Understanding Angles
|
|
- Annice Norton
- 6 years ago
- Views:
Transcription
1 SKILL BUILDER 5.2 Understanding Angles SKILL BUILDER A triangle has three angles and no angle can e equal to or greater than 18. Consider what happens when an angle is not part of a triangle ut is in the - plane. Angles and Their Location in the - Plane An angle is formed when a ra is rotated aout a fied point called the erte. The ra is called the at the eginning of the angle and the at the end of the angle. Angles are often laelled with Greek letters, such as theta, a alpha, and eta. An angle is in standard position if the erte of the angle is at the origin and the lies along the positie -ais. The can e anwhere on the arc of rotation. Standard Form Not Standard Form Not Standard Form erte erte erte An angle can e positie or negatie. A positie angle is formed a counterclockwise rotation of the. A negatie angle is formed a clockwise rotation of the. a positie angle a negatie angle The - plane is diided into four s the - and -aes. If is a positie angle, then the lies in I when < < 9 II when 9 < < 18 III when 18 < < 27 = 18 II III = 9 I IV = or 36 IV when 27 < < 36 = CHAPTER 5 MODELLING PERIODIC FUNCTIONS
2 Let P (, ) e a point on the of an angle in standard position. Since P can e anwhere in the - plane, the angle can terminate anwhere in the - plane. P(, ) P(, ) 9 < 1 < < 2 < 27 P (, ) lies in the negatie -ais. 1 terminates in II. 2 terminates in III Coterminal angles share the same and the same. As an eample, here are four different angles with the same and the same. P(, ) If 1 12, then The principal angle is the angle etween and 36. The coterminal angles of 48, 84, and 24 all share the same principal angle of 12. The related acute angle is the angle formed the of an angle in standard position and the -ais. The related acute angle is alwas positie and lies etween and 9. In this eample, represents the related acute angle for. 5.2 UNDERSTANDING ANGLES 419
3 Eample 1 Determine the principal angle and the related acute angle for 225. Solution Sketch 225 terminating in II. Lael the principal angle and the related acute angle. related acute angle = -225 principal angle The principal angle is the smallest positie angle that is coterminal to 225. In this case, The related acute angle lies etween the and the -ais. It is positie ut less than 9. In this case, 225 ( 18 ) 45. Or, using the principal angle, Eample 2 Determine the net two consecutie positie coterminal angles and the first negatie coterminal angle for 43. Solution Sketch each situation, showing the principal angle of (a) The first positie coterminal angle for 43 is () The second coterminal angle is (c) The first negatie coterminal angle is Eample 3 Point P ( 3, 4) is on the of an angle in standard position. (a) Sketch the principal angle,. () Determine the alue of the related acute angle to the nearest degree. (c) Solution (a) What is the measure of to the nearest degree? Point P ( 3, 4) is in II, so the principal angle,, terminates in II. P(-3, 4) CHAPTER 5 MODELLING PERIODIC FUNCTIONS
4 () The related acute angle,, is in the right triangle. P(-3, 4) (c) The opposite side and the adjacent side are known so the tangent ratio can e used. tan o ad tan 4 3 pposite jacent tan Sustitute known alues. Focus 5.2 Ke Ideas Angles can e located anwhere in the - plane. The - and -aes diide the - plane into four s. The erte of an angle in standard position is at the origin, and the of the angle is along the positie -ais. The of the angle can lie anwhere in the - plane. The of an angle rotates to its terminal position, either in a positie, counterclockwise direction or a negatie, clockwise direction. The principal angle is the first positie angle less than 36. The of an angle defines an infinite numer of coterminal angles. These can e positie or negatie and are defined in terms of the principal angle. The are multiples of 36 ; that is, 36 n, where n I. The related acute angle is the positie angle etween the and the -ais. It is alwas less than 9. An angle in standard position can e epressed in terms of its related acute angle. 5.2 UNDERSTANDING ANGLES 421
