There are three standard ways of measuring angles: degrees, radians and grads.

Size: px
Start display at page:

Download "There are three standard ways of measuring angles: degrees, radians and grads."

Transcription

1 CHAPTER Trigonometry Section. Angles and Their Measures Degrees and Radians (and Gradians) There are three standard ways of measuring angles: degrees, radians and grads. Degree measurement is based on dividing the central angle of a circle in 0 equal arts. Radian measurement is based on dividing the central angle of a circle in equal arts. Gradian measurement is based on dividing the central angle of a circle in 400 equal arts. It is not commonly used and will be ignored here. Because of these definitions, we have the following relationshi. radians = 0 () = 0 = 0 Eamle.: Convert into degrees. Multilying by 0 gives 0 =. Eamle.: Convert 0 into radians. Multilying by 0 gives 0 0 = 9. Circular Sector A circular sector is a ortion of a circle, with the center of the circle as the verte of the angle q. r s q r The length of the arc s is given by s = rq, where q is measured in radians. The area A of the sector is given by A = r q, where q is measured in radians. Eamle.: A car travels along a circular arc with radius 40 feet and angle. How far did the car travel? Since the angle q must be in radians, we have s = feet 80 Eamle.4: A lawn srinkler srays along a circular arc with radius 4 feet and angle. What area is covered by the srinkler? Since the angle q must be in radians, we have Angular and Linear Velocity A = (4) 94 feet 80 For the uroses of this class, linear velocity is the same as seed. Using the standard method from beginning algebra, linear velocity = distance time Similarly, angular velocity is defined as =) v = s t The letter w is lower case omega. angular velocity = angle time =) w = q t Eamle.: Find the angular velocity of the minute hand of a clock. The minute hand of a clock makes a comlete revolution in 0 minutes, therefore w = 0 rad /min = 0 rad /min

2 Eamle.: The head of a foot long golf club rotates through an angle of 00 in 0. sec. Find the seed of the head of the golf club. The radius r of the circular ath is ft. The angular velocity is w = 00 (/80) 0.. rad/sec. Therefore, the seed of the club is v = rw = (.) 7 feet/sec 7 miles/hour Eamle.7: A car travels at 0 mh three-fourths the way around a circular ath. If the radius of the ath is 0 feet, find the angular velocity of the car. Since w = q t, and q = s r, w = q t = s/r t = r s t = v r Converting the seed 0 mh to 88 feet/sec, we have Section. Si Trigonometric Functions w = v r = rad/sec Right Triangle Trigonometry hyotenuse oosite side A adjacent side cos A = adjacent hyotenuse sec A = hyotenuse adjacent sin A = oosite hyotenuse csc A = hyotenuse oosite tan A = oosite adjacent cot A = adjacent oosite This section involves using trigonometry to solve word roblems. Two terms used in the homework are angle of elevation and angle of deression. The angle of elevation is an angle that starts at a horizontal line and oens uward. Similarly, the angle of deression is an angle that starts at a horizontal line and oens downward. Eamle.8: The angle of elevation from your eye to the to of a ainting is. The angle of elevation from your eye to the bottom of the ainting is 8. If you are standing feet from the wall holding the ainting, how tall is the ainting? y 8 When looking at the bottom of the ainting, we have tan 8 = =) = tan 8 When looking at the to of the ainting, we have tan = + y =) + y = tan Therefore, the height of the ainting is y = tan tan 8.8 feet Section. The Unit Circle An angle in standard osition is two rays, etending from the origin with the initial side of the angle along the ositive -aes. The terminal side of an angle is the other ray of the angle. terminal side reference angle A reference angle is the angle between the terminal side of the given angle and -ais.

3 Eamle.9: Find the reference angle for. Drawing the angle in the second quadrant leaves gives a reference angle of 80 = 0. A coterminal angle with a given angle is an angle whose terminal side is at the same location. Eamle.: Find two angles coterminal with. Drawing the angle in the third quadrant, we find the reference angle. angle = reference angle = 0 coterminal angle = Secial Angles Since the reference angle is 0, we have a coterminal angle of. In general, coterminal angles can be found by adding multiles of 0. Therefore, + 0 = 70 is another coterminal angle sin 0 cos 0 tan 0 undefined Eamle.: Find the eact value of sin and cos. Start by drawing the angle and noticing the angle is in the second quadrant and has a reference angle of. From the secial angles, we know that sin = cos =. and Since the sine function is ositive in the second quadrant and the cosine function is negative in the second quadrant, we have sin = and cos = Section.4 Grahs of Sines and Cosines This section focusses on the grahs of the sine and cosine functions. y = A cos(b + C)+D and y = A sin(b + C)+D For both of these functions, amlitude = A eriod = B hase shift = C B vertical shift = D Eamle.: Find all the imortant information for f () = sin + + and sketch the grah.

