While you wait: For a-d: use a calculator to evaluate: Fill in the blank.

Size: px
Start display at page:

Download "While you wait: For a-d: use a calculator to evaluate: Fill in the blank."

Transcription

1 While you wait: For a-d: use a calculator to evaluate: a) sin 50 o, cos 40 o b) sin 25 o, cos65 o c) cos o, sin 79 o d) sin 83 o, cos 7 o Fill in the blank. a) sin30 = cos b) cos57 = sin

2 Trigonometric Identities and Equations Section 8.4

3 Cofuntion Relationships

4 UC revisited y (,y) y Pythagorean Theorem: 2 + y 2 =

5 UC revisited y (cos,sin ) sin θ cos Pythagorean Theorem:cos 2 θ + sin 2 θ =

6 The trig relationships: cos 2 θ + sin 2 θ = cos 2 θ = sin 2 θ sin 2 θ = cos 2 θ

7 An identity is an equation that is true for all values of the variables. Difference between identity and equation: An identity is true for any value of the variable, but an equation is not. For eample the equation 3=2 is true only when =4, so it is an equation, but not an identity.

8 What are identities used for? They are used in simplifying or rearranging algebraic epressions. By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time. In this section we will study identities with trig functions.

9 The trigonometry identities There are dozens of identities in the field of trigonometry. Many websites list the trig identities. Many websites will also eplain why identities are true. i.e. prove the identities. For an eample of such a site: click here

10 Trigonometric Identities Quotient Identities tan sin cos cot cos sin Reciprocal Identities sin csc Pythagorean Identities cos sec tan cot sin 2 + cos 2 = tan 2 + = sec 2 cot 2 + = csc 2 sin 2 = - cos 2 cos 2 = - sin 2 tan 2 = sec 2 - cot 2 = csc

11 Where did our pythagorean identities come from?? Do you remember the Unit Circle? What is the equation for the unit circle? 2 + y 2 = What does =? What does y =? (in terms of trig functions) sin 2 θ + cos 2 θ = Pythagorean Identity!

12 Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos 2 θ sin 2 θ + cos 2 θ =. cos 2 θ cos 2 θ cos 2 θ tan 2 θ + = sec 2 θ Quotient Identity another Pythagorean Identity Reciprocal Identity

13 Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin 2 θ sin 2 θ + cos 2 θ =. sin 2 θ sin 2 θ sin 2 θ + cot 2 θ = csc 2 θ Quotient Identity a third Pythagorean Identity Reciprocal Identity

14 Simplifying Trigonometric Epressions Identities can be used to simplify trigonometric epressions. Simplify. a) cos sin tan cos sin sin cos cos sin2 cos cos 2 sin 2 cos cos b) cot 2 sin 2 cos 2 sin 2 cos 2 cos 2 sin 2 cos 2 sin 2 sec csc

15 Practice Problems for Day : refer to class handout.

16 While you wait Factor: a) 2 4 b) 2 36 c) 2 d) 2 Identify as True or False: A. cos θ = cos (θ) B. sin θ = sin(θ) C. tan θ = tan(θ)

17

18 Proving a Trigonometric Identity:. Transform the right side of the identity into the left side, 2. Vice versa (Left side to Right ) We do not want to use properties from algebra that involve both sides of the identity.

19 Guidelines for Proving Identities:. It is usually best to work on the more complicated side first. 2. Look for trigonometric substitutions involving the basic identities that may help simplify things. 3. Look for algebraic operations, such as adding fractions, the distributive property, or factoring, that may simplify the side you are working with or that will at least lead to an epression that will be easier to simplify.

20 4. If you cannot think of anything else to do, change everything to sines and cosines and see if that helps. 5. Always keep an eye on the side you are not working with to be sure you are working toward it. There is a certain sense of direction that accompanies a successful proof. 6. Practice, practice, practice!

