5.1Angles and Their Measure


 Lewis Hunt
 2 years ago
 Views:
Transcription
1 5.1Angles and Their Measure Objectives: 1. Find the degree or radian measure of coterminal angles. 2. Convert between degrees minutes and seconds and decimal degrees. 3. Convert between degrees and radians. 4. Find the length of an arc and area of a sector. 5. Find linear and angular velocity Overview: Trigonometry was first studied by the Greeks, Egyptians, and Babylonians and used in surveying, navigation and astronomy. Using trigonometry, they had a powerful tool for finding areas of triangular plots of land as well as lengths of sides and measures of angles, without physically measuring them. Basic Terminology: A Ray is a point on a line together with all the points of the line on one side of that point. An Angle is the union of two rays with a common endpoint, the vertex. An angle is formed by rotating one ray away from a fixed ray. The fixed ray is the initial side and the rotated ray is the terminal side. An angle whose vertex is the center of a circle is a central angle and the arc of the circle through which the terminal side moves is the intercepted arc. An angle in standard position is located in a rectangular coordinate system with the vertex at the origin and the initial side on the positive xaxis.
2 The measure m(α ) of an angle indicates the amount of rotation of the terminal side from the initial side. It is found using any circle. The circle is divided into 36 equal arcs and each arc is one degree. Degree Measure of an The degree measure of an angle Angle: is the number of degrees in the intercepted arc of a circle centered at the vertex. The degree measure is positive if the rotation is counterclockwise and negative if the rotation is clockwise. Angles are typically denoted using Greek letters such as, Alphaα, Beta β or Thetaθ. Types of Angles: Acute angles are between and 9 degrees. Obtuse angles are between 9 and 18 degrees. Right angles are 9 degrees. Straight angles are 18 degrees.
3 The rectangular coordinate system is divided into four quadrants as shown in the diagram to the right. The terminal side of an angle will either lie in one of the four quadrants or on one of the axes. Quadrantal angles have the terminal side on one of the axes. Example: Draw each of the following angles and identify the quadrant the terminal side lies in. a. 45 b. 9 c. 225 d. 45 Solution: a. Quadrant 1 b. Quadrantal c. Quadrant III d. Quadrant I
4 Coterminal Angles: Coterminal angles are angles in standard position that have the same initial side and the same terminal side. Any two coterminal angles have degree measures that differ by a multiple of 36 degrees. 27 and 9 are coterminal. Notice that they differ by 36 degrees (9) = 36 Example: Name two angles, one positive and one negative that are coterminal to 32 degrees. Solution: Add and subtract 36 degrees from the angles measure = 4 degrees = 68 degrees Example: Determine if the angles 12 and 6 are coterminal. Solution: If these two angles differ by a multiple of 36, then they are coterminal. If n is an integer in the equationα + 36 n = β, then the angles are coterminal n = 6 36n = 72 n = 2 Since n is an integer the angles are coterminal. Example: Determine if the angles 96 and 14 are coterminal. Solution: If these two angles differ by a multiple of 36, then they are coterminal. If n is an integer in the equationα + 36 n = β, then the angles are coterminal n = 14 36n = 2 n = Since n is not an integer the angles are not coterminal.
