2. Be able to evaluate a trig function at a particular degree measure. Example: cos. again, just use the unit circle!

Size: px
Start display at page:

Download "2. Be able to evaluate a trig function at a particular degree measure. Example: cos. again, just use the unit circle!"

Transcription

1 Study Guide for PART II of the Fall 18 MAT187 Final Exam NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will be provided with two sheets of scratch paper which must both be turned in with you exam regardless of whether or not you use the scratch paper. You may NOT use your own scratch paper. Pencils ONLY! (Pen = - 5%). You may use YOUR single x 5 index card (turn in index card with test). This portion of the exam covers the Trigonometry portion of this course (.1.1). You should focus on your notes and homework from those sections and Tests #5 and #6. Of course some of you may benefit from utilizing the supplemental reading and videos on the class Help page from these sections too! BE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!! Here is a brief overview of what you should be able to do! 1. Be able to convert a degree measure into a radian measure (where will be part of your answer). Try 5 5 radians 18. Be able to evaluate a trig function at a particular degree measure. Example: cos 1 1 just use the unit circle!. Be able to utilize you unit circle to find the value of trig expressions like cos sin 1 cos sin 1 again, just use the unit circle!. Know which trig functions are positive and which ones are negative in each quadrant! S A 55 T C 5. Given a standard angle on the unit circle, be able to identify the coordinates. Having a solid understanding of the unit circle will help you on MANY of these problems! 6. Be able to find the value of an inverse trig function expression. Example: and arcsin problems from your notes and homework too. arctan angle is arctan. Be sure to practice arcos draw the partial unit circle from to and go find the only place where the tangent of that 6, note the answer is NOT Be able to find the value of composed trig functions and inverse trig functions. Example: Find others in your notes and homework to practice! 5 1 sin arccos sin 6 sin arccos.

2 8. Be able to evaluate various trig functions at certain degree measures. Example: cos, cos1,sin 6, sin 6 Answers are,,, respectively 9. Be able to calculate the length of an arc when given the radius and angle (in radians). See section. S r, in radians! 1. Can you solve tan cot? Think about the unit circle! x y Since tan and cot we just need to find any places on the unit circle where those quotients of the y x coordinates match! 11. Be able to simplify a basic trigonometric expression. Example: cos cot sin cos sin cos csc 1 sin 1 sin 1. Can you simplify 1 sin x1 sin x? 1 sin 1 sin 1 sin sin sin 1 sin cos x x x x x x x cot csc 1. What is sin 6 cos cos6 sin equivalent to? cos1,sin1,cos,sin. sin 6 cos cos6 sin sin 6 sin Make sure that you know the trig identities that I told you were important! You should recognize THIS example as an example of the sine of the difference of two angles. Be sure to think about what this problem would look like if it led to the sine of the sum of two angles or the cosine of the sum or difference of two angles. 5 sin A and cos B and A and B are both acute angles what does cos A B? 1. If 5 1 What about sin A B? To solve this problem draw two right triangles, one for angle A and one for angle B. On the triangle with angle A in it you can solve for the missing side and get x = and the one with angle B in it and get y = 1. Then just remember the correct sum and difference formulas cos A B cos Acos B sin Asin B sin A B sin Acos B cos Asin B Be able to simplify trigonometric expressions involving double angles. (go look some up!) Be sure that you are familiar with the double angle identities for sine and cosine (# and #1). Be sure that you know all three versions of #1!

3 16. Be able to solve basic trig equations in restricted domains. Example: Solve tan x in 18,6 tan x tan x x Use the unit circle on the given interval to look up the only place where y/x is C 17. Given a modified Sine equation be able to figure out the amplitude. y AsinBx D B Amplitude = A 18. Given a modified Sine equation be able to figure out the period. Period = B 19. Given a modified Sine equation be able to figure out the horizontal shift. Horizontal shift = C B. C Note if is negative (after you have rewritten the equation in "standard form") B C then the horizontal shift is to the left. B. Given the graph of a modified Sine function be able to select the corresponding equation. 1. Given a modified Sine equation be able to pick out the correct graph.. Make sure that you know the domains of Sine, Cosine and Tangent. Domain of both the sine and cosine function is, k Domain of the tangent function is all real numbers except where k is any odd integer. Remember that the tangent function has vertical asymptotes at any location where the cosine function =. Given two sides of a triangle and the included angle, be able to estimate the area Area absin C bc sin A ac sin B Also remember your decimal approximations for. Given two angles and one side of an angle, be able to find one of the remaining sides. Law of sines (unless it is a SAS in which case use law of cosines). b a b sin B a sin B sin A sin A 5. If cos A. What is the value of sin A? Make sure that YOU know all three versions of identity #1! 5 cos A 1 sin A 1 sin A 1 sin A sin A sin A sin A sin A