4 Alying the above concets to the given equation, we have the following figure. 7 Figure.: Grah of f () = sin + + using ale ale, ale y ale The eriod is found by using eriod = B = = and the hase shift is found from hase shift = C B = / = Reading directly from the function, we have an amlitude of and a vertical shift of. Periodic behavior Periodic behavior is a attern that reeats itself. For eamle, the temerature in a city is eriodic, both on an annual scale (cold every winter, warm every summer) and on a daily scale (cooler at night, warmer in the day.) Certain tyes of eriodic behavior can be modeled by a sine or cosine function. This tye of modeling can be done without using a calculator. Eamle.: The daily summer temerature in Fremont can be modeled by a sine function. If at :00 AM the temerature reaches is coldest value of 48 F and at :00 PM, the temerature reaches it highest value of 78 F, find a cosine function to model the temerature in the room. Let t = 0 reresent midnight. Since the temerature varies from 48 to 78, these reresent the high and low oints on the sine grah. Also, we know that the low temerature occurs at time t =. This information is summarized in Figure.. 80 Temerature (in F) Time (in hours after midnight) Figure.: Time versus Temerature With this grah, we can now generate the constants in the equation T(t) =A cos(bt + C)+D The total height of the wave is = 0 so the amlitude of the wave is. Since the grah is an inverted cosine wave, we have A =. Since the vertical center of the wave is T =, the vertical shift is given by D =. The temerature has a eriod of 4 hours. Therefore, eriod = B = 4 =) B = Since the hase shift is t =, we can set determine value of C by substituting known values into B + C = 0. ()+C = 0 =) C = Therefore, the equation is given by T(t) = cos t +

5 Section. Grahs of Other Trig Functions The four functions tangent, cotangent, secant and cosecant can all be written in terms of sine and cosine. tan A = sin A cos A cot A = cos A sin A sec A = cos A csc A = sin A Since each of these four functions has a denominator that becomes zero for certain values of A, each of the functions must contain vertical asymtotes. In all cases, the vertical asymtotes occur when the denominator has a value of zero. Eamle.4: Use the grah of y = cos and grah of y = tan. Start by sketching the grah of f () =cos to determine the asymtotes. 4 Figure.: Grah of f () =cos( ) using ale ale, ale y ale tiongrah of f () =cos using ale ale, ale y ale At each lace the grah of cosine crosses the -ais, draw a vertical line for the asymtotes of tangent grah. 4 Figure.4: Asymtotes of f () =tan( ) using ale ale 4 Figure.: Grah of f () =tan( ) using 4 ale ale 4 Figure.: Grah of f () =tan( ) using 4 ale ale

6 Section. Inverse Trig Functions The inverse trigonometric functions are written in two forms. For eamle, the inverse function for f () =sin is written as either f () =arcsin or f () =sin. Note: sin = (sin ) Because of the misleading notation, I much refer the arcsin notation. Since none of the trig functions are one-to-one, to find the inverses, we must use a restricted domain. The effect of this is to restrict the range of the inverse trig function. Eamle.: Restrict the domain of y = sin so that the resulting grah shows a one-to-one function. Below is the grah of f () =sin on the interval ale ale drawn as a dashed curve. To have a one-to-one function, use only the ortion of the grah in the interval ale ale. Figure.7: Grah of f () =sin Eamle.: Sketch a grah of y = arcsin and use the grah to state the range. The grah of an inverse function y = f () can be found by switching the and y value of y = f (). We also know the range of f () is the same as the domain of f (). Therefore, the range of the inverse sine function is the same as the domain of the the restricted sine function. Below is the grah of y = arcsin on the interval ale y ale drawn as a dashed curve. To ensure that arcsine is a function, use only the ortion of the grah in the interval ale y ale Figure.8: Grah of f () =arcsin For the reasons outlined in the revious eamles, the range of each of the inverse trig functions is restricted to a secific interval. The three most common are given below. function range arccos arcsin arctan 0 ale y ale ale y ale < y < One of the tyes of roblems in which you will see the inverse trig functions is demonstrated in the net eamle. Eamle.7: Write tan arccos without trig functions.