21 Prove cota( + tan 2 A) tana = csc 2 A cota(sec 2 A) tana = csc 2 A Pythagorean Relationship

22 cosa sina ( cos 2 A ) sina cosa =csc 2 A Definition of trig Functions sinacosa sina cosa = csc 2 A Reduce

23 cosa sin 2 AcosA =csc2 A Reduce sin 2 A = csc2 A Def of trig function. csc 2 A = csc 2 A

24 Practice Problems Day 2 Sec 8- Written Eercises page 32 #3-9 odds; odds Eit Question: #3b the handout. A complete, step by step solution must be included.

25 Using the identities you now know, find the trig value..) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ. sec cos sin 2 cos 2 sin sin sin sin 4 5 csc sin

26 3.) sinθ = -/3, find tanθ tan 2 sec 2 tan 2 (3) 2 tan 2 8 tan ) secθ = -7/5, find sinθ tan 2 8

27 Simplifing Trigonometric Epressions c) ( + tan ) 2-2 sin sec ( tan) 2 2 sin cos 2 tan tan 2 2 sin cos tan 2 2tan 2 tan sec 2 d) csc tan cot sin sin cos cos sin sin sin 2 cos 2 sincos sin sin cos sin cos sin cos

28 Simplify each epression. sin cos sin sin sin cos cos sec cos sin sin cos cos cos sin sin cos 2 sin sin2 sin cos 2 sin 2 sin sin csc

29 Simplifying trig Identity Eample: simplify tancos sin tan cos cos tancos = sin

30 Simplifying trig Identity Eample2: simplify sec csc cos sec csc cos sin sin = cos = = tan sin

31 Simplifying trig Identity Eample2: simplify cos 2 - sin 2 cos cos 2 - sin 2 cos = sec

32 Eample Simplify: = cot (csc 2 - ) Factor out cot = cot (cot 2 ) Use pythagorean identi = cot 3 Simplify

33 Simplify: Eample = sin (sin ) + cos cos = sin 2 + (cos ) cos cos cos = sin 2 + cos 2 cos = cos = sec Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean iden Use reciprocal identity

34 Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

35 Practice

36 One way to use identities is to simplify epressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let s see an eample of this: tan sin cos substitute using each identity csc sin simplify sin cos sin cos Simplify: tan csc sec cos cos sec cos

37 Another way to use identities is to write one function in terms of another function. Let s see an eample of this: Write the following epression in terms of only one trig function: 2 cos sin 2 = sin sin 2 = sin sin 2 This epression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute. 2 2 sin cos cos sin 2 2

38 (E) Eamples Prove tan() cos() = sin() LS tan cos LS sin cos cos LS sin LS RS 38

39 (E) Eamples Prove tan 2 () = sin 2 () cos -2 () RS RS RS RS RS RS RS sin 2 sin cos 2 sin 2 cos 2 sin sin cos tan 2 LS cos cos

40 (E) Eamples Prove tan tan sin cos LS LS LS LS LS LS LS tan tan sin cos sin cos sin cos cos sin sin sin cos cos cos sin 2 2 sin cos cos sin cos sin RS 40

41 (E) Eamples Prove 2 sin cos cos LS LS LS LS 2 sin cos 2 cos cos ( cos )( cos ) ( cos ) cos LS RS 4

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

Principles of Mathematics 12: Explained!

Principles of Mathematics 12: Explained! Principles of Mathematics : Eplained! www.math.com PART I MULTIPLICATION & DIVISION IDENTITLES Algebraic proofs of trigonometric identities In this lesson, we will look at various strategies for proving

More information

Section 2.7 Proving Trigonometric Identities

Section 2.7 Proving Trigonometric Identities Sec. 2.7 Proving Trigonometric Identities 87 Section 2.7 Proving Trigonometric Identities In this section, we use the identities presented in Section 2.6 to do two different tasks: ) to simplify a trigonometric

More information

Math 10/11 Honors Section 3.6 Basic Trigonometric Identities

Math 10/11 Honors Section 3.6 Basic Trigonometric Identities Math 0/ Honors Section 3.6 Basic Trigonometric Identities 0-0 - SECTION 3.6 BASIC TRIGONOMETRIC IDENTITIES Copright all rights reserved to Homework Depot: www.bcmath.ca I) WHAT IS A TRIGONOMETRIC IDENTITY?

