4.8 Solving Problems with Trigonometry Pre-Calculus
|
|
- Brianne Lawson
- 7 years ago
- Views:
Transcription
1 4.8 Solving Problems with Trigonometry Pre-Calculus Learning Targets 1. Set up and solve application problems involving right triangle trigonometry. 2. Use overlapping right triangles to solve word problems including the use of indirect measurement. Next we will apply what we know about the trigonometric functions, and their inverses, to solve real world application problems. The most important part of these types of questions is an accurate, detailed picture. Angle of Elevation: The acute angle measured from a horizontal line UP to an object. Angle of Depression: The acute angle measured from a horizontal line DOWN to an object. Angle of Depression Angle of Elevation Example 1: From a boat on the lake, the angle of elevation to the top of a cliff is If the base of the cliff is 1394 feet from the boat, how high is the cliff (to the nearest foot)? Example 2: The bearings of two points on the shore from a boat are 115 and 123. Assume the two points are 855 feet apart. How far is the boat from the nearest point on shore if the shore is straight and runs north-south? Unit 5 1
2 4.1 Angles and Their Measures Pre-Calculus Learning Targets 1. Draw an angle in Radians by thinking in radians (without converting to degrees). 2. List the reference angle of the radian angle. 3. Convert Degrees Radians and vice versa. 4. Apply arc length formulas in Radians when given a radian angle or a degree angle. 5. Convert linear speed to angular speed & vice versa 6. Relate definitions of bearing and heading to real world problems. 7. Convert DMS to decimal degrees and vice versa There are multiple ways to measure angles: degrees, revolutions, bearings and radians Degrees: From Geometry: straight angle = 180 degrees and a circle = 360 degrees Revolutions: 1 revolution = 1 full turn around a circle y Radians: An angle of 1 radian is defined to be the angle at the center of a circle which spans an arc of length equal to the radius of that circle. 1 radian Arc length = 1 Radius x Standard Position: For degree angles, revolutions and radian angles, we draw the angles with their initial side along the positive x-axis of the coordinate axes. The terminal (end) side of the angle is then measured in a counterclockwise direction. NOTE: Bearings work differently and will be covered in class. Reference Angles: The acute angle between the terminal side of an angle and the x-axis. Example 1: Draw the following Radian angles. Then list their reference angle. a) 2 b) 11 6 c) 4 3 d) 2.9 Unit 5 2
3 Example 2: Convert from degrees to radians or radians to degrees: a) 200 b) 7 8 radians ArcLength: The length of a portion of the circumference of a circle is S = θrwhere S is the arc length, θ is the angle measured in radians, and r is the radius of the circle. Example 3: Find the length of an arc. Express the answer in terms of π. 5π a) θ = ; r = 4 cm b) θ = 125 ; r = 1.5 mm 6 Angular and Linear Motion Angular Speed: How fast something is spinning. Linear Speed: How fast something is travelling in one direction. Example 4: A turntable rotates at 50 revolutions per minute. What is its angular speed in radians per second? Unit 5 3
4 4.2 Trig. Functions of Acute Angles Pre-Calculus Learning Targets 1. Know and apply the six trigonometric ratios 2. Solve right triangles using the six trig. ratios 3. Know the ratios of the sides of the special right triangle 4. Know the ratios of the sides of the special right triangle 5. Apply the ratios of the special right triangles to real life application questions. Next let s put our angles inside triangles specifically right triangles. Three Basic Trigonometric Ratios side opposite θ side adjacent to θ side opposite θ sin θ =, cos θ =, tanθ = hypotenuse hypotenuse side adjacent to θ o Remember the ratios are used on an acute angle. o We memorized. The Reciprocal Ratios:, and 1 cscθ = sinθ 1 secθ = cosθ 1 cotθ = tanθ Example 1: Using the triangle at the right, find all six trigonometric functions of the angle θ θ Example 2: Given tan θ = 5, find the remaining trigonometric functions. 12 Unit 5 4