5 Convert Between Decimal and DMS Measures: Angles may be written in degrees minutes and seconds or as decimal degrees. To convert from one unit to the other we use the following relationships: 1 degree = 6 minutes 1 minute = 6 seconds 1 degree = 36 seconds Example: Convert ' to decimal degrees. Solution: Use the above relationships as ratios to convert 45 minutes to degrees. 1deg min = deg =.75 deg 6 min 6 Therefore ' = ' "" Example: Convert to decimal degrees. Solution: Use the above relationships as ratios to convert 37 seconds to minutes. 1min sec = min =.617 min 6 sec 6 ' "" ' = Now convert minutes to degrees. ' "" ' = = 1deg min = deg =.877 deg 6 min Example: Convert to dms. Solution: 6 min.243deg = min 1deg ' = Now convert.58 to seconds. ' = = 154 6sec.58 min = 34.8sec 1min ' ''
6 Radian Measure of Angles: The radian measure of the angle α in standard position is the directed length of the intercepted arc on the unit circle. Radian measure is used extensively in scientific fields and results in simpler formulas in trigonometry and calculus. Radian measure is a directed length because it is positive or negative depending on the direction of the terminal side. Radian measure is a real number without any dimension. Converting Between Radian Measure and Degrees: The circumference of the unit circle is C = 2 rπ = 2π. If the terminal side rotates 36 degrees the length of the intercepted arc is 2 π. Therefore an angle measure of 36 degrees has a radian measure of 2π radians. This relation simplifies as follows: 18 = π. rad. This conversion factor can be used to convert between radian measure and degree measure. Example: Convert 45 degrees to radians Solution: Use the conversion factor 18 = π. rad. as a ratio such that units cross cancel. π. rad 45deg 18 deg π = rad 4 Example: Convert 12 degrees to radians Solution: Use the conversion factor 18 = π. rad. as a ratio such that units cross cancel. π. rad 12 deg 18deg 2π = rad 3
7 Example: Convert 5π radians to degrees 6 Solution: Use the conversion factor 18 = π. rad. as a ratio such that units cross cancel. 5π 18 deg rad 6 πrad = 15 deg Example: Convert 7π radians to degrees 2 Solution: Use the conversion factor 18 = π. rad. as a ratio such that units cross cancel. 7π 18 deg rad 2 πrad = 63 deg Finding Coterminal Angles using Radian Measure: Any two coterminal angles in radian measure differ by a multiple of 2 π. Notice that they differ by a multiple of 2π radians. 3π π 4π = = 2π π π and 2 2 are coterminal. Example: Name two angles, one positive and one negative that are coterminal to Solution: Add and subtract 2π radians from the angles measure. 3π radians π π = π radians π 5 2π = π radians 4 4 Example: Determine if the angles 5π radians and 4 11π radians are coterminal. 4 Solution: If these two angles differ by a multiple of 2π then they are coterminal. If n is an integer in the equationα + 2 π. n = β, then the angles are coterminal. 5π 11π + 2π. n = π 2π. n = 4 n = 2π Since n is an integer the angles are coterminal.
8 Radian Measures of Common Angles: The following diagram presents the radian measure of angles that are frequently used in trigonometry. It is strongly advised that you keep this diagram available when working on trigonometry problems. The angles in the Quadrant I and the Quadrantal Angles should be memorized. Arc Length: Radian measure of a central angle of a circle can be used to easily find the length of the intercepted arc of the circle. The length s of an arc intercepted by a central angle of α radians on a circle of radius r is given by: s = αr Where s is the arc length, r is the radius and α is the measure of central angle. In general, a central angle in a circle of radius r intercepts an arc whose length is a fraction of the circumference of the circle.
9 7π Example: Find the length of the arc intercepted by a central angle of 9 Round to two decimal places. in a circle of radius 7 cm. Solution: Use the arc length formula. s = αr 7π s = 7 9 s = 17.1 The measure of the arc is 17.1 cm. Example: Find the radian measure of the central angle α which intercepts a 3 cm arc in a circle of radius 1 cm. Round to one decimal place, if necessary. Solution: Use the arc length formula to solve for α The measure of the central angle is 3 radians. s = αr 3 = α 1 α = 3 Example: A car wheel has a 16inch radius. Through what angle (to the nearest tenth of a degree) does the wheel turn when the car rolls forward 3 ft? Solution: Use the arc length formula to solve for α in radians. Now convert from radians to degrees. s = αr 36 = α 16 α = deg α = 2.25rad = πrad When the car rolls forward 3 ft, the wheel turns through an angle of degrees.