7 Start by searating the roblem into two arts by making the substitution q = arccos. The question can then be restated as Find tan q if q = arccos From q = arccos, we have cos q = q From this triangle, we can use the Pythagorean Theorem to find the length of the vertical segment (call it y). + y = =) y = 4 4 q From this new triangle, we can now write tan arccos 4 = tan q =

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.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 information

Trigonometric Functions: The Unit Circle

Trigonometric 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 information

Algebra. 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. 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 information

Section 5-9 Inverse Trigonometric Functions

Section 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 information

Trigonometric Functions and Triangles

Trigonometric 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 information

RIGHT TRIANGLE TRIGONOMETRY

RIGHT 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 information

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Angles 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 information

Math Placement Test Practice Problems

Math 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 information

Section 6-3 Double-Angle and Half-Angle Identities

Section 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 information

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

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 information

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

Trigonometry 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 information

Algebra and Geometry Review (61 topics, no due date)

Algebra 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 information

Chapter 5: Trigonometric Functions of Angles

Chapter 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 information

Right Triangle Trigonometry

Right 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 information

Give an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179

Give 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 information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL 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 information

1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic

1. 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 information

Solutions to Exercises, Section 5.1

Solutions 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 information

opp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles

opp (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 information

Graphing Trigonometric Skills

Graphing 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 information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell 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 information

Lesson Plan. Students will be able to define sine and cosine functions based on a right triangle

Lesson 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 information

4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles

4.3 & 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 information

Trigonometry. An easy way to remember trigonometric properties is:

Trigonometry. 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 information

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Prentice 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 information

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Geometry 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 information

5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.

5.3 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 information

2312 test 2 Fall 2010 Form B

2312 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 information

Functions and their Graphs

Functions 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 information

General Physics 1. Class Goals

General Physics 1. Class Goals General Physics 1 Class Goals Develop problem solving skills Learn the basic concepts of mechanics and learn how to apply these concepts to solve problems Build on your understanding of how the world works

More information

Higher Education Math Placement

Higher 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 information

Trigonometry Hard Problems

Trigonometry 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 information

Unit 6 Trigonometric Identities, Equations, and Applications

Unit 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 information

Trigonometry 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 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 information

Radian Measure and the Unit Circle Approach

Radian Measure and the Unit Circle Approach Radian Measure and the Unit Circle Aroach How does an odometer or seedometer on an automobile work? The transmission counts how many times the tires rotate (how many full revolutions take lace) er second.

More information

Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring

Right 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 information

Core Maths C3. Revision Notes

Core Maths C3. Revision Notes Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

More information

Measuring relative phase between two waveforms using an oscilloscope

Measuring relative phase between two waveforms using an oscilloscope Measuring relative hase between two waveforms using an oscilloscoe Overview There are a number of ways to measure the hase difference between two voltage waveforms using an oscilloscoe. This document covers

More information

Trigonometry Review Workshop 1

Trigonometry 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 information

Lesson 1: Exploring Trigonometric Ratios

Lesson 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 information

Chapter 5 Resource Masters

Chapter 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 information

SAT Subject Math Level 2 Facts & Formulas

SAT 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

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South 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

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.

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. 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 information

Evaluating trigonometric functions

Evaluating 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 information

PRE-CALCULUS GRADE 12

PRE-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 information

Self-Paced Study Guide in Trigonometry. March 31, 2011

Self-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 information

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

Georgia 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 information

Friday, January 29, 2016 9:15 a.m. to 12:15 p.m., only

Friday, 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 information

Trigonometry for AC circuits

Trigonometry 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 information

United Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane

United Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane United Arab Emirates University College of Sciences Deartment of Mathematical Sciences HOMEWORK 1 SOLUTION Section 10.1 Vectors in the Plane Calculus II for Engineering MATH 110 SECTION 0 CRN 510 :00 :00

More information

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM. 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 information

Semester 2, Unit 4: Activity 21

Semester 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 information

ALGEBRA 2/TRIGONOMETRY

ALGEBRA 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 information

Mathematics Placement Examination (MPE)