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

( 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

3.2 Proving Identities

3.2 Proving Identities 3.. Proving Identities www.ck.org 3. Proving Identities Learning Objectives Prove identities using several techniques. Working with Trigonometric Identities During the course, you will see complex trigonometric

More information

Trig/AP Calc A. Created by James Feng. Semester 1 Version fengerprints.weebly.com

Trig/AP Calc A. Created by James Feng. Semester 1 Version fengerprints.weebly.com Trig/AP Calc A Semester Version 0.. Created by James Feng fengerprints.weebly.com Trig/AP Calc A - Semester Handy-dandy Identities Know these like the back of your hand. "But I don't know the back of my

More information

HONORS PRECALCULUS Prove the following identities- ( ) x x x x x x. cos x cos x cos x cos x 1 sin x cos x 1 sin x

HONORS PRECALCULUS Prove the following identities- ( ) x x x x x x. cos x cos x cos x cos x 1 sin x cos x 1 sin x HONORS PRECALCULUS Prove the following identities-.) ( ) cos sin cos cos sin + sin sin + cos sin cos sin cos.).) ( ) ( sin) ( ) ( ) sin sin + + sin sin tan + sec + cos cos cos cos sin cos sin cos cos cos

More information

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

Trigonometric Identities. Copyright 2017, 2013, 2009 Pearson Education, Inc. 5 Trigonometric Identities Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 5.5 Double-Angle Double-Angle Identities An Application Product-to-Sum and Sum-to-Product Identities Copyright 2017, 2013,

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

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

Exercise 1. Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ.

Exercise 1. Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ. 1 Radian Measures Exercise 1 Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ. 1. Suppose I know the radian measure of the

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

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

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

MATH Week 10. Ferenc Balogh Winter. Concordia University

MATH Week 10. Ferenc Balogh Winter. Concordia University MATH 20 - Week 0 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics for Calculus, 5th Edition, Thomson All figures and videos

More information

PROVING IDENTITIES TRIGONOMETRY 4. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Proving identities 1/ 7 Adrian Jannetta

PROVING IDENTITIES TRIGONOMETRY 4. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Proving identities 1/ 7 Adrian Jannetta PROVING IDENTITIES TRIGONOMETRY 4 INU05/55 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Proving identities / 7 Adrian Jannetta Proving an identity Proving an identity is a process which starts with the

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

Mathematics UNIT FIVE Trigonometry II. Unit. Student Workbook. Lesson 1: Trigonometric Equations Approximate Completion Time: 4 Days

Mathematics UNIT FIVE Trigonometry II. Unit. Student Workbook. Lesson 1: Trigonometric Equations Approximate Completion Time: 4 Days Mathematics 0- Student Workbook Unit 5 Lesson : Trigonometric Equations Approximate Completion Time: 4 Days Lesson : Trigonometric Identities I Approximate Completion Time: 4 Days Lesson : 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

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

Trig Identities Packet

Trig Identities Packet Advanced Math Name Trig Identities Packet = = = = = = = = cos 2 θ + sin 2 θ = sin 2 θ = cos 2 θ cos 2 θ = sin 2 θ + tan 2 θ = sec 2 θ tan 2 θ = sec 2 θ tan 2 θ = sec 2 θ + cot 2 θ = csc 2 θ cot 2 θ = csc

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

JUST THE MATHS SLIDES NUMBER 3.5. TRIGONOMETRY 5 (Trigonometric identities & wave-forms) A.J.Hobson

JUST THE MATHS SLIDES NUMBER 3.5. TRIGONOMETRY 5 (Trigonometric identities & wave-forms) A.J.Hobson JUST THE MATHS SLIDES NUMBER 3.5 TRIGONOMETRY 5 (Trigonometric identities & wave-forms by A.J.Hobson 3.5.1 Trigonometric identities 3.5. Amplitude, wave-length, frequency and phase-angle UNIT 3.5 - TRIGONOMETRY