4 6. Be able to use the Law of Sines to solve problems like. In triangle ABC, B 6 c 8 and sin C Find the length of side b 1 b c c 8 b sin B sin 6 16 sin B sin C sin C 1 7. Be able to solve a SSS triangle for the largest angle. Since you are not allowed a calculator then obviously the side lengths must be chosen in such a way to yield an angle that you can obtain WITHOUT the use of a calculator! 8. Be able to solve a SAS triangle for the missing side. You should be able to estimate the answer without a calculator given your knowledge of the few basic decimals I told you to know Be able to find an exact value for a trig expression involving lesser used trig functions. Example: csc6 cot csc 5 1 cos csc 6 cot csc 5 sin 6 sin sin 5 1 or. Be able to compute compositions of regular trig functions with inverse trig functions. Like #7 BUT different trig and inverse trig functions AND you will need to draw a triangle rather than use the unit circle! 1. Given an angle in standard position and the coordinates of a point on the terminal side be able to obtain the value of any trig function for that angle.. Can you find the exact value of sin 15 tan15? sin 15 tan or. If cos and is in quadrant IV, find the value of sin. Draw a right triangle in quadrant IV with in standard position. The reference angle for will be one of the angles in your right triangle. Label the side adjacent to (the side on the positive x-axis) and label the hypotenuse. Use the Pythagorean theorem to find the y value and obtain 7. Then use your trig identity for sin. 7 7 sin sin cos 8

5 . Given a modified cosine equation, be able to find the period. 5. What is cos7x cosx sin 7x sin x equivalent to? Be sure you know the sum and difference identities for both sine and cosine. cos 7x cos x sin 7x sin x cos 7x x cos 1x 6. Be able to simplify a trig expression involving cos Remember cos cos sin cos 1 1 sin 7. Make sure that you know identity #9 and the several other versions of it that we have used in this course. cos sin 1 cos 1sin sin 1 cos 8. Make sure that you understand the relationship between a trigonometric function and its inverse. 1 If cos A x then cos x A 9. Be able to find a reference angle for a given degree measure.. Can you find an angle x in the third quadrant where x cot tan x? What about in the other quadrants? Go look at your unit circle! Remember that cotangent is x/y and tangent is y/x. Can you find an angle whose tangent matches the cotangent of an angle degrees less? (Answer 1. Can you find an angle coterminal to another angle? x 15 ). What about the other quadrants? Be sure to know for what values sin,sin,cos,cos, tan and tan are defined. i.e. What are the domains of each? sine and cosine both have domain,. tangent has domain all real numbers except 1 1 sin and cos both have domain 1,1 1 tan has domain, k where k is any odd integer.. Be able to find a solution to a trig equation involving sine and cosine Can you find cot 1? sec? cot 1 tan tan (1) sec cos 5. Be able to solve a very basic sine or cosine equation in the interval,6 6. Can you simplify basic expressions like cot x cos x? cos x cot x sin x cos x 1 1 csc x cos x cos x sin x cos x sin x 1

6 7. Be sure that you know the word definitions for each of the six regular trig functions. i.e. opposite sin hypotenuse 8. Be sure that you know the range of each of arcsin x, arc cos x and arctan x,,, respectively! 9. Know when you should use each of the following to find the missing side of a triangle.. Pythagorean theorem, Law of Sines, Law of Cosines, Similar triangles. 5. Be able to calculate the area of a triangle when you are given two sides and the included angle Area absin C bc sin A ac sin B

Math 1205 Trigonometry Review

Math 1205 Trigonometry Review Math 105 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x + y =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of

More information

Chapter 4 Trigonometric Functions

Chapter 4 Trigonometric Functions Chapter 4 Trigonometric Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Section 8 Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry

More information

Trigonometry Review Page 1 of 14

Trigonometry Review Page 1 of 14 Trigonometry Review Page of 4 Appendix D has a trigonometric review. This material is meant to outline some of the proofs of identities, help you remember the values of the trig functions at special values,

More information

Basic Trigonometry You Should Know (Not only for this class but also for calculus)

Basic Trigonometry You Should Know (Not only for this class but also for calculus) Angle measurement: degrees and radians. Basic Trigonometry You Should Know (Not only for this class but also for calculus) There are 360 degrees in a full circle. If the circle has radius 1, then the circumference

More information

Trigonometry. An Overview of Important Topics

Trigonometry. An Overview of Important Topics Trigonometry An Overview of Important Topics 1 Contents Trigonometry An Overview of Important Topics... 4 UNDERSTAND HOW ANGLES ARE MEASURED... 6 Degrees... 7 Radians... 7 Unit Circle... 9 Practice Problems...

