tanθ = Reciprocals: Hint: Every function pair has a co in it. sinθ = cscθ = sinθ = cscθ = cosθ = secθ = cosθ
More informationFind the component form and magnitude of the vector where P = (3,4), Q = (5, 2), R = (1, 3) and S = (4, 7)
PRECALCULUS: by Finney,Demana,Watts and Kennedy Chapter 6: Applications of Trigonometry 6.1: Vectors in the Plane What you'll Learn About Two Dimensional Vectors/Vector Operations/Unit Vectors Direction
More informationMultiple Choice Questions Circle the letter of the correct answer. 7 points each. is:
This Math 114 final exam was administered in the Fall of 008. This is a sample final exam. The problems are not exhaustive. Be prepared for ALL CONCEPTS for the actual final exam. Multiple Choice Questions
More informationSemester 2 Review Units 4, 5, and 6
Precalculus Semester 2 Review Units 4, 5, and 6 NAME: Period: UNIT 4 Simplify each expression. 1) (sec θ tan θ)(1 + tan θ) 2) cos θ sin 2 θ 1 3) 1+tan θ 1+cot θ 4) cos 2θ cosθ sin θ 5) sec 2 x sec 2 x
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 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 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 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 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 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 informationCLEP PreCalculus. 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 PreCalculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. An online graphing calculator (noncas) is allowed to be used for this section..
More informationDay 4 Trig Applications HOMEWORK
Day 4 Trig Applications HOMEWORK 1. In ΔABC, a = 0, b = 1, and mc = 44º a) Find the length of side c to the nearest integer. b) Find the area of ΔABC to the nearest tenth.. In ΔABC, ma = 50º, a = 40, b
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 informationis a plane curve and the equations are parametric equations for the curve, with parameter t.
MATH 2412 Sections 6.3, 6.4, and 6.5 Parametric Equations and Polar Coordinates. Plane Curves and Parametric Equations Suppose t is contained in some interval I of the real numbers, and = f( t), = gt (
More information12 Analyzing Graphs of Functions and Relations
Use the graph of each function to estimate the indicated function values. Then confirm the estimate algebraically. Round to the nearest hundredth, if necessary. The function value at x = 1 appears to be
More information10.7. Polar Coordinates. Introduction. What you should learn. Why you should learn it. Example 1. Plotting Points on the Polar Coordinate System
_7.qxd /8/5 9: AM Page 779 Section.7 Polar Coordinates 779.7 Polar Coordinates What ou should learn Plot points on the polar coordinate sstem. Convert points from rectangular to polar form and vice versa.
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 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 informationPolar Coordinates. Calculus 2 Lia Vas. If P = (x, y) is a point in the xyplane and O denotes the origin, let
Calculus Lia Vas Polar Coordinates If P = (x, y) is a point in the xyplane and O denotes the origin, let r denote the distance from the origin O to the point P = (x, y). Thus, x + y = r ; θ be the angle
More informationInvestigating the Sine and Cosine Functions Part 1
Investigating the Sine and Cosine Functions Part 1 Name: Period: Date: SetUp 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 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 informationMATH STUDENT BOOK. 12th Grade Unit 7
MATH STUDENT BOOK 1th Grade Unit 7 Unit 7 POLAR COORDINATES MATH 107 POLAR COORDINATES INTRODUCTION 1. POLAR EQUATIONS 5 INTRODUCTION TO POLAR COORDINATES 5 POLAR EQUATIONS 1 POLAR CURVES 19 POLAR FORMS
More information1. (10 pts.) Find and simplify the difference quotient, h 0for the given function
MATH 1113/ FALL 016 FINAL EXAM Section: Grade: Name: Instructor: f ( x h) f ( x) 1. (10 pts.) Find and simplify the difference quotient, h 0for the given function h f ( x) x 5. (10 pts.) The graph of the
More information14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio.
14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio. Using the space below, draw at least right triangles, each of which has one
More informationChapter 5.6: The Other Trig Functions
Chapter 5.6: The Other Trig Functions The other four trig functions, tangent, cotangent, cosecant, and secant are not sinusoids, although they are still periodic functions. Each of the graphs of these
More informationBasic Graphs of the Sine and Cosine Functions
Chapter 4: Graphs of the Circular Functions 1 TRIGFall 2011Jordan Trigonometry, 9 th edition, Lial/Hornsby/Schneider, Pearson, 2009 Section 4.1 Graphs of the Sine and Cosine Functions Basic Graphs of
More informationMATH 1112 Trigonometry Final Exam Review
MATH 1112 Trigonometry Final Exam Review 1. Convert 105 to exact radian measure. 2. Convert 2 to radian measure to the nearest hundredth of a radian. 3. Find the length of the arc that subtends an central
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Review for Test 2 MATH 116 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the right triangle. If two sides are given, give angles in degrees and
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 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 informationWarmUp: Final Review #1. A rectangular pen is made from 80 feet of fencing. What is the maximum area the pen can be?
WarmUp: Final Review #1 A rectangular pen is made from 80 feet of fencing. What is the maximum area the pen can be? WarmUp: Final Review #2 1) Find distance (2, 4) (6, 3) 2) Find roots y = x 46x 2
More informationAlgebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions
Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions 20151217 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Coterminal angles Arc
More information5.1 Angles & Their Measures. Measurement of angle is amount of rotation from initial side to terminal side. radians = 60 degrees
.1 Angles & Their Measures An angle is determined by rotating array at its endpoint. Starting side is initial ending side is terminal Endpoint of ray is the vertex of angle. Origin = vertex Standard Position:
More informationAlgebra II Trigonometric Functions
Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 20151217 www.njctl.org Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Coterminal angles Arc
More informationPractice Test  Chapter 7
Write an equation for an ellipse with each set of characteristics. 1. vertices (7, 4), ( 3, 4); foci (6, 4), ( 2, 4) The distance between the vertices is 2a. 2a = 7 ( 3) a = 5; a 2 = 25 The distance between
More information