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

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

Math 3 Trigonometry Part 2 Waves & Laws

Math 3 Trigonometry Part 2 Waves & Laws Math 3 Trigonometry Part 2 Waves & Laws GRAPHING SINE AND COSINE Graph of sine function: Plotting every angle and its corresponding sine value, which is the y-coordinate, for different angles on the unit

More information

Name Date Class. Identify whether each function is periodic. If the function is periodic, give the period

Name Date Class. Identify whether each function is periodic. If the function is periodic, give the period Name Date Class 14-1 Practice A Graphs of Sine and Cosine Identify whether each function is periodic. If the function is periodic, give the period. 1.. Use f(x) = sinx or g(x) = cosx as a guide. Identify

More information

MHF4U. Advanced Functions Grade 12 University Mitchell District High School. Unit 4 Radian Measure 5 Video Lessons

MHF4U. Advanced Functions Grade 12 University Mitchell District High School. Unit 4 Radian Measure 5 Video Lessons MHF4U Advanced Functions Grade 12 University Mitchell District High School Unit 4 Radian Measure 5 Video Lessons Allow no more than 1 class days for this unit! This includes time for review and to write

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

Ferris Wheel Activity. Student Instructions:

Ferris Wheel Activity. Student Instructions: Ferris Wheel Activity Student Instructions: Today we are going to start our unit on trigonometry with a Ferris wheel activity. This Ferris wheel will be used throughout the unit. Be sure to hold on to

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

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

= tanθ 3) cos2 θ. = tan θ. = 3cosθ 6) sinθ + cosθcotθ = cscθ. = 3cosθ. = 3cosθ sinθ

= tanθ 3) cos2 θ. = tan θ. = 3cosθ 6) sinθ + cosθcotθ = cscθ. = 3cosθ. = 3cosθ sinθ PRE-CALCULUS/TRIGONOMETRY 3 Name 5.-5.5 REVIEW Date: Block Verify. ) cscθ secθ = cotθ 2) sec2 θ tanθ = tanθ 3) cos2 θ +sin θ = Use RIs sin θ = cotθ tan 2 θ tanθ = tan θ sin 2 θ +sin θ = Multiply by reciprocal

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

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

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

1 Trigonometric Identities

1 Trigonometric Identities MTH 120 Spring 2008 Essex County College Division of Mathematics Handout Version 6 1 January 29, 2008 1 Trigonometric Identities 1.1 Review of The Circular Functions At this point in your mathematical

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

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

Prerequisite Knowledge: Definitions of the trigonometric ratios for acute angles

Prerequisite Knowledge: Definitions of the trigonometric ratios for acute angles easures, hape & pace EXEMPLAR 28 Trigonometric Identities Objective: To explore some relations of trigonometric ratios Key Stage: 3 Learning Unit: Trigonometric Ratios and Using Trigonometry Materials

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

Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes. Name: Date: Block:

Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes. Name: Date: Block: Algebra 2/Trig AIIT.13 AIIT.15 AIIT.16 Reference Angles/Unit Circle Notes Mrs. Grieser Name: Date: Block: Trig Functions in a Circle Circle with radius r, centered around origin (x 2 + y 2 = r 2 ) Drop

More information

One of the classes that I have taught over the past few years is a technology course for

One of the classes that I have taught over the past few years is a technology course for Trigonometric Functions through Right Triangle Similarities Todd O. Moyer, Towson University Abstract: This article presents an introduction to the trigonometric functions tangent, cosecant, secant, and

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

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

Trigonometric Identities. Copyright 2017, 2013, 2009 Pearson Education, Inc. 5 Trigonometric Identities Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 5.3 Sum and Difference Identities Difference Identity for Cosine Sum Identity for Cosine Cofunction Identities Applications