5 Special Right Triangles: #1: Right Triangle (MEMORIZE THESE RATIOS) A a) If AC = 1, and m A 45, solve for the remaining parts of the triangle. B C b) Find the sine, cosine and tangent values of 45. #2: Right Triangle (MEMORIZE THESE RATIOS) A a) ABC is equilateral, so each angle is. b) Draw the altitude of the triangle from A to BC. Call the point of intersection D. c) Therefore, m BAD. d) Suppose AB = 2. Solve for the remaining parts of BAD. B C e) Find the sine, cosine and tangent values of 30 and 60. Example 4: A ladder is extended to reach the top floor of an 84 foot tall burning building. The fire fighters see someone who needs rescuing in a window 8 feet below the roof. How far should the ladder be extended to reach the window if the ladder must be placed at the optimum operating angle of 60? Unit 5 5
6 4.3 Trig. Extended: The Circular Functions Pre-Calculus Learning Targets 1. Graph Radian and Degree angles in standard position 2. Find reference angles for Radian and Degree angles in standard position 3. Identify positive and negative angles that are Coterminal with a given angle. 4. Find the exact value of the six trig. ratios of an angle in standard position 5. Find the exact value of the six trig. ratios of quadrantal angles 6. Find the exact value of the six trig. ratios of non-quadrantal angles. Day 1 Day 2 DAY 1: SohCahToa works well for acute angles; but what if we need angles 90 or larger? What if we need negative angles? Standard Position: By placing angles in standard position, we can extend the terminal sides past the first quadrant. Remember each of the following facts Positive Angles are measured counterclockwise. Negative Angles are measured clockwise. Reference Angles are still the acute angle measured to the x-axis (just as in lesson 4.1) Coterminal Angles: Angles with the same initial and terminal sides, but different measures. The measures differ by integral multiple(s) of 2π or 360. Example1: Find and draw one positive and one negative angle that is co terminal with the given angle. a) 210 b) 13 4 Unit 5 6
7 Trigonometric Functions Redefined: Let θ be any angle in standard position and let Pxybe (, ) any point on the terminal side of the angle. Then r is the distance from the origin to Pxyor (, ) the radius of a circle and 2 2 r = x + y. The six Trigonometric Ratios are sinθ = cosθ = tanθ = cscθ = secθ = cotθ = Notice if r = 1, then we have a Unit Circle which is a circle with radius 1. sinθ = cosθ = tanθ = Example 2: Find the six trigonometric ratios of θ whose terminal side passes through the given point. a) (3, 4) b) (3, 4) A Shortcut for the Signs of Trigonometric Functions: Now we know the trig functions when given a point. We can also predict the of the trig functions if we know the. But how do we find the trig functions when given an angle? Unit 5 7
8 DAY 2: Quadrantal Angles: Any angle in standard position whose terminal side is on the x-axis or the y-axis. Example 3: Find the exact value of each given trig function using the given angle. a) sin π b) cos ( 360 ) c) 3π csc 2 d) 11π sec 2 Remember that you cannot divide by zero, so sometimes & are undefined! Non Quadrantal Angles: Any angle in standard position whose terminal side is NOT on the x-axis or the y-axis. To find a trig. ratio for a non-quadrantal angle, we need three things: 1. : We identify the quadrant of the angle according to its terminal side. The quadrant tells us the sign of the trigonometric ratio. 2. : Reference triangles are right triangles created by drawing a perpendicular from the terminal side of a non-quadrantal angle to the x-axis. The reference angles we used in Lesson 4.1 are inside this right triangle and for our purposes here, the reference angle will always be a multiple of the angles found in the special right triangles from Lesson : Remember we memorized these ratios (or the sine, cosine and tangent of these angles) in lesson 4.2. Unit 5 8
9 Example 4: Draw the given angle and list the quadrant in which it lies. List the measure of θ ref. Then, evaluate the indicated trig value. a) csc( 60 ) b) 7π cos 4 5π c) sin 6 d) tan 120 Now you have all the pieces to solve any puzzle Example 5: Find cos θ and cot θ if sin θ = 1 4 and tan θ < 0. Unit 5 9
10 Graphing Trig. Functions: Tangents and Reciprocals Pre-Calculus Learning Targets 8. Graph the six parent trigonometric functions. 9. Apply transformations to the six parent trigonometric functions. 2sin 1 = Warm Up: Graph y ( x π ) Tangent Function: f( x) = tan x Period: Amplitude: Vertical Asymptotes: Key points: Example 1: Graph at least one period of each function below.. Be sure to label your axes and clearly identify the asymptotes. 1 x a) y = tan π b) y = 2 tan 3x 2 Unit 5 10