5 Practise, Appl, Sole 5.2 A 1. Sketch each angle in standard position. (a) 135 () 21 (c) 315 (d) 3 (e) 225 (f) 33 (g) 15 (h) 12 (i) 15 (j) 163 (k) 321 (l) Determine the related acute angle for each angle in question Sketch each angle in standard position. (a) 379 () 491 (c) 545 (d) 64 (e) Determine whether each pair of angles is coterminal or not. (a) 23, 383 () 41, 421 (c) 5, 31 (d) 38, 398 (e) 19, 39 (f) 41, 319 (g) 28, 232 (h) 15, 465 (i) 123, 237 (j) 19, Calculate the net two positie coterminal angles. (a) 132 () 275 (c) 35 (d) 73 (e) Calculate the net two negatie coterminal angles. (a) 53 () 138 (c) 299 (d) 18 (e) Match each angle with its diagram. (a) 15 () 12 (c) 765 (d) 65 (e) 22 (f) 29 (g) 56 (h) 38 i. ii. iii. i.. i. 422 CHAPTER 5 MODELLING PERIODIC FUNCTIONS
6 ii. iii. 8. Determine the principal angle. (a) 187 () 41 (c) 67 (d) 95 (e) 282 (f) 73 (g) 135 (h) State the principal angle for the gien related acute angle and gien. (a) 24, II () 35, III (c) 19, IV (d) 63, I B 1. State all alues of, where n I as shown. (a) n, 4 n 6 () n, 1 n 2 (c) n, 2 n (d) n, 5 n Point P ( 9, 4) is on the of an angle in standard position. (a) Sketch the principal angle,. () What is the measure of the related acute angle to the nearest degree? (c) What is the measure of to the nearest degree? 12. Point P (7, 24) is on the of an angle in standard position. (a) Sketch the principal angle,. () What is the measure of the related acute angle to the nearest degree? (c) What is the measure of to the nearest degree? 13. Point P ( 5, 3) is on the of an angle,, in standard position. (a) Sketch the principal angle,. () What is the measure of the related acute angle to the nearest degree? (c) What is the measure of to the nearest degree? (d) What is the measure of the first negatie coterminal angle? 14. Check Your Understanding: Point P ( 5, 9) is on the of an angle in standard position. Eplain the role of the right triangle and the related acute angle in determining the principal alue of. C 15. Point P ( 5, 8) is on the of an angle,, in standard position. Determine all alues of for UNDERSTANDING ANGLES 423
Transition to College Math
Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: F-TF.2 Learning Target: Eplain
More informationAn angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.
Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationUnit 6 Introduction to Trigonometry Degrees and Radians (Unit 6.2)
Unit 6 Introduction to Trigonometr Degrees and Radians (Unit 6.2) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Understand an angle
More informationDISTRIBUTED LEARNING
DISTRIBUTED LEARNING RAVEN S WNCP GRADE 12 MATHEMATICS BC Pre Calculus Math 12 Alberta Mathematics 0 1 Saskatchewan Pre Calculus Math 0 Manitoba Pre Calculus Math 40S STUDENT GUIDE AND RESOURCE BOOK The
More informationTrigonometric Functions. Copyright Cengage Learning. All rights reserved.
4 Trigonometric Functions Copyright Cengage Learning. All rights reserved. 4.1 Radian and Degree Measure Copyright Cengage Learning. All rights reserved. What You Should Learn Describe angles. Use radian
More information1.1 Angles and Degree Measure
J. Jenkins - Math 060 Notes. Angles and Degree Measure An angle is often thought of as being formed b rotating one ra awa from a fied ra indicated b an arrow. The fied ra is the initial side and the rotated
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More information27 ft 3 adequately describes the volume of a cube with side 3. ft F adequately describes the temperature of a person.
VECTORS The stud of ectors is closel related to the stud of such phsical properties as force, motion, elocit, and other related topics. Vectors allow us to model certain characteristics of these phenomena
More information4.1 Angles and Angle Measure.notebook. Chapter 4: Trigonometry and the Unit Circle
Chapter 4: Trigonometry and the Unit Circle 1 Chapter 4 4.1 Angles and Angle Measure Pages 166 179 How many radii are there around any circle??? 2 There are radii around the circumference of any circle.
More informationThroughout this chapter you will need: pencil ruler protractor. 7.1 Relationship Between Sides in Rightangled. 13 cm 10.5 cm
7. Trigonometry In this chapter you will learn aout: the relationship etween the ratio of the sides in a right-angled triangle solving prolems using the trigonometric ratios finding the lengths of unknown
More informationI. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.
0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.