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

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

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

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

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

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

Chapter 6.2: Trig Proofs

Chapter 6.2: Trig Proofs Chapter 6.2: Trig Proofs Proofs are fun, simply because they can be so challenging. No two are alike. While there are several common strategies for analytically proofing non-fundamental trig identities,

More information

#9: Fundamentals of Trigonometry, Part II

#9: Fundamentals of Trigonometry, Part II #9: Fundamentals of Trigonometry, Part II November 1, 2008 do not panic. In the last assignment, you learned general definitions of the sine and cosine functions. This week, we will explore some of the

More information

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

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

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

Grade 10 Trigonometry

Grade 10 Trigonometry ID : ww-10-trigonometry [1] Grade 10 Trigonometry For more such worksheets visit www.edugain.com Answer t he quest ions (1) If - 0, f ind value of sin 4 θ - cos 4 θ. (2) Simplif y 3(sin 4 θ cos 4 θ) -

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

cos 2 x + sin 2 x = 1 cos(u v) = cos u cos v + sin u sin v sin(u + v) = sin u cos v + cos u sin v

cos 2 x + sin 2 x = 1 cos(u v) = cos u cos v + sin u sin v sin(u + v) = sin u cos v + cos u sin v Concepts: Double Angle Identities, Power Reducing Identities, Half Angle Identities. Memorized: cos x + sin x 1 cos(u v) cos u cos v + sin v sin(u + v) cos v + cos u sin v Derive other identities you need

More information

4.3. Trigonometric Identities. Introduction. Prerequisites. Learning Outcomes

4.3. Trigonometric Identities. Introduction. Prerequisites. Learning Outcomes Trigonometric Identities 4.3 Introduction trigonometric identity is a relation between trigonometric expressions which is true for all values of the variables (usually angles. There are a very large number

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

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

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

5.4 Multiple-Angle Identities

5.4 Multiple-Angle Identities 4 CHAPTER 5 Analytic Trigonometry 5.4 Multiple-Angle Identities What you ll learn about Double-Angle Identities Power-Reducing Identities Half-Angle Identities Solving Trigonometric Equations... and why

More information

# 1,5,9,13,...37 (hw link has all odds)

# 1,5,9,13,...37 (hw link has all odds) February 8, 17 Goals: 1. Recognize trig functions and their integrals.. Learn trig identities useful for integration. 3. Understand which identities work and when. a) identities enable substitution by

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

Math 123 Discussion Session Week 4 Notes April 25, 2017

Math 123 Discussion Session Week 4 Notes April 25, 2017 Math 23 Discussion Session Week 4 Notes April 25, 207 Some trigonometry Today we want to approach trigonometry in the same way we ve approached geometry so far this quarter: we re relatively familiar with

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

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

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

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

Right Triangle Trigonometry (Section 4-3)

Right Triangle Trigonometry (Section 4-3) Right Triangle Trigonometry (Section 4-3) Essential Question: How does the Pythagorean Theorem apply to right triangle trigonometry? Students will write a summary describing the relationship between the

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

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

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

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

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

Exactly Evaluating Even More Trig Functions

Exactly Evaluating Even More Trig Functions Exactly Evaluating Even More Trig Functions Pre/Calculus 11, Veritas Prep. We know how to find trig functions of certain, special angles. Using our unit circle definition of the trig functions, as well

More information

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

Trigonometric Identities

Trigonometric Identities Trigonometric Identities Scott N. Walck September 1, 010 1 Prerequisites You should know the cosine and sine of 0, π/6, π/4, π/, and π/. Memorize these if you do not already know them. cos 0 = 1 sin 0

More information

Chapter 3, Part 1: Intro to the Trigonometric Functions

Chapter 3, Part 1: Intro to the Trigonometric Functions Haberman MTH 11 Section I: The Trigonometric Functions Chapter 3, Part 1: Intro to the Trigonometric Functions In Example 4 in Section I: Chapter, we observed that a circle rotating about its center (i.e.,

More information

Trigonometry. David R. Wilkins

Trigonometry. David R. Wilkins Trigonometry David R. Wilkins 1. Trigonometry 1. Trigonometry 1.1. Trigonometric Functions There are six standard trigonometric functions. They are the sine function (sin), the cosine function (cos), the

More information

Class 10 Trigonometry

Class 10 Trigonometry ID : in-10-trigonometry [1] Class 10 Trigonometry For more such worksheets visit www.edugain.com Answer t he quest ions (1) An equilateral triangle width side of length 18 3 cm is inscribed in a circle.