11 So, what about the reciprocal functions?? The best way to create their graph is based on the original trig. functions. Let s start with y = csc x which is based on. From last chapter, we know there are certain quadrantal angles which make the reciprocal function undefined Answer the following questions When csc x is undefined,sin x =? Where on the unit circle does sin x = this value? What angles make us land there on the unit circle? Where are those angles on the sine graph provided? Since y = csc xundefined at those x values, what should y = csc x we draw on the graph at the x-intercepts of sine? Add them to your graph. Now let s find some other values o What angles make sin x = csc x? o Add these points to your graph. o Remember that sin 1 π = what is 6 2 csc π? Add this point to your graph. 6 1 o Try some other values when sin x =, what is csc x? When 3 1 sin x =, what is csc x? 4 o What pattern do you see with the y-values of csc x when x is between the asymptotes? We can repeat this process for the other reciprocal functions Graph y = sec xbelow. Graph y = cot xbelow. Unit 5 11
12 4.7 Inverse Trigonometric Functions Pre-Calculus Learning Targets 1. Use the appropriate notation for inverse trigonometric functions. 2. Graph the inverse Sine, Cosine and Tangent functions. 3. List the correct Domain and Range of the inverse functions. 4. Find an exact solution to an expression involving an inverse sine, cosine or tangent. 5. Find the composition of trig functions and their inverses. 6. Solve equations with inverse trig expressions (Calculator and non) Inverse Sine Inverse Cosine Inverse Tangent y = sin 1 x y = arcsin x y = cos 1 x y = arccos x y = tan 1 x y = arctan x D: D: D: R: R: R: Example 1: Find the exact value (in radians). a) cos -1 0 b) sin -1 0 c) arcsin 1 2 d) arctan 1 e) cos f) tan ( 3) g) 3 arcsin 2 h) arccos 3 2 Unit 5 12
13 Compositions with Inverse Functions Work from the inside out. Remember domain and range restrictions. Example 2: Evaluate each expression. a) sin arctan ( 3) b) cos 1 5π cos 3 Example 3: Find the algebraic expression equivalent to the given expression. 1 a) sin ( cos x) 1 b) cot ( sin 2x) Unit 5 13
Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.
SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.
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 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 informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
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 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 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 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 informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
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 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 informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More information2312 test 2 Fall 2010 Form B
2312 test 2 Fall 2010 Form B 1. Write the slope-intercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function
More informationDear Accelerated Pre-Calculus Student:
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
More information6.1 Basic Right Triangle Trigonometry
6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationTrigonometry Hard Problems
Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More information4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles
4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationSection 5-9 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.
5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationRight Triangle Trigonometry
Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationEvaluating trigonometric functions
MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More informationChapter 5 Resource Masters
Chapter Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois StudentWorks TM This CD-ROM includes the entire Student Edition along with the Study Guide, Practice,
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationTrigonometry. An easy way to remember trigonometric properties is:
Trigonometry It is possible to solve many force and velocity problems by drawing vector diagrams. However, the degree of accuracy is dependent upon the exactness of the person doing the drawing and measuring.