More informationChapter 3 Motion in a Plane
Chapter 3 Motion in a Plane Introduce ectors and scalars. Vectors hae direction as well as magnitude. The are represented b arrows. The arrow points in the direction of the ector and its length is related
More informationLesson 3: Free fall, Vectors, Motion in a plane (sections )
Lesson 3: Free fall, Vectors, Motion in a plane (sections.6-3.5) Last time we looked at position s. time and acceleration s. time graphs. Since the instantaneous elocit is lim t 0 t the (instantaneous)
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More informationNumber Plane Graphs and Coordinate Geometry
Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot:
More information1. Trigonometry.notebook. September 29, Trigonometry. hypotenuse opposite. Recall: adjacent
Trigonometry Recall: hypotenuse opposite adjacent 1 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2 Example: Determine the value of x. a)
More informationPractice Test - Chapter 4
Find the value of x. Round to the nearest tenth, if necessary. Find the measure of angle θ. Round to the nearest degree, if necessary. 1. An acute angle measure and the length of the hypotenuse are given,
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationLesson 10.2 Radian Measure and Arc Length
Lesson 10.1 Defining the Circular Functions 1. Find the eact value of each epression. a. sin 0 b. cos 5 c. sin 150 d. cos 5 e. sin(0 ) f. sin(10 ) g. sin 15 h. cos 0 i. sin(0 ) j. sin 90 k. sin 70 l. sin
More information9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes
Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we
More information7.4. The Primary Trigonometric Ratios. LEARN ABOUT the Math. Connecting an angle to the ratios of the sides in a right triangle. Tip.
The Primary Trigonometric Ratios GOL Determine the values of the sine, cosine, and tangent ratios for a specific acute angle in a right triangle. LERN OUT the Math Nadia wants to know the slope of a ski
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationGZW. How can you find exact trigonometric ratios?
4. Special Angles Aircraft pilots often cannot see other nearb planes because of clouds, fog, or visual obstructions. Air Traffic Control uses software to track the location of aircraft to ensure that
More informationHigher. Functions and Graphs. Functions and Graphs 15
Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 5 Set Theor 5 Functions 6 Inverse Functions 9 4 Eponential Functions 0 5 Introduction to Logarithms 0 6 Radians 7 Eact Values
More informationFundamentals of Mathematics (MATH 1510)
Fundamentals of Mathematics () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University March 14-18, 2016 Outline 1 2 s An angle AOB consists of two rays R 1 and R
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Polnomials and Quadratics Contents Polnomials and Quadratics 1 1 Quadratics EF 1 The Discriminant EF Completing the Square EF Sketching Paraolas EF 7 5 Determining the Equation of a
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More informationMath 20C. Lecture Examples.
Math 20C. Lecture Eamples. (8//08) Section 2.. Vectors in the plane Definition A ector represents a nonnegatie number and, if the number is not zero, a direction. The number associated ith the ector is
More informationExercise Set 4.3: Unit Circle Trigonometry
Eercise Set.: Unit Circle Trigonometr Sketch each of the following angles in standard position. (Do not use a protractor; just draw a quick sketch of each angle. Sketch each of the following angles in
More informationName These exercises cover topics from Algebra I and Algebra II. Complete each question the best you can.
Name These eercises cover topics from Algebra I and Algebra II. Complete each question the best you can. Multiple Choice: Place through the letter of the correct answer. You may only use your calculator
More informationQuadratic Equation. ax bx c =. = + + =. Example 2. = + = + = 3 or. The solutions are -7/3 and 1.
Quadratic Equation A quadratic equation is any equation that is equialent to the equation in fmat a c + + = 0 (1.1) where a,, and c are coefficients and a 0. The ariale name is ut the same fmat applies
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 111.
Algera Chapter : Polnomial and Rational Functions Chapter : Polnomial and Rational Functions - Polnomial Functions and Their Graphs Polnomial Functions: - a function that consists of a polnomial epression
More informationMATRIX TRANSFORMATIONS
CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B
More informationSection 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
More informationdifferent formulas, depending on whether or not the vector is in two dimensions or three dimensions.
ectors The word ector comes from the Latin word ectus which means carried. It is best to think of a ector as the displacement from an initial point P to a terminal point Q. Such a ector is expressed as
More informationModule 2: Trigonometry
Principles of Mathematics 1 Contents 1 Module : Trigonometr Section 1 Trigonometric Functions 3 Lesson 1 The Trigonometric Values for θ, 0 θ 360 5 Lesson Solving Trigonometric Equations, 0 θ 360 9 Lesson
More informationGeometry The Unit Circle
Geometry The Unit Circle Day Date Class Homework F 3/10 N: Area & Circumference M 3/13 Trig Test T 3/14 N: Sketching Angles (Degrees) WKS: Angles (Degrees) W 3/15 N: Arc Length & Converting Measures WKS:
More informationSpace Coordinates and Vectors in Space. Coordinates in Space
0_110.qd 11//0 : PM Page 77 SECTION 11. Space Coordinates and Vectors in Space 77 -plane Section 11. -plane -plane The three-dimensional coordinate sstem Figure 11.1 Space Coordinates and Vectors in Space
More informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationMath 20C. Lecture Examples.