10 Example: A satellite photographs a path on the surface of the Earth that is 25 miles wide. Find the measure of the central angle in degrees that intercepts an arc 25 miles on the surface of the Earth (radius 395 miles). Solution: To determine the measure of the central angle, you can use the fact that in a circle of radius r, a central angle of α radians intercepts an arc length of s. Use the formula: s = αr Let s = 25 and r = 395. s = αr 25 = α(395) α = α = Remember, the angle is measured in radians. 18 To obtain the answer in degrees, multiply by. π = π Therefore, the measure of the central angle is degrees. Linear Velocity: If a point is in motion on a circle of radius r through an angle of α radians in time t, then its linear velocity v is given by where s is the arc length determined by s = αr. v = Linear Velocity is a measure linear distance along an arc over a period of time. s t
11 Angular Velocity: If a point is in motion on a circle through an angle of α radians in time t, then its angular velocity ω is given by α ω = t Angular velocity is a measure of how a central angle changes over a period of time. Example: A windmill for generating electricity has a blade that is 3 feet long. Depending on the wind, it rotates at various velocities. Find the angular velocity in rad/sec. for the top of the blade if the windmill is rotating at a rate of 5 rev/sec. Solution: Use the cancellation of units and the fact that 2 π radians = 1 rev. α ω = t 5rev 2π. rad ω = sec. rev ω = rad / sec 1π. rad = sec. The angular velocity is radians second. Linear Velocity in Terms of Angular Velocity: If v is the linear velocity of a point on a circle of radius r, and ω is its angular velocity, then v = rω Example: A wheel is rotating at 2 radians/sec, and the wheel has a 24inch diameter. To the nearest foot per minute, what is the velocity of a point on the rim? Solution: We are given the angular velocity ω = 2rad / sec and the radius r = 12. v = rω v = 12(2) = 24in / sec This value needs to be converted into the appropriate units of feet/min. 24in 1 ft sec 12in 6sec 1min = 12 ft / min
12 Example: The blade on a typical table saw rotates at 3,6 rpm. What is the difference in linear velocity in inches per second of a point on the edge of an 18inch diameter blade and a 6inch diameter blade? Solution: Find the angular velocity of the blade given that the blade rotates at 3,6 rpm. α ω = t 3,6. rev 2π. rad ω = min. rev ω = 12. π. rad / sec. 1min 6sec Now use the formula v = r. ω to obtain the linear velocity for each point. v = r. ω v = r. ω v = 9(12π ) v = 3(12π ) v = 1,8π. in./ sec. v = 36π. in./ sec. Find the difference between these two velocities: The difference in linear velocity is 72. in./ sec. 18π 36π = 72. in./ sec. Problem: The propeller of a Cessna Caravan is 16inches in diameter and rotates at 1,8 rpm under normal cruising flight. Determine the angular velocity of the propeller. What is the difference in linear velocity in inches per second of a point on the tip of the propeller and a point 6inch from the propeller hub (vertex)? The solution should be written in exact form. Solution: Find the angular velocity of the blade given that the blade rotates at 1,8 rpm. α ω = t 1,8. rev 2π. rad ω = min rev ω = 6π. rad / sec. 1min 6sec Now use the formula v = r. ω to obtain the linear velocity for each point. v = r. ω v = r. ω v = 53(6π ) v = 6(6π ) v = 3,18 π. in./ sec. v = 36π. in./ sec. Find the difference between these two velocities: The difference in linear velocity is 2,82. π. in./ sec. 3,18 36 = 2,82. π. in./ sec.
Chapter 4. Trigonometric Functions. 4.1 Angle and Radian Measure /45
Chapter 4 Trigonometric Functions 4.1 Angle and Radian Measure 1 /45 Chapter 4 Homework 4.1 p290 13, 17, 21, 35, 39, 43, 49, 53, 57, 65, 73, 75, 79, 83, 89, 91 2 /45 Chapter 4 Objectives Recognize and
More information4.1: Angles and Radian Measure
4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is
More informationTrigonometry Basics. Angle. vertex at the and initial side along the. Standard position. Positive and Negative Angles
Name: Date: Pd: Trigonometry Basics Angle terminal side direction of rotation arrow CCW is positive CW is negative vertex initial side Standard position vertex at the and initial side along the Positive
More information6.1: Angle Measure in degrees
6.1: Angle Measure in degrees How to measure angles Numbers on protractor = angle measure in degrees 1 full rotation = 360 degrees = 360 half rotation = quarter rotation = 1/8 rotation = 1 = Right angle
More informationLecture Two Trigonometric
Lecture Two Trigonometric Section.1 Degrees, Radians, Angles and Triangles Basic Terminology Two distinct points determine line AB. Line segment AB: portion of the line between A and B. Ray AB: portion
More information5.1 Angles and Their Measure. Objectives
Objectives 1. Convert between decimal degrees and degrees, minutes, seconds measures of angles. 2. Find the length of an arc of a circle. 3. Convert from degrees to radians and from radians to degrees.
More informationvertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector
vertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector Convert Between DMS and Decimal Degree Form A. Write 329.125 in DMS form. First, convert 0.125
More informationUnit 1  Radian and Degree Measure Classwork
Unit  Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side  starting and ending Position of the ray Standard position origin if the vertex,
More informationMA Lesson 19 Summer 2016 Angles and Trigonometric Functions
DEFINITIONS: An angle is defined as the set of points determined by two rays, or halflines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common
More informationFLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A.
Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is
More informationUnit 1  Radian and Degree Measure Classwork
Unit 1  Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side  starting and ending Position of the ray Standard position origin if the
More informationTrigonometry Chapter 3 Lecture Notes
Ch Notes Morrison Trigonometry Chapter Lecture Notes Section. Radian Measure I. Radian Measure A. Terminology When a central angle (θ) intercepts the circumference of a circle, the length of the piece
More informationTrigonometry LESSON ONE  Degrees and Radians Lesson Notes
210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:
More informationANGLE CALCULATIONS. plus 76 hundredths of 1
Learning Assistance Center Southern Maine Community College ANGLE CALCULATIONS An angle is formed by rotating a given ray about its endpoint to some terminal position. The original ray is the initial side
More informationDETAILED SOLUTIONS AND CONCEPTS  INTRODUCTION TO ANGLES
DETAILED SOLUTIONS AND CONCEPTS  INTRODUCTION TO ANGLES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationTEST #3 RADIAN MEASURE AND CIRCULAR FUNCTIONS, PRACTICE. 1) Convert each degree measure to radians. Leave answers as multiples of π. cos 4.
PRECALCULUS B TEST # RADIAN MEASURE AND CIRCULAR FUNCTIONS, PRACTICE SECTION. Radian Measure ) Convert each degree measure to radians. Leave answers as multiples of. A) 45 B) 60 C) 0 D) 50 E) 75 F) 40
More information42 Degrees and Radians
Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest thousandth. 3. 141.549 First, convert 0. 549 into minutes and seconds. Next, convert 0.94' into
More informationSection 6.1 Angle Measure
Section 6.1 Angle Measure An angle AOB consists of two rays R 1 and R 2 with a common vertex O (see the Figures below. We often interpret an angle as a rotation of the ray R 1 onto R 2. In this case, R
More informationChapter 6 Trigonometric Functions of Angles
6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.
More information3. Right Triangle Trigonometry
. Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities . Reference Angle Reference Angle Reference angle
More informationSection 3.1 Radian Measure
Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle
More informationSect 10.1 Angles and Triangles
75 Sect 0. Angles and Triangles Objective : Converting Angle Units Each degree can be divided into 60 minutes, denoted 60', and each minute can be divided into 60 seconds, denoted 60". Hence, = 60' and
More informationWho uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)
1 The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150
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. ) 56
More information4.1 Radian and Degree Measure
4. Radian and Degree Meaure An angle AOB (notation: AOB ) conit of two ray R and R with a common vertex O (ee Figure below). We often interpret an angle a a rotation of the ray R onto R. In thi cae, R
More informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
More informationθ. The angle is denoted in two ways: angle θ
1.1 Angles, Degrees and Special Triangles (1 of 24) 1.1 Angles, Degrees and Special Triangles Definitions An angle is formed by two rays with the same end point. The common endpoint is called the vertex
More informationTrigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011
Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles
More informationMidChapter Quiz: Lessons 41 through 44
Find the exact values of the six trigonometric functions of θ. Find the value of x. Round to the nearest tenth if necessary. 1. The length of the side opposite is 24, the length of the side adjacent to
More information4.1 Radian and Degree Measure
Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position
More informationCHAPTER 13 LINEAR AND ANGULAR MOTION
CHAPTER 13 LINEAR AND ANGULAR MOTION EXERCISE 73, Page 17 1. A pulley driving a belt has a diameter of 360 mm and is turning at 700/π revolutions per minute. Find the angular velocity of the pulley and
More informationy = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)
5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If
More informationWheels Diameter / Distance Traveled
Mechanics Teacher Note to the teacher On these pages, students will learn about the relationships between wheel radius, diameter, circumference, revolutions and distance. Students will use formulas relating
More informationLinear Speed and Angular Speed
Preliminaries and Objectives Preliminaries: Circumference of a circle Conversion factors (dimensional analysis) Objectives: Given the central angle and radius (or diameter) of a circle, find the arc length
More informationSolutions to Exercises, Section 5.2
Instructor s Solutions Manual, Section 5.2 Exercise 1 Solutions to Exercises, Section 5.2 In Exercises 1 8, convert each angle to radians. 1. 15 Start with the equation Divide both sides by 360 to obtain
More informationObjectives After completing this section, you should be able to:
Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding
More information1) Convert 13 32' 47" to decimal degrees. Round your answer to four decimal places.