More information

MATH 1113 Exam 3 Review. Fall 2017

MATH 1113 Exam 3 Review. Fall 2017 MATH 1113 Exam 3 Review Fall 2017 Topics Covered Section 4.1: Angles and Their Measure Section 4.2: Trigonometric Functions Defined on the Unit Circle Section 4.3: Right Triangle Geometry Section 4.4:

More information

Chapter 1 and Section 2.1

Chapter 1 and Section 2.1 Chapter 1 and Section 2.1 Diana Pell Section 1.1: Angles, Degrees, and Special Triangles Angles Degree Measure Angles that measure 90 are called right angles. Angles that measure between 0 and 90 are called

More information

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values

More information

Math 102 Key Ideas. 1 Chapter 1: Triangle Trigonometry. 1. Consider the following right triangle: c b

Math 102 Key Ideas. 1 Chapter 1: Triangle Trigonometry. 1. Consider the following right triangle: c b Math 10 Key Ideas 1 Chapter 1: Triangle Trigonometry 1. Consider the following right triangle: A c b B θ C a sin θ = b length of side opposite angle θ = c length of hypotenuse cosθ = a length of side adjacent

More information

6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function.

6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function. Math 160 www.timetodare.com Periods of trigonometric functions Definition A function y f ( t) f ( t p) f ( t) 6.4 & 6.5 Graphing Trigonometric Functions = is periodic if there is a positive number p such

More information

Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions. By Dr. Mohammed Ramidh

Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions. By Dr. Mohammed Ramidh Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions By Dr. Mohammed Ramidh Trigonometric Functions This section reviews the basic trigonometric functions. Trigonometric functions are important because

More information

SECTION 1.5: TRIGONOMETRIC FUNCTIONS

SECTION 1.5: TRIGONOMETRIC FUNCTIONS SECTION.5: TRIGONOMETRIC FUNCTIONS The Unit Circle The unit circle is the set of all points in the xy-plane for which x + y =. Def: A radian is a unit for measuring angles other than degrees and is measured

More information

Algebra 2/Trigonometry Review Sessions 1 & 2: Trigonometry Mega-Session. The Unit Circle

Algebra 2/Trigonometry Review Sessions 1 & 2: Trigonometry Mega-Session. The Unit Circle Algebra /Trigonometry Review Sessions 1 & : Trigonometry Mega-Session Trigonometry (Definition) - The branch of mathematics that deals with the relationships between the sides and the angles of triangles

More information

Unit 5. Algebra 2. Name:

Unit 5. Algebra 2. Name: Unit 5 Algebra 2 Name: 12.1 Day 1: Trigonometric Functions in Right Triangles Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Theta Opposite Side of an Angle Adjacent Side of

More information

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values

More information

Math 104 Final Exam Review

Math 104 Final Exam Review Math 04 Final Exam Review. Find all six trigonometric functions of θ if (, 7) is on the terminal side of θ.. Find cosθ and sinθ if the terminal side of θ lies along the line y = x in quadrant IV.. Find

More information

The reciprocal identities are obvious from the definitions of the six trigonometric functions.

The reciprocal identities are obvious from the definitions of the six trigonometric functions. The Fundamental Identities: (1) The reciprocal identities: csc = 1 sec = 1 (2) The tangent and cotangent identities: tan = cot = cot = 1 tan (3) The Pythagorean identities: sin 2 + cos 2 =1 1+ tan 2 =

More information

Trigonometry Review Tutorial Shorter Version

Trigonometry Review Tutorial Shorter Version Author: Michael Migdail-Smith Originally developed: 007 Last updated: June 4, 0 Tutorial Shorter Version Avery Point Academic Center Trigonometric Functions The unit circle. Radians vs. Degrees Computing

More information

Algebra2/Trig Chapter 10 Packet

Algebra2/Trig Chapter 10 Packet Algebra2/Trig Chapter 10 Packet In this unit, students will be able to: Convert angle measures from degrees to radians and radians to degrees. Find the measure of an angle given the lengths of the intercepted

More information

7.1 INTRODUCTION TO PERIODIC FUNCTIONS

7.1 INTRODUCTION TO PERIODIC FUNCTIONS 7.1 INTRODUCTION TO PERIODIC FUNCTIONS *SECTION: 6.1 DCP List: periodic functions period midline amplitude Pg 247- LECTURE EXAMPLES: Ferris wheel, 14,16,20, eplain 23, 28, 32 *SECTION: 6.2 DCP List: unit

More information

Section 5.1 Angles and Radian Measure. Ever Feel Like You re Just Going in Circles?