Mathematics 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 information

Trigonometric Functions

Trigonometric 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 information

Additional Topics in Math

Additional 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 information

Physics 201 Homework 8

Physics 201 Homework 8 Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

More information

Dear Accelerated Pre-Calculus Student:

Dear 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 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. 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 information

How To Solve The Pythagorean Triangle

How 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 information

Function Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015

Function Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015 Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax

More information

INVERSE TRIGONOMETRIC FUNCTIONS. Colin Cox

INVERSE TRIGONOMETRIC FUNCTIONS. Colin Cox INVERSE TRIGONOMETRIC FUNCTIONS Colin Cox WHAT IS AN INVERSE TRIG FUNCTION? Used to solve for the angle when you know two sides of a right triangle. For example if a ramp is resting against a trailer,

More information

ALGEBRA 2/TRIGONOMETRY

ALGEBRA 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 information

6.1 Basic Right Triangle Trigonometry

6.1 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 information

Birmingham City Schools

Birmingham City Schools Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 Pre-Assessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:

More information

Triangle Trigonometry and Circles

Triangle 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 information

SOLVING TRIGONOMETRIC EQUATIONS

SOLVING 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 information

Biggar 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 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 information

SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS) By Nghi H. Nguyen

SOLVING 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 information

Core Maths C2. Revision Notes

Core 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 information

9 Right Triangle Trigonometry

9 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 information

Chapter 8 Geometry We will discuss following concepts in this chapter.

Chapter 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 information

TRIGONOMETRY Compound & Double angle formulae

TRIGONOMETRY Compound & Double angle formulae TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae

More information

Estimated Pre Calculus Pacing Timeline

Estimated 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 information

Section 6.1 Angle Measure

Section 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 information

alternate interior angles

alternate interior angles alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

More information

Right Triangles 4 A = 144 A = 16 12 5 A = 64

Right 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 information

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

1 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 information

Advanced Math Study Guide

Advanced 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

DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2

DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2 DOE-HDBK-1014/2-92 JUNE 1992 DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2 U.S. Department of Energy Washington, D.C. 20585 FSC-6910 Distribution Statement A. Approved for public release; distribution

More information

The Deadly Sins of Algebra

The Deadly Sins of Algebra The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.

More information

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point. 6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.

More information

Using the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º.

Using the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º. Using the Quadrant Eye Piece Protractor Handle You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements 90º. Plumb Bob ø

More information

Geometry Notes PERIMETER AND AREA

Geometry 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 information

Pre Calculus Math 40S: Explained!

Pre Calculus Math 40S: Explained! www.math40s.com 7 Part I Ferris Wheels One of the most common application questions for graphing trigonometric functions involves Ferris wheels, since the up and down motion of a rider follows the shape

More information

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens.

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens. Pinhole Otics Science, at bottom, is really anti-intellectual. It always distrusts ure reason and demands the roduction of the objective fact. H. L. Mencken (1880-1956) OBJECTIVES To study the formation

More information

Course outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems)

Course outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Functions, domain and range Domain and range of rational and algebraic

More information

8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections 8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

More information

Unit 1 - Radian and Degree Measure Classwork

Unit 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 information

Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

Parallel 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 information

Coordinate Transformation

Coordinate Transformation Coordinate Transformation Coordinate Transformations In this chater, we exlore maings where a maing is a function that "mas" one set to another, usually in a way that reserves at least some of the underlyign

More information

Week 13 Trigonometric Form of Complex Numbers

Week 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 information

Chapter 7 Outline Math 236 Spring 2001

Chapter 7 Outline Math 236 Spring 2001 Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

More information

With the Tan function, you can calculate the angle of a triangle with one corner of 90 degrees, when the smallest sides of the triangle are given:

With the Tan function, you can calculate the angle of a triangle with one corner of 90 degrees, when the smallest sides of the triangle are given: Page 1 In game development, there are a lot of situations where you need to use the trigonometric functions. The functions are used to calculate an angle of a triangle with one corner of 90 degrees. By

More information

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Exam #3 Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

More information

How to Graph Trigonometric Functions

How 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 information

Introduction and Mathematical Concepts

Introduction and Mathematical Concepts CHAPTER 1 Introduction and Mathematical Concepts PREVIEW In this chapter you will be introduced to the physical units most frequently encountered in physics. After completion of the chapter you will be

More information

GRE Prep: Precalculus

GRE 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 information

The Primary Trigonometric Ratios Word Problems

The 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 information