PRECLCULUS FINL EXM, PRCTICE UNIT ONE TRIGONOMETRIC FUNCTIONS ) Convert ' 47" to decimal degrees. Round your answer to four decimal places. ) Convert 5.6875 to degrees, minutes, and seconds. Round to the
More information( 1, 0 ) Trigonometry Exact Value, Laws, and Vectors. rad deg π π 180 Coterminal Angles. Conversions 180
Trigonometry Exact Value, Laws, and Vectors Conversions 180 rad deg π π deg rad 180 Coterminal Angles θ + 60 n, n Ζ Arc length and Area of a Sector Arc length: s = θr 1 Area of a sector: Area = θr Only
More informationAngles and Their Measure
Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two
More informationMAC 1114: Trigonometry Notes
MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant
More informationMPE Review Section II: Trigonometry
MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the
More informationMAT 0950 Course Objectives
MAT 0950 Course Objectives 5/15/20134/27/2009 A student should be able to R1. Do long division. R2. Divide by multiples of 10. R3. Use multiplication to check quotients. 1. Identify whole numbers. 2. Identify
More informationCIRCLES AND VOLUME Lesson 4: Finding Arc Lengths and Areas of Sectors
CIRCLES AND VOLUME Lesson : Finding Arc Lengths and Areas of Sectors Common Core Georgia Performance Standard MCC9 12.G.C.5 Essential Questions 1. How is radian measure related to degree measure? 2. How
More informationCHAPTER 28 THE CIRCLE AND ITS PROPERTIES
CHAPTER 8 THE CIRCLE AND ITS PROPERTIES EXERCISE 118 Page 77 1. Calculate the length of the circumference of a circle of radius 7. cm. Circumference, c = r = (7.) = 45.4 cm. If the diameter of a circle
More informationDepartment of Natural Sciences Clayton State University. Physics 1111 Quiz 8. a. Find the average angular acceleration of the windmill.
November 5, 2007 Physics 1111 Quiz 8 Name SOLUTION 1. Express 27.5 o angle in radians. (27.5 o ) /(180 o ) = 0.480 rad 2. As the wind dies, a windmill that was rotating at 3.60 rad/s comes to a full stop
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationPhysics 111. Lecture 20 (Walker: ) Rotational Motion Oct.19, Rotational Motion. Angular Position θ
Physics 111 Lecture 0 (Walker: 10.13) Rotational Motion Oct.19, 009 Rotational Motion Study rotational motion of solid object with fixed axis of rotation (axle) Have angular versions of the quantities
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationSolutions to Chapter 3 Exercise Problems
Solutions to hapter Exercise Problems Problem.1 In the figure below, points and have the same horizontal coordinate, and ω = 0 rad/s. Draw and dimension the velocity polygon. Identify the sliding velocity
More informationLinear Motion vs. Rotational Motion
Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a
More informationMath 115 Spring 2014 Written Homework 10SOLUTIONS Due Friday, April 25
Math 115 Spring 014 Written Homework 10SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain
More informationAngles are what trigonometry is all about. This is where it all started, way back when.
Chapter 1 Tackling Technical Trig In This Chapter Acquainting yourself with angles Identifying angles in triangles Taking apart circles Angles are what trigonometry is all about. This is where it all started,
More informationPhysics 100A Homework 9 Chapter 10 (part 1)
Physics 1A Homework 9 Chapter 1 (part 1) 1.1) The following angles are given in degrees. Convert them to radians. 1. Picture the Problem: This is a units conversion problem. π radians Strategy: Multiply
More informationIUPUI Department of Mathematical Sciences Departmental Final Examination PRACTICE FINAL EXAM VERSION #1 MATH Algebra & Trigonometry II
IUPUI Department of Mathematical Sciences Departmental Final Examination PRACTICE FINAL EXAM VERSION #1 MATH 15400 Algebra & Trigonometry II Exam directions similar to those on the departmental final.