Section 5.1 Angles and Radian Measure. Ever Feel Like You re Just Going in Circles? Section 5.1 Angles and Radian Measure Ever Feel Like You re Just Going in Circles? You re riding on a Ferris wheel and wonder how fast you are traveling. Before you got on the ride, the operator told you

More information

Trigonometric identities

Trigonometric identities Trigonometric identities An identity is an equation that is satisfied by all the values of the variable(s) in the equation. For example, the equation (1 + x) = 1 + x + x is an identity. If you replace

More information

MATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

MATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. MATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) (sin x + cos x) 1 + sin x cos x =? 1) ) sec 4 x + sec x tan x - tan 4 x =? ) ) cos

More information

Honors Algebra 2 w/ Trigonometry Chapter 14: Trigonometric Identities & Equations Target Goals

Honors Algebra 2 w/ Trigonometry Chapter 14: Trigonometric Identities & Equations Target Goals Honors Algebra w/ Trigonometry Chapter 14: Trigonometric Identities & Equations Target Goals By the end of this chapter, you should be able to Identify trigonometric identities. (14.1) Factor trigonometric

More information

Pythagorean Identity. Sum and Difference Identities. Double Angle Identities. Law of Sines. Law of Cosines

Pythagorean Identity. Sum and Difference Identities. Double Angle Identities. Law of Sines. Law of Cosines Review for Math 111 Final Exam The final exam is worth 30% (150/500 points). It consists of 26 multiple choice questions, 4 graph matching questions, and 4 short answer questions. Partial credit will be

More information

Geometry Problem Solving Drill 11: Right Triangle

Geometry Problem Solving Drill 11: Right Triangle Geometry Problem Solving Drill 11: Right Triangle Question No. 1 of 10 Which of the following points lies on the unit circle? Question #01 A. (1/2, 1/2) B. (1/2, 2/2) C. ( 2/2, 2/2) D. ( 2/2, 3/2) The

More information

Math Section 4.3 Unit Circle Trigonometry

Math Section 4.3 Unit Circle Trigonometry Math 0 - Section 4. Unit Circle Trigonometr An angle is in standard position if its verte is at the origin and its initial side is along the positive ais. Positive angles are measured counterclockwise

More information

Chapter 8. Analytic Trigonometry. 8.1 Trigonometric Identities

Chapter 8. Analytic Trigonometry. 8.1 Trigonometric Identities Chapter 8. Analytic Trigonometry 8.1 Trigonometric Identities Fundamental Identities Reciprocal Identities: 1 csc = sin sec = 1 cos cot = 1 tan tan = 1 cot tan = sin cos cot = cos sin Pythagorean Identities:

More information

Trigonometric Equations

Trigonometric Equations Chapter Three Trigonometric Equations Solving Simple Trigonometric Equations Algebraically Solving Complicated Trigonometric Equations Algebraically Graphs of Sine and Cosine Functions Solving Trigonometric

More information

MATH STUDENT BOOK. 12th Grade Unit 5

MATH STUDENT BOOK. 12th Grade Unit 5 MATH STUDENT BOOK 12th Grade Unit 5 Unit 5 ANALYTIC TRIGONOMETRY MATH 1205 ANALYTIC TRIGONOMETRY INTRODUCTION 3 1. IDENTITIES AND ADDITION FORMULAS 5 FUNDAMENTAL TRIGONOMETRIC IDENTITIES 5 PROVING IDENTITIES

More information

Math 180 Chapter 6 Lecture Notes. Professor Miguel Ornelas

Math 180 Chapter 6 Lecture Notes. Professor Miguel Ornelas Math 180 Chapter 6 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 6.1 Section 6.1 Verifying Trigonometric Identities Verify the identity. a. sin x + cos x cot x = csc

More information

Unit 3 Unit Circle and Trigonometry + Graphs

Unit 3 Unit Circle and Trigonometry + Graphs HARTFIELD PRECALCULUS UNIT 3 NOTES PAGE 1 Unit 3 Unit Circle and Trigonometry + Graphs (2) The Unit Circle (3) Displacement and Terminal Points (5) Significant t-values Coterminal Values of t (7) Reference

More information

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle. We noticed how the x and y

More information

PREREQUISITE/PRE-CALCULUS REVIEW

PREREQUISITE/PRE-CALCULUS REVIEW PREREQUISITE/PRE-CALCULUS REVIEW Introduction This review sheet is a summary of most of the main topics that you should already be familiar with from your pre-calculus and trigonometry course(s), and which

More information

MAC 1114 REVIEW FOR EXAM #2 Chapters 3 & 4

MAC 1114 REVIEW FOR EXAM #2 Chapters 3 & 4 MAC 111 REVIEW FOR EXAM # Chapters & This review is intended to aid you in studying for the exam. This should not be the only thing that you do to prepare. Be sure to also look over your notes, textbook,