More information

Lesson 27: Sine and Cosine of Complementary and Special Angles

Lesson 27: Sine and Cosine of Complementary and Special Angles Lesson 7 M Classwork Example 1 If α and β are the measurements of complementary angles, then we are going to show that sin α = cos β. In right triangle ABC, the measurement of acute angle A is denoted

More information

Figure 5.1. sin θ = AB. cos θ = OB. tan θ = AB OB = sin θ. sec θ = 1. cotan θ = 1

Figure 5.1. sin θ = AB. cos θ = OB. tan θ = AB OB = sin θ. sec θ = 1. cotan θ = 1 5 Trigonometric functions Trigonometry is the mathematics of triangles. A right-angle triangle is one in which one angle is 90, as shown in Figure 5.1. The thir angle in the triangle is φ = (90 θ). Figure

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

GRAPHING TRIGONOMETRIC FUNCTIONS

GRAPHING TRIGONOMETRIC FUNCTIONS GRAPHING TRIGONOMETRIC FUNCTIONS Section.6B Precalculus PreAP/Dual, Revised 7 viet.dang@humbleisd.net 8//8 : AM.6B: Graphing Trig Functions REVIEW OF GRAPHS 8//8 : AM.6B: Graphing Trig Functions A. Equation:

More information

Date Lesson Text TOPIC Homework. Periodic Functions Hula Hoop Sheet WS 6.1. Graphing Sinusoidal Functions II WS 6.3

Date Lesson Text TOPIC Homework. Periodic Functions Hula Hoop Sheet WS 6.1. Graphing Sinusoidal Functions II WS 6.3 UNIT 6 SINUSOIDAL FUNCTIONS Date Lesson Text TOPIC Homework Ma 0 6. (6) 6. Periodic Functions Hula Hoop Sheet WS 6. Ma 4 6. (6) 6. Graphing Sinusoidal Functions Complete lesson shell WS 6. Ma 5 6. (6)

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

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

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 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

ASSIGNMENT ON TRIGONOMETRY LEVEL 1 (CBSE/NCERT/STATE BOARDS) Find the degree measure corresponding to the following radian measures :

ASSIGNMENT ON TRIGONOMETRY LEVEL 1 (CBSE/NCERT/STATE BOARDS) Find the degree measure corresponding to the following radian measures : ASSIGNMENT ON TRIGONOMETRY LEVEL 1 (CBSE/NCERT/STATE BOARDS) Find the degree measure corresponding to the following radian measures : (i) c 1 (ii) - c (iii) 6 c (iv) c 11 16 Find the length of an arc of

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

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

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

Unit 7 Trigonometric Identities and Equations 7.1 Exploring Equivalent Trig Functions

Unit 7 Trigonometric Identities and Equations 7.1 Exploring Equivalent Trig Functions Unit 7 Trigonometric Identities and Equations 7.1 Exploring Equivalent Trig Functions When we look at the graphs of sine, cosine, tangent and their reciprocals, it is clear that there will be points where

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

Pre-Calculus Unit 3 Standards-Based Worksheet

Pre-Calculus Unit 3 Standards-Based Worksheet Pre-Calculus Unit 3 Standards-Based Worksheet District of Columbia Public Schools Mathematics STANDARD PCT.P.9. Derive and apply basic trigonometric identities (e.g., sin 2 θ+cos 2 θ= 1,tan 2 θ + 1 = sec

More information