Math 20C. Lecture Eamples. (8/1/08) Section 12.2. Vectors in three dimensions To use rectangular z-coordinates in three-dimensional space, e introduce mutuall perpendicular -, -, and z-aes intersecting
More informationLesson 6.2 Exercises, pages
Lesson 6.2 Eercises, pages 448 48 A. Sketch each angle in standard position. a) 7 b) 40 Since the angle is between Since the angle is between 0 and 90, the terminal 90 and 80, the terminal arm is in Quadrant.
More informationAlgebra/Trigonometry Review Notes
Algebra/Trigonometry Review Notes MAC 41 Calculus for Life Sciences Instructor: Brooke Quinlan Hillsborough Community College ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 1: POLYNOMIAL BASICS, POLYNOMIAL END BEHAVIOR,
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More informationTRIGONOMETRIC FUNCTIONS
TRIGNMETRIC FUNCTINS INTRDUCTIN In general, there are two approaches to trigonometry ne approach centres around the study of triangles to which you have already been introduced in high school ther one
More informationLesson 6: Apparent weight, Radial acceleration (sections 4:9-5.2)
Beore we start the new material we will do another Newton s second law problem. A bloc is being pulled by a rope as shown in the picture. The coeicient o static riction is 0.7 and the coeicient o inetic
More informationExponentials and Logarithms Review Part 2: Exponentials
Eponentials and Logaritms Review Part : Eponentials Notice te difference etween te functions: g( ) and f ( ) In te function g( ), te variale is te ase and te eponent is a constant. Tis is called a power
More informationChapter 1. The word trigonometry comes from two Greek words, trigonon, meaning triangle, and. Trigonometric Ideas COPYRIGHTED MATERIAL
Chapter Trigonometric Ideas The word trigonometr comes from two Greek words, trigonon, meaning triangle, and metria, meaning measurement This is the branch of mathematics that deals with the ratios between
More information11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
More informationSection 6: PRISMATIC BEAMS. Beam Theory
Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam
More informationRadian Measure and Angles on the Cartesian Plane
. Radian Measure and Angles on the Cartesian Plane GOAL Use the Cartesian lane to evaluate the trigonometric ratios for angles between and. LEARN ABOUT the Math Recall that the secial triangles shown can
More informationChapter 13: Trigonometry Unit 1
Chapter 13: Trigonometry Unit 1 Lesson 1: Radian Measure Lesson 2: Coterminal Angles Lesson 3: Reference Angles Lesson 4: The Unit Circle Lesson 5: Trig Exact Values Lesson 6: Trig Exact Values, Radian
More informationPearson Physics Level 20 Unit I Kinematics: Chapter 2 Solutions
Pearson Phsics Leel 0 Unit I Kinematics: Chapter Solutions Student Book page 71 Skills Practice Students answers will ar but ma consist of: (a) scale 1 cm : 1 m; ector will be 5 cm long scale 1 m forward
More informationA BRIEF REVIEW OF ALGEBRA AND TRIGONOMETRY
A BRIEF REVIEW OF ALGEBRA AND TRIGONOMETR Some Key Concepts:. The slope and the equation of a straight line. Functions and functional notation. The average rate of change of a function and the DIFFERENCE-
More informationStrain Transformation and Rosette Gage Theory
Strain Transformation and Rosette Gage Theor It is often desired to measure the full state of strain on the surface of a part, that is to measure not onl the two etensional strains, and, but also the shear
More informationPractice Test - Chapter 4
Find the value of x. Round to the nearest tenth, if necessary. 1. An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the length of the side opposite.