More information

7.1 INTRODUCTION TO PERIODIC FUNCTIONS

7.1 INTRODUCTION TO PERIODIC FUNCTIONS 7.1 INTRODUCTION TO PERIODIC FUNCTIONS Ferris Wheel Height As a Function of Time The London Eye Ferris Wheel measures 450 feet in diameter and turns continuously, completing a single rotation once every

More information

Graphs of other Trigonometric Functions

Graphs of other Trigonometric Functions Graphs of other Trigonometric Functions Now we will look at other types of graphs: secant. tan x, cot x, csc x, sec x. We will start with the cosecant and y csc x In order to draw this graph we will first

More information

Precalculus Second Semester Final Review

Precalculus Second Semester Final Review Precalculus Second Semester Final Review This packet will prepare you for your second semester final exam. You will find a formula sheet on the back page; these are the same formulas you will receive for

More information

Unit 6 Test REVIEW Algebra 2 Honors

Unit 6 Test REVIEW Algebra 2 Honors Unit Test REVIEW Algebra 2 Honors Multiple Choice Portion SHOW ALL WORK! 1. How many radians are in 1800? 10 10π Name: Per: 180 180π 2. On the unit circle shown, which radian measure is located at ( 2,

More information

Copyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1

Copyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1 8.3-1 Transformation of sine and cosine functions Sections 8.2 and 8.3 Revisit: Page 142; chapter 4 Section 8.2 and 8.3 Graphs of Transformed Sine and Cosine Functions Graph transformations of y = sin

More information

MATH 130 FINAL REVIEW version2

MATH 130 FINAL REVIEW version2 MATH 130 FINAL REVIEW version2 Problems 1 3 refer to triangle ABC, with =. Find the remaining angle(s) and side(s). 1. =50, =25 a) =40,=32.6,=21.0 b) =50,=21.0,=32.6 c) =40,=21.0,=32.6 d) =50,=32.6,=21.0

More information

In this section, you will learn the basic trigonometric identities and how to use them to prove other identities.

In this section, you will learn the basic trigonometric identities and how to use them to prove other identities. 4.6 Trigonometric Identities Solutions to equations that arise from real-world problems sometimes include trigonometric terms. One example is a trajectory problem. If a volleyball player serves a ball

More information

How to work out trig functions of angles without a scientific calculator

How to work out trig functions of angles without a scientific calculator Before starting, you will need to understand how to use SOH CAH TOA. How to work out trig functions of angles without a scientific calculator Task 1 sine and cosine Work out sin 23 and cos 23 by constructing

More information

Graphing Trig Functions. Objectives: Students will be able to graph sine, cosine and tangent functions and translations of these functions.

Graphing Trig Functions. Objectives: Students will be able to graph sine, cosine and tangent functions and translations of these functions. Graphing Trig Functions Name: Objectives: Students will be able to graph sine, cosine and tangent functions and translations of these functions. y = sinx (0,) x 0 sinx (,0) (0, ) (,0) /2 3/2 /2 3/2 2 x

More information

2009 A-level Maths Tutor All Rights Reserved

2009 A-level Maths Tutor All Rights Reserved 2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents radians 3 sine, cosine & tangent 7 cosecant, secant & cotangent

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Trigonometry Final Exam Study Guide Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a polar equation is given. Select the polar

More information

13.4 Chapter 13: Trigonometric Ratios and Functions. Section 13.4

13.4 Chapter 13: Trigonometric Ratios and Functions. Section 13.4 13.4 Chapter 13: Trigonometric Ratios and Functions Section 13.4 1 13.4 Chapter 13: Trigonometric Ratios and Functions Section 13.4 2 Key Concept Section 13.4 3 Key Concept Section 13.4 4 Key Concept Section

More information

1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle

1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle Pre- Calculus Mathematics 12 5.1 Trigonometric Functions Goal: 1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle Measuring Angles: Angles in Standard

More information

Unit 8 Trigonometry. Math III Mrs. Valentine

Unit 8 Trigonometry. Math III Mrs. Valentine Unit 8 Trigonometry Math III Mrs. Valentine 8A.1 Angles and Periodic Data * Identifying Cycles and Periods * A periodic function is a function that repeats a pattern of y- values (outputs) at regular intervals.