More informationChapter 5: Trigonometric Functions of Angles
Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has
More informationName Class Date 1 E. Finding the Ratio of Arc Length to Radius
Name Class Date 0 RightAngle Trigonometry Connection: Radian Measure Essential question: What is radian, and how are radians related to degrees? CC.9 2.G.C.5 EXPLORE Finding the Ratio of Arc Length to
More informationSA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid
Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More information9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration
Ch 9 Rotation 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Q: What is angular velocity? Angular speed? What symbols are used to denote each? What units are used? Q: What is linear
More informationLinear and angular kinematics
Linear and angular kinematics How far? Describing change in linear or angular position Distance (scalar): length of path Displacement (vector): difference between starting and finishing positions; independent
More informationTrigonometry Lesson Objectives
Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the
More informationDrexelSDP GK12 LESSON
Lesson: Torqued DrexelSDP GK12 LESSON Subject Area(s) Measurement, Number & Operations, Physical Science, Science & Technology Associated Unit Forget the Chedda! Lesson Title Torqued Header A force from
More information42 Degrees and Radians
Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest thousandth. 1.11.773 First, convert 0. 773 into minutes and seconds. Next, convert 0.38' into seconds.
More informationChapter 7  Rotational Motion w./ QuickCheck Questions
Chapter 7  Rotational Motion w./ QuickCheck Questions 2015 Pearson Education, Inc. Anastasia Ierides Department of Physics and Astronomy University of New Mexico October 1, 2015 Review of Last Time Uniform
More informationEVERY DAY: Practise and develop oral and mental skills (e.g. counting, mental strategies, rapid recall of +, and x facts)
Year 6/Band 6 Autumn Term Maths Curriculum Progression and Assessment Criteria EVERY DAY: Practise and develop oral and mental skills (e.g. counting, mental strategies, rapid recall of +, and x facts)
More informationNOTE: Please DO NOT write in exam booklet. Use the answer sheet for your answers.
ELECTRICIAN PREAPPRENTICESHIP MATH ENTRANCE EXAM NOTE: Please DO NOT write in exam booklet. Use the answer sheet for your answers. NOTE: DO NOT MARK SECTION A. PLACE YOUR ANSWERS ON THE SHEET PROVIDED
More information1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft
2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite
More informationMaths Targets Year 1 Addition and Subtraction Measures. N / A in year 1.
Number and place value Maths Targets Year 1 Addition and Subtraction Count to and across 100, forwards and backwards beginning with 0 or 1 or from any given number. Count, read and write numbers to 100
More informationTrigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:
First, a couple of things to help out: Page 1 of 24 Use periodic properties of the trigonometric functions to find the exact value of the expression. 1. cos 2. sin cos sin 2cos 4sin 3. cot cot 2 cot Sin
More informationHW 7 Q 14,20,20,23 P 3,4,8,6,8. Chapter 7. Rotational Motion of the Object. Dr. Armen Kocharian
HW 7 Q 14,20,20,23 P 3,4,8,6,8 Chapter 7 Rotational Motion of the Object Dr. Armen Kocharian Axis of Rotation The radian is a unit of angular measure The radian can be defined as the arc length s along
More informationYear 6 Maths Objectives
Year 6 Maths Objectives Place Value COUNTING COMPARING NUMBERS IDENTIFYING, REPRESENTING & ESTIMATING NUMBERS READING & WRITING NUMBERS UNDERSTANDING PLACE VALUE ROUNDING PROBLEM SOLVING use negative numbers
More informationApplications for Triangles
Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given
More informationWorking in 2 & 3 dimensions Revision Guide
Tips for Revising Working in 2 & 3 dimensions Make sure you know what you will be tested on. The main topics are listed below. The examples show you what to do. List the topics and plan a revision timetable.
More informationDegrees, seconds, minutes changed to decimal degree by hand using the TI82 or TI83 Plus
Angles ANGLE Topics Coterminal Angles Defintion of an angle Decimal degrees to degrees, minutes, seconds by hand using the TI82 or TI83 Plus Degrees, seconds, minutes changed to decimal degree by hand
More informationRugs. This problem gives you the chance to: find perimeters of shapes use Pythagoras Rule. Hank works at a factory that makes rugs.