More informationLesson 1: Exploring Trigonometric Ratios
Lesson 1: Exploring Trigonometric Ratios Common Core Georgia Performance Standards MCC9 12.G.SRT.6 MCC9 12.G.SRT.7 Essential Questions 1. How are the properties of similar triangles used to create trigonometric
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationTrigonometry for AC circuits
Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 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
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationTrigonometric Functions
Trigonometric Functions 13A Trigonometry and Angles 13-1 Right-Angle Trigonometry 13- Angles of Rotation Lab Explore the Unit Circle 13-3 The Unit Circle 13-4 Inverses of Trigonometric Functions 13B Applying
More informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationThe Primary Trigonometric Ratios Word Problems
The Primary Trigonometric Ratios Word Problems. etermining the measures of the sides and angles of right triangles using the primary ratios When we want to measure the height of an inaccessible object
More informationHow to Graph Trigonometric Functions
How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle
More informationSelf-Paced Study Guide in Trigonometry. March 31, 2011
Self-Paced Study Guide in Trigonometry March 1, 011 1 CONTENTS TRIGONOMETRY Contents 1 How to Use the Self-Paced Review Module Trigonometry Self-Paced Review Module 4.1 Right Triangles..........................
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 informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More informationTriangle Trigonometry and Circles
Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More information1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic
1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single
More information9 Right Triangle Trigonometry
www.ck12.org CHAPTER 9 Right Triangle Trigonometry Chapter Outline 9.1 THE PYTHAGOREAN THEOREM 9.2 CONVERSE OF THE PYTHAGOREAN THEOREM 9.3 USING SIMILAR RIGHT TRIANGLES 9.4 SPECIAL RIGHT TRIANGLES 9.5
More informationQuestion Bank Trigonometry
Question Bank Trigonometry 3 3 3 3 cos A sin A cos A sin A 1. Prove that cos A sina cos A sina 3 3 3 3 cos A sin A cos A sin A L.H.S. cos A sina cos A sina (cosa sina) (cos A sin A cosa sina) (cosa sina)
More informationAdvanced Math Study Guide
Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular
More information(15.) To find the distance from point A to point B across. a river, a base line AC is extablished. AC is 495 meters
(15.) To find the distance from point A to point B across a river, a base line AC is extablished. AC is 495 meters long. Angles
More informationIndirect Measurement Technique: Using Trigonometric Ratios Grade Nine
Ohio Standards Connections Measurement Benchmark D Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More information1. Introduction circular definition Remark 1 inverse trigonometric functions
1. Introduction In Lesson 2 the six trigonometric functions were defined using angles determined by points on the unit circle. This is frequently referred to as the circular definition of the trigonometric
More informationSet 4: Special Congruent Triangles Instruction
Instruction Goal: To provide opportunities for students to develop concepts and skills related to proving right, isosceles, and equilateral triangles congruent using real-world problems Common Core Standards
More informationTRIGONOMETRY FOR ANIMATION
TRIGONOMETRY FOR ANIMATION What is Trigonometry? Trigonometry is basically the study of triangles and the relationship of their sides and angles. For example, if you take any triangle and make one of the
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationPRACTICE PROBLEMS IN ALGEBRA, TRIGONOMETRY, AND ANALYTIC GEOMETRY
PRACTICE PROLEMS IN ALGERA, TRIGONOMETRY, AND ANALYTIC GEOMETRY The accompanying problems from the subjects covered on the Mathematics Placement Examination can be used by students to identify subject
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationLesson Plan. Students will be able to define sine and cosine functions based on a right triangle
Lesson Plan Header: Name: Unit Title: Right Triangle Trig without the Unit Circle (Unit in 007860867) Lesson title: Solving Right Triangles Date: Duration of Lesson: 90 min. Day Number: Grade Level: 11th/1th
More informationPage. Trigonometry Sine Law and Cosine Law. push
Trigonometry Sine Law and Cosine Law Page Trigonometry can be used to calculate the side lengths and angle measures of triangles. Triangular shapes are used in construction to create rigid structures.
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More information(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its
(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:
More informationSOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions
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 informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationName: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More information6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationSolutions to Practice Problems
Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles
More informationPHYSICS 151 Notes for Online Lecture #6
PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities
More information