More information

C.3 Review of Trigonometric Functions

C.3 Review of Trigonometric Functions C. Review of Trigonometric Functions C7 C. Review of Trigonometric Functions Describe angles and use degree measure. Use radian measure. Understand the definitions of the si trigonometric functions. Evaluate

More information

Chapter 3, Part 4: Intro to the Trigonometric Functions

Chapter 3, Part 4: Intro to the Trigonometric Functions Haberman MTH Section I: The Trigonometric Functions Chapter, Part : Intro to the Trigonometric Functions Recall that the sine and cosine function represent the coordinates of points in the circumference

More information

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

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D. Review of Trigonometric Functions D7 APPENDIX D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving

More information

( x "1) 2 = 25, x 3 " 2x 2 + 5x "12 " 0, 2sin" =1.

( x 1) 2 = 25, x 3  2x 2 + 5x 12  0, 2sin =1. Unit Analytical Trigonometry Classwork A) Verifying Trig Identities: Definitions to know: Equality: a statement that is always true. example:, + 7, 6 6, ( + ) 6 +0. Equation: a statement that is conditionally

More information

Mod E - Trigonometry. Wednesday, July 27, M132-Blank NotesMOM Page 1

Mod E - Trigonometry. Wednesday, July 27, M132-Blank NotesMOM Page 1 M132-Blank NotesMOM Page 1 Mod E - Trigonometry Wednesday, July 27, 2016 12:13 PM E.0. Circles E.1. Angles E.2. Right Triangle Trigonometry E.3. Points on Circles Using Sine and Cosine E.4. The Other Trigonometric

More information

2. (8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given

2. (8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given Trigonometry Joysheet 1 MAT 145, Spring 2017 D. Ivanšić Name: Covers: 6.1, 6.2 Show all your work! 1. 8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given that sin

More information

13.2 Define General Angles and Use Radian Measure. standard position:

13.2 Define General Angles and Use Radian Measure. standard position: 3.2 Define General Angles and Use Radian Measure standard position: Examples: Draw an angle with the given measure in standard position..) 240 o 2.) 500 o 3.) -50 o Apr 7 9:55 AM coterminal angles: Examples:

More information

1 Trigonometry. Copyright Cengage Learning. All rights reserved.

1 Trigonometry. Copyright Cengage Learning. All rights reserved. 1 Trigonometry Copyright Cengage Learning. All rights reserved. 1.2 Trigonometric Functions: The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives Identify a unit circle and describe

More information

WARM UP. 1. Expand the expression (x 2 + 3) Factor the expression x 2 2x Find the roots of 4x 2 x + 1 by graphing.

WARM UP. 1. Expand the expression (x 2 + 3) Factor the expression x 2 2x Find the roots of 4x 2 x + 1 by graphing. WARM UP Monday, December 8, 2014 1. Expand the expression (x 2 + 3) 2 2. Factor the expression x 2 2x 8 3. Find the roots of 4x 2 x + 1 by graphing. 1 2 3 4 5 6 7 8 9 10 Objectives Distinguish between

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

Introduction to Trigonometry. Algebra 2

Introduction to Trigonometry. Algebra 2 Introduction to Trigonometry Algebra 2 Angle Rotation Angle formed by the starting and ending positions of a ray that rotates about its endpoint Use θ to represent the angle measure Greek letter theta

More information

You found trigonometric values using the unit circle. (Lesson 4-3)

You found trigonometric values using the unit circle. (Lesson 4-3) You found trigonometric values using the unit circle. (Lesson 4-3) LEQ: How do we identify and use basic trigonometric identities to find trigonometric values & use basic trigonometric identities to simplify

More information

Unit 5 Graphing Trigonmetric Functions

Unit 5 Graphing Trigonmetric Functions HARTFIELD PRECALCULUS UNIT 5 NOTES PAGE 1 Unit 5 Graphing Trigonmetric Functions This is a BASIC CALCULATORS ONLY unit. (2) Periodic Functions (3) Graph of the Sine Function (4) Graph of the Cosine Function

More information

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2

More information

Section 5.2 Graphs of the Sine and Cosine Functions

Section 5.2 Graphs of the Sine and Cosine Functions A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in

More information

Unit Circle: Sine and Cosine

Unit Circle: Sine and Cosine Unit Circle: Sine and Cosine Functions By: OpenStaxCollege The Singapore Flyer is the world s tallest Ferris wheel. (credit: Vibin JK /Flickr) Looking for a thrill? Then consider a ride on the Singapore

More information

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes 8 = 6 Trigonometry LESSON ONE - Degrees and Radians Example : Define each term or phrase and draw a sample angle. Angle in standard position. b) Positive and negative angles. Draw. c) Reference angle.