Rugs This problem gives you the chance to: find perimeters of shapes use Pythagoras Rule Hank works at a factory that makes rugs. The edge of each rug is bound with braid. Hank s job is to cut the correct
More informationThe Six Trigonometric Functions
CHAPTER 1 The Six Trigonometric Functions Copyright Cengage Learning. All rights reserved. SECTION 1.1 Angles, Degrees, and Special Triangles Copyright Cengage Learning. All rights reserved. Learning Objectives
More informationLecture 14. More on rotation. Rotational Kinematics. Rolling Motion. Torque. Cutnell+Johnson: , 9.1. More on rotation
Lecture 14 More on rotation Rotational Kinematics Rolling Motion Torque Cutnell+Johnson: 8.18.6, 9.1 More on rotation We ve done a little on rotation, discussing objects moving in a circle at constant
More informationChapter. Trigonometry. By the end of this chapter, you will. 4, _ π 3, π_, and their multiples less
Chapter Trigonometry You may be surprised to learn that scientists, engineers, designers, and other professionals who use angles in their daily work generally do not measure the angles in degrees. In this
More informationMath 366 Lecture Notes Section 11.1 Basic Notions (of Geometry)
Math 366 Lecture Notes Section. Basic Notions (of Geometry) The fundamental building blocks of geometry are points, lines, and planes. These terms are not formally defined, but are described intuitively.
More informationSpherical coordinates 1
Spherical coordinates 1 Spherical coordinates Both the earth s surface and the celestial sphere have long been modeled as perfect spheres. In fact, neither is really a sphere! The earth is close to spherical,
More information6.1  Introduction to Periodic Functions
6.1  Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: A function f is periodic if its values repeat at regular intervals. Graphically, this means that
More informationCircumference CHAPTER. www.ck12.org 1
www.ck12.org 1 CHAPTER 1 Circumference Here you ll learn how to find the distance around, or the circumference of, a circle. What if you were given the radius or diameter of a circle? How could you find
More informationReview for Chapter Five Exam
Write the decimal in words. 1. 4.00927 2. 4.79 Write the decimal in numbers 3. One hundred and twotenths Review for Chapter Five Exam 4. In his qualifying time trial, a race car driver averages a speed
More informationMathematics Book Two. Trigonometry One Trigonometry Two Permutations and Combinations
Mathematics 0 Book Two Trigonometry One Trigonometry Two Permutations and Combinations A workbook and animated series by Barry Mabillard Copyright 04 This page has been left blank for correct workbook
More informationPlane Trigonometry  Fall 1996 Test File
Plane Trigonometry  Fall 1996 Test File Test #1 1.) Fill in the blanks in the two tables with the EXACT values of the given trigonometric functions. The total point value for the tables is 10 points.
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationMATH122 Final Exam Review If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71.
MATH1 Final Exam Review 5. 1. If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71. 5.1. If 8 secθ = and θ is an acute angle, find sinθ exactly. 3 5.1 3a. Convert
More informationBiggar High School Mathematics Department. National 4 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 4 Learning Intentions & Success Criteria: Assessing My Progress Expressions and Formulae Topic Learning Intention Success Criteria I understand this Algebra
More informationTypes of Angles acute right obtuse straight Types of Triangles acute right obtuse hypotenuse legs
MTH 065 Class Notes Lecture 18 (4.5 and 4.6) Lesson 4.5: Triangles and the Pythagorean Theorem Types of Triangles Triangles can be classified either by their sides or by their angles. Types of Angles An
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of prealgebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationConcepts of Physics. Wednesday, October 7th
1206  Concepts of Physics Wednesday, October 7th Center of mass Last class, we talked about collision  in many cases objects were interacting with each other  for examples two skaters pushing off. In
More informationName Revision Sheet 1
Name Revision Sheet 1 1 What is 8? Show your working 11 Solve the equation y 1 Round 79 to the nearest 10. 1 Expand ( x 1 0 ) Use BIDMAS to work out 5 1 How many lines of symmetry does a square have? 1
More informationCampus Academic Resource Program Introduction to Radians
Handout Objectives: This handout will: Campus Academic Resource Program Introduce the unit circle. Define and describe the unit of radians. Show how to convert between radians to degrees. The Unit Circle:
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationSixth Grade Math Pacing Guide Page County Public Schools MATH 6/7 1st Nine Weeks: Days Unit: Decimals B
Sixth Grade Math Pacing Guide MATH 6/7 1 st Nine Weeks: Unit: Decimals 6.4 Compare and order whole numbers and decimals using concrete materials, drawings, pictures and mathematical symbols. 6.6B Find
More informationUnit 1, Review Transitioning from Previous Mathematics Instructional Resources: McDougal Littell: Course 1
Unit 1, Review Transitioning from Previous Mathematics Transitioning from previous mathematics to Sixth Grade Mathematics Understand the relationship between decimals, fractions and percents and demonstrate
More information