More information7-1. Basic Trigonometric Identities
7- BJECTIVE Identif and use reciprocal identities, quotient identities, Pthagorean identities, smmetr identities, and opposite-angle identities. Basic Trigonometric Identities PTICS Man sunglasses have
More informationAFM Midterm Review I Fall Determine if the relation is a function. 1,6, 2. Determine the domain of the function. . x x
AFM Midterm Review I Fall 06. Determine if the relation is a function.,6,,, 5,. Determine the domain of the function 7 h ( ). 4. Sketch the graph of f 4. Sketch the graph of f 5. Sketch the graph of f
More information5.1 Angles and Their Measurements
Graduate T.A. Department of Mathematics Dnamical Sstems and Chaos San Diego State Universit November 8, 2011 A ra is the set of points which are part of a line which is finite in one direction, but infinite
More informationWhen two letters name a vector, the first indicates the and the second indicates the of the vector.
8-8 Chapter 8 Applications of Trigonometry 8.3 Vectors, Operations, and the Dot Product Basic Terminology Algeraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle etween Vectors
More informationUnit 6: 10 3x 2. Semester 2 Final Review Name: Date: Advanced Algebra
Semester Final Review Name: Date: Advanced Algebra Unit 6: # : Find the inverse of: 0 ) f ( ) = ) f ( ) Finding Inverses, Graphing Radical Functions, Simplifying Radical Epressions, & Solving Radical Equations
More information4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS MR. FORTIER 1. Trig Functions of Any Angle We now extend the definitions of the six basic trig functions beyond triangles so that we do not have to restrict
More informationEssential Question How can you find a trigonometric function of an acute angle θ? opp. hyp. opp. adj. sec θ = hyp. adj.
. Right Triangle Trigonometry Essential Question How can you find a trigonometric function of an acute angle? Consider one of the acute angles of a right triangle. Ratios of a right triangle s side lengths
More informationTrigonometry Math 076
Trigonometry Math 076 133 Right ngle Trigonometry Trigonometry provides us with a way to relate the length of sides of a triangle to the measure of its angles. There are three important trigonometric functions
More informationBlue and purple vectors have same magnitude and direction so they are equal. Blue and green vectors have same direction but different magnitude.
A ector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the ector represents the magnitude and the arrow indicates the direction of the ector. Blue and
More informationMth 133 Trigonometry Review Problems for the Final Examination
Mth 1 Trigonometry Review Problems for the Final Examination Thomas W. Judson Stephen F. Austin State University Fall 017 Final Exam Details The final exam for MTH 1 will is comprehensive and will cover
More informationDerivatives 2: The Derivative at a Point
Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned
More informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012
The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,
More informationREVIEW, pages
REVIEW, pages 5 5.. Determine the value of each trigonometric ratio. Use eact values where possible; otherwise write the value to the nearest thousandth. a) tan (5 ) b) cos c) sec ( ) cos º cos ( ) cos
More information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More information192 Calculus and Structures
9 Calculus and Structures CHAPTER PRODUCT, QUOTIENT, CHAIN RULE, AND TRIG FUNCTIONS Calculus and Structures 9 Copyright Chapter PRODUCT, QUTIENT, CHAIN RULE AND TRIG FUNTIONS. NEW FUNCTIONS FROM OLD ONES
More information9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
More information5Higher-degree ONLINE PAGE PROOFS. polynomials
5Higher-degree polnomials 5. Kick off with CAS 5.2 Quartic polnomials 5.3 Families of polnomials 5.4 Numerical approimations to roots of polnomial equations 5.5 Review 5. Kick off with CAS Quartic transformations
More informationThe Dot Product
The Dot Product 1-9-017 If = ( 1,, 3 ) and = ( 1,, 3 ) are ectors, the dot product of and is defined algebraically as = 1 1 + + 3 3. Example. (a) Compute the dot product (,3, 7) ( 3,,0). (b) Compute the
More informationTrigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric
Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric
More informationMATH 120-Vectors, Law of Sinesw, Law of Cosines (20 )
MATH 120-Vectors, Law of Sinesw, Law of Cosines (20 ) *Before we get into solving for oblique triangles, let's have a quick refresher on solving for right triangles' problems: Solving a Right Triangle
More informationDefine General Angles and Use Radian Measure
1.2 a.1, a.4, a.5; P..E TEKS Define General Angles and Use Radian Measure Before You used acute angles measured in degrees. Now You will use general angles that ma be measured in radians. Wh? So ou can
More information- 5π 2. a. a. b. b. In 5 7, convert to a radian measure without using a calculator
4-1 Skills Objective A In 1 and, the measure of a rotation is given. a. Convert the measure to revolutions. b. On the circle draw a central angle showing the given rotation. 1. 5. radians - a. a. b. b.