More information

Double-Angle, Half-Angle, and Reduction Formulas

Double-Angle, Half-Angle, and Reduction Formulas Double-Angle, Half-Angle, and Reduction Formulas By: OpenStaxCollege Bicycle ramps for advanced riders have a steeper incline than those designed for novices. Bicycle ramps made for competition (see [link])

More information

http://www.math.utah.edu/~palais/sine.html http://www.ies.co.jp/math/java/trig/index.html http://www.analyzemath.com/function/periodic.html http://math.usask.ca/maclean/sincosslider/sincosslider.html http://www.analyzemath.com/unitcircle/unitcircle.html

More information

Chapter 1. Trigonometry Week 6 pp

Chapter 1. Trigonometry Week 6 pp Fall, Triginometry 5-, Week -7 Chapter. Trigonometry Week pp.-8 What is the TRIGONOMETRY o TrigonometryAngle+ Three sides + triangle + circle. Trigonometry: Measurement of Triangles (derived form Greek

More information

Year 10 Term 1 Homework

Year 10 Term 1 Homework Yimin Math Centre Year 10 Term 1 Homework Student Name: Grade: Date: Score: Table of contents 6 Year 10 Term 1 Week 6 Homework 1 6.1 Triangle trigonometry................................... 1 6.1.1 The

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Draw the given angle in standard position. Draw an arrow representing the correct amount of rotation.

More information

Name: Period: Date: Math Lab: Explore Transformations of Trig Functions

Name: Period: Date: Math Lab: Explore Transformations of Trig Functions Name: Period: Date: Math Lab: Explore Transformations of Trig Functions EXPLORE VERTICAL DISPLACEMENT 1] Graph 2] Explain what happens to the parent graph when a constant is added to the sine function.

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

Pre-Calc Chapter 4 Sample Test. 1. Determine the quadrant in which the angle lies. (The angle measure is given in radians.) π

Pre-Calc Chapter 4 Sample Test. 1. Determine the quadrant in which the angle lies. (The angle measure is given in radians.) π Pre-Calc Chapter Sample Test 1. Determine the quadrant in which the angle lies. (The angle measure is given in radians.) π 8 I B) II C) III D) IV E) The angle lies on a coordinate axis.. Sketch the angle

More information

Unit 5 Investigating Trigonometry Graphs

Unit 5 Investigating Trigonometry Graphs Mathematics IV Frameworks Student Edition Unit 5 Investigating Trigonometry Graphs 1 st Edition Table of Contents INTRODUCTION:... 3 What s Your Temperature? Learning Task... Error! Bookmark not defined.

More information

Section 8.1 Radians and Arc Length

Section 8.1 Radians and Arc Length Section 8. Radians and Arc Length Definition. An angle of radian is defined to be the angle, in the counterclockwise direction, at the center of a unit circle which spans an arc of length. Conversion Factors:

More information

θ = = 45 What is the measure of this reference angle?

θ = = 45 What is the measure of this reference angle? OF GENERAL ANGLES Our method of using right triangles only works for acute angles. Now we will see how we can find the trig function values of any angle. To do this we'll place angles on a rectangular

More information

13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. SOLUTION: 2. If, find cos θ.

13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. SOLUTION: 2. If, find cos θ. Find the exact value of each expression if 0 < θ < 90 1. If cot θ = 2, find tan θ. 2. If, find cos θ. Since is in the first quadrant, is positive. Thus,. 3. If, find sin θ. Since is in the first quadrant,

More information

7.3 The Unit Circle Finding Trig Functions Using The Unit Circle Defining Sine and Cosine Functions from the Unit Circle

7.3 The Unit Circle Finding Trig Functions Using The Unit Circle Defining Sine and Cosine Functions from the Unit Circle 7.3 The Unit Circle Finding Trig Functions Using The Unit Circle For any angle t, we can label the intersection of the terminal side and the unit circle as by its coordinates,(x,y).the coordinates x and

More information

Name: A Trigonometric Review June 2012

Name: A Trigonometric Review June 2012 Name: A Trigonometric Review June 202 This homework will prepare you for in-class work tomorrow on describing oscillations. If you need help, there are several resources: tutoring on the third floor of

More information

In Exercises 1-12, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function.

In Exercises 1-12, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function. 0.5 Graphs of the Trigonometric Functions 809 0.5. Eercises In Eercises -, graph one ccle of the given function. State the period, amplitude, phase shift and vertical shift of the function.. = sin. = sin.

More information

Section 5.2 Graphs of the Sine and Cosine Functions

Section 5.2 Graphs of the Sine and Cosine Functions Section 5.2 Graphs of the Sine and Cosine Functions We know from previously studying the periodicity of the trigonometric functions that the sine and cosine functions repeat themselves after 2 radians.