More informationTrigonometric Functions
TrigonometricReview.nb Trigonometric Functions The trigonometric (or trig) functions are ver important in our stud of calculus because the are periodic (meaning these functions repeat their values in a
More informationMEI Core 2. Sequences and series. Section 1: Definitions and Notation
Notes and Eamples MEI Core Sequences and series Section : Definitions and Notation In this section you will learn definitions and notation involving sequences and series, and some different ways in which
More information1.1 Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. 1) 162
Math 00 Midterm Review Dugopolski Trigonometr Edition, Chapter and. Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. ) ) - ) For the given angle,
More informationa by a factor of = 294 requires 1/T, so to increase 1.4 h 294 = h
IDENTIFY: If the centripetal acceleration matches g, no contact force is required to support an object on the spinning earth s surface. Calculate the centripetal (radial) acceleration /R using = πr/t to
More informationTrigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,
More informationMOTION IN 2-DIMENSION (Projectile & Circular motion And Vectors)
MOTION IN -DIMENSION (Projectile & Circular motion nd Vectors) INTRODUCTION The motion of an object is called two dimensional, if two of the three co-ordinates required to specif the position of the object
More informationVectors in Two Dimensions
Vectors in Two Dimensions Introduction In engineering, phsics, and mathematics, vectors are a mathematical or graphical representation of a phsical quantit that has a magnitude as well as a direction.
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 informationInverse Trigonometric Functions. inverse sine, inverse cosine, and inverse tangent are given below. where tan = a and º π 2 < < π 2 (or º90 < < 90 ).
Page 1 of 7 1. Inverse Trigonometric Functions What ou should learn GOAL 1 Evaluate inverse trigonometric functions. GOAL Use inverse trigonometric functions to solve real-life problems, such as finding
More informationAPPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I
APPENDIXES A Numbers, Inequalities, and Absolute Values B Coordinate Geometr and Lines C Graphs of Second-Degree Equations D Trigonometr E F Sigma Notation Proofs of Theorems G The Logarithm Defined as
More informationA11.1 Areas under curves
Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium.
More informationGraphs and polynomials
1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions
More informationAlgebra 1 Skills Needed for Success in Math
Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif
More informationTrigonometric Ratios. θ + k 360
Trigonometric Ratios These notes are intended as a summary of section 6.1 (p. 466 474) in your workbook. You should also read the section for more complete explanations and additional examples. Coterminal
More informationCHAPTER 3: Kinematics in Two Dimensions; Vectors
HAPTER 3: Kinematics in Two Dimensions; Vectors Solution Guide to WebAssign Problems 3.1 [] The truck has a displacement of 18 + (16) blocks north and 1 blocks east. The resultant has a magnitude of +
More informationScalars distance speed mass time volume temperature work and energy
Scalars and Vectors scalar is a quantit which has no direction associated with it, such as mass, volume, time, and temperature. We sa that scalars have onl magnitude, or size. mass ma have a magnitude
More informationragsdale (zdr82) HW7 ditmire (58335) 1 The magnetic force is
ragsdale (zdr8) HW7 ditmire (585) This print-out should have 8 questions. Multiple-choice questions ma continue on the net column or page find all choices efore answering. 00 0.0 points A wire carring
More informationc arc length radius a r radians degrees The proportion can be used to
Advanced Functions Page of Radian Measures Angles can be measured using degrees or radians. Radian is the measure of an angle. It is defined as the angle subtended at the centre of the circle in the ratio
More information10.5. Polar Coordinates. 714 Chapter 10: Conic Sections and Polar Coordinates. Definition of Polar Coordinates
71 Chapter 1: Conic Sections and Polar Coordinates 1.5 Polar Coordinates rigin (pole) r P(r, ) Initial ra FIGURE 1.5 To define polar coordinates for the plane, we start with an origin, called the pole,
More informationPractice Questions for Midterm 2 - Math 1060Q - Fall 2013
Eam Review Practice Questions for Midterm - Math 060Q - Fall 0 The following is a selection of problems to help prepare ou for the second midterm eam. Please note the following: anthing from Module/Chapter
More informationIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
Maima and minima In this unit we show how differentiation can be used to find the maimum and minimum values of a function. Because the derivative provides information about the gradient or slope of the
More information