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Math 1316 Ch.1-2 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Find the supplement of an angle whose

More information

Analytic Geometry/ Trigonometry

Analytic Geometry/ Trigonometry Analytic Geometry/ Trigonometry Course Numbers 1206330, 1211300 Lake County School Curriculum Map Released 2010-2011 Page 1 of 33 PREFACE Teams of Lake County teachers created the curriculum maps in order

More information

Algebra and Trig. I. In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position.

Algebra and Trig. I. In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position. Algebra and Trig. I 4.4 Trigonometric Functions of Any Angle In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position. IN this section we

More information

Solutions to Exercises, Section 5.6

Solutions to Exercises, Section 5.6 Instructor s Solutions Manual, Section 5.6 Exercise 1 Solutions to Exercises, Section 5.6 1. For θ = 7, evaluate each of the following: (a) cos 2 θ (b) cos(θ 2 ) [Exercises 1 and 2 emphasize that cos 2

More information

Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine

Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine 14A Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine Find these vocabulary words in Lesson 14-1 and the Multilingual Glossary. Vocabulary periodic function cycle period amplitude frequency

More information

Pythagorean Theorem: Trigonometry Packet #1 S O H C A H T O A. Examples Evaluate the six trig functions of the angle θ. 1.) 2.)

Pythagorean Theorem: Trigonometry Packet #1 S O H C A H T O A. Examples Evaluate the six trig functions of the angle θ. 1.) 2.) Trigonometry Packet #1 opposite side hypotenuse Name: Objectives: Students will be able to solve triangles using trig ratios and find trig ratios of a given angle. S O H C A H T O A adjacent side θ Right

More information

5.3-The Graphs of the Sine and Cosine Functions

5.3-The Graphs of the Sine and Cosine Functions 5.3-The Graphs of the Sine and Cosine Functions Objectives: 1. Graph the sine and cosine functions. 2. Determine the amplitude, period and phase shift of the sine and cosine functions. 3. Find equations

More information

Module 5 Trigonometric Identities I

Module 5 Trigonometric Identities I MAC 1114 Module 5 Trigonometric Identities I Learning Objectives Upon completing this module, you should be able to: 1. Recognize the fundamental identities: reciprocal identities, quotient identities,

More information

of the whole circumference.

of the whole circumference. TRIGONOMETRY WEEK 13 ARC LENGTH AND AREAS OF SECTORS If the complete circumference of a circle can be calculated using C = 2πr then the length of an arc, (a portion of the circumference) can be found by

More information

5-5 Multiple-Angle and Product-to-Sum Identities

5-5 Multiple-Angle and Product-to-Sum Identities Find the values of sin 2, cos 2, and tan 2 for the given value and interval. 1. cos =, (270, 360 ) Since on the interval (270, 360 ), one point on the terminal side of θ has x-coordinate 3 and a distance

More information

Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3

Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3 Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan Review Problems for Test #3 Exercise 1 The following is one cycle of a trigonometric function. Find an equation of this graph. Exercise

More information

Section 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

Section 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions Section 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions In this section, we will look at the graphs of the other four trigonometric functions. We will start by examining the tangent

More information

6.1 - Introduction to Periodic Functions

6.1 - Introduction to Periodic Functions 6.1 - Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: A function f is periodic if its values repeat at regular intervals. Graphically, this means that

More information

Chapter 4/5 Part 2- Trig Identities and Equations

Chapter 4/5 Part 2- Trig Identities and Equations Chapter 4/5 Part 2- Trig Identities and Equations Lesson Package MHF4U Chapter 4/5 Part 2 Outline Unit Goal: By the end of this unit, you will be able to solve trig equations and prove trig identities.

More information

MATH 1112 FINAL EXAM REVIEW e. None of these. d. 1 e. None of these. d. 1 e. None of these. e. None of these. e. None of these.

MATH 1112 FINAL EXAM REVIEW e. None of these. d. 1 e. None of these. d. 1 e. None of these. e. None of these. e. None of these. I. State the equation of the unit circle. MATH 111 FINAL EXAM REVIEW x y y = 1 x+ y = 1 x = 1 x + y = 1 II. III. If 1 tan x =, find sin x for x in Quadrant IV. 1 1 1 Give the exact value of each expression.

More information

Trigonometric Integrals Section 5.7

Trigonometric Integrals Section 5.7 A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Trigonometric Integrals Section 5.7 Dr. John Ehrke Department of Mathematics Spring 2013 Eliminating Powers From Trig Functions

More information

Multiple-Angle and Product-to-Sum Formulas

Multiple-Angle and Product-to-Sum Formulas Multiple-Angle and Product-to-Sum Formulas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: use multiple-angle formulas to rewrite

More information

Trigonometric Functions. Copyright 2017, 2013, 2009 Pearson Education, Inc.

Trigonometric Functions. Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 Trigonometric Functions Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean

More information