You found trigonometric values using the unit circle. (Lesson 4-3)
|
|
- Lester Kristopher Dickerson
- 5 years ago
- Views:
Transcription
1
2 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 and rewrite trigonometric expressions?
3 identity trigonometric identity cofunction odd-even identities
4
5 Use Reciprocal and Quotient Identities A. If, find sec θ. Reciprocal Identity Divide. Answer:
6 Use Reciprocal and Quotient Identities A. If, find sec θ. Reciprocal Identity Divide. Answer:
7 Use Reciprocal and Quotient Identities B. If and, find sin x. Reciprocal Identity Quotient Identity Substitute for cos x.
8 Use Reciprocal and Quotient Identities Divide. Multiply each side by. Simplify. Answer:
9 Use Reciprocal and Quotient Identities Divide. Multiply each side by. Simplify. Answer:
10 If, find sin. A. B. C. D.
11 If, find sin. A. B. C. D.
12
13 Use Pythagorean Identities If cot θ = 2 and cos θ < 0, find sin θ and cos θ. Use the Pythagorean Identity that involves cot θ. cot 2 θ + 1 = csc 2 θ (2) = csc 2 θ cot θ = 2 5 = csc 2 θ Simplify. Pythagorean Identity = csc θ Take the square root of each side. Reciprocal Identity Solve for sin θ.
14 Use Pythagorean Identities Since is positive and cos θ < 0, sin θ must be negative. So. You can then use this quotient identity again to find cos θ. Quotient Identity cot θ = 2 and Multiply each side by.
15 Use Pythagorean Identities So, Answer:
16 Use Pythagorean Identities So, Answer: Check sin 2 θ + cos 2 θ = 1 Pythagorean Identity Simplify.
17 Find the value of csc and cot if cos < 0. and A. B. C. D.
18 Find the value of csc and cot if cos < 0. and A. B. C. D.
19
20
21 If cos x = 0.75, find Use Cofunction and Odd-Even Identities Factor. Odd-Even Identity Cofunction Identity cos x = 0.75 Simplify.
22 So, = Use Cofunction and Odd-Even Identities Answer:
23 So, = Use Cofunction and Odd-Even Identities Answer: 0.75
24 If cos x = 0.73, find. A B C D. 0.73
25 If cos x = 0.73, find. A B C D. 0.73
26 Solve Algebraically Simplify by Rewriting Using Only Sine and Cosine Simplify. Pythagorean Identity Multiply. Simplify. So, = cos x.
27 Simplify by Rewriting Using Only Sine and Cosine Support Graphically The graphs of appear to be identical. and y = cos x Answer:
28 Simplify by Rewriting Using Only Sine and Cosine Support Graphically The graphs of appear to be identical. and y = cos x Answer: cos x
29 Simplify csc x cos x cot x. A. cot x B. tan x C. cos x D. sin x
30 Simplify csc x cos x cot x. A. cot x B. tan x C. cos x D. sin x
31 Simplify by Factoring Simplify cos x tan x sin x cos 2 x. Solve Algebraically cos x tan x sin x cos 2 x = sin 3 x Original expression Quotient Identity Multiply. Factor. So, cos x tan x sin x cos 2 x = sin 3 x. Pythagorean Identity Simplify.
32 Support Graphically Simplify by Factoring The graphs below appear to be identical. Answer:
33 Support Graphically Simplify by Factoring The graphs below appear to be identical. Answer: sin 3 x
34 Simplify cos 2 x sin x cos(90 x). A. sin 3 x B. sin 3 x C. cos 2 x 1 D. sin x cos x
35 Simplify cos 2 x sin x cos(90 x). A. sin 3 x B. sin 3 x C. cos 2 x 1 D. sin x cos x
36 Simplify by Combining Fractions Simplify. Common denominator Multiply. Add the numerators. Simplify. Pythagorean Identity
37 Simplify by Combining Fractions Reciprocal Identity Reciprocal and Quotient Identities Divide out common factor. 2csc 2 x. Reciprocal Identity Answer:
38 Simplify by Combining Fractions Reciprocal Identity Reciprocal and Quotient Identities Divide out common factor. 2csc 2 x. Reciprocal Identity Answer: 2 sec 2 x 2 sec 2 x
39 Simplify. A. cos x B cos x C. 2 sin x D. 2 csc x
40 Simplify. A. cos x B cos x C. 2 sin x D. 2 csc x
41 Rewrite to Eliminate Fractions Rewrite as an expression that does not involve a fraction. Pythagorean Identity Reciprocal Identity Reciprocal Identity Quotient Identity
42 So, = tan 2 x. Rewrite to Eliminate Fractions Answer:
43 So, = tan 2 x. Rewrite to Eliminate Fractions Answer: tan 2 x
44 Rewrite as an expression that does not involve a fraction. A. 2 tan 2 x B. 1+ sin x C. 1 cos x D. 2 sin x
45 Rewrite as an expression that does not involve a fraction. A. 2 tan 2 x B. 1+ sin x C. 1 cos x D. 2 sin x
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 informationHonors 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 informationMATH 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 informationTrigonometric 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 information1 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 informationChapter 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 informationMath 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 informationMATH 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 informationWhile you wait: For a-d: use a calculator to evaluate: Fill in the blank.
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 Trigonometric
More informationcos 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 informationPrinciples 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 informationTrig 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 informationSection 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 informationIn 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 information4-3 Trigonometric Functions on the Unit Circle
Find the exact values of the five remaining trigonometric functions of θ. 33. tan θ = 2, where sin θ > 0 and cos θ > 0 To find the other function values, you must find the coordinates of a point on the
More informationMATH 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 informationThe 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# 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 informationTrigonometric Equations
Chapter Three Trigonometric Equations Solving Simple Trigonometric Equations Algebraically Solving Complicated Trigonometric Equations Algebraically Graphs of Sine and Cosine Functions Solving Trigonometric
More information5-5 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 informationAlgebra2/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 informationTrigonometric 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 informationDouble-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 informationGraphs 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 informationTrigonometry. 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 informationMath 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 informationGeometry 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 information3.2 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 informationTrigonometric 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( 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 informationMath 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 informationExercise 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 informationTrigonometric 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 informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationBasic 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 informationMathematics 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 informationRight 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 informationChapter 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 informationTrigonometric 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 informationUnit 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 informationChapter 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 informationcos sin sin 2 60 = 1.
Name: Class: Date: Use the definitions to evaluate the six trigonometric functions of. In cases in which a radical occurs in a denominator, rationalize the denominator. Suppose that ABC is a right triangle
More informationReady 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 informationDouble-Angle and Half-Angle Identities
7-4 OBJECTIVE Use the doubleand half-angle identities for the sine, ine, and tangent functions. Double-Angle and Half-Angle Identities ARCHITECTURE Mike MacDonald is an architect who designs water fountains.
More informationMHF4U. 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 information5-5 Multiple-Angle and Product-to-Sum Identities
Find the values of sin 2, cos 2, tan 2 1 cos for the given value interval, (270, 360 ) Since on the interval (270, 360 ), one point on the terminal side of θ has x-coordinate 3 a distance of 5 units from
More informationChapter 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 informationFind the exact values of the indicated trigonometric functions. Write fractions in lowest terms. 1)
MAC 1114 Review for Exam 1 Name Find the exact values of the indicated trigonometric functions. Write fractions in lowest terms. 1) 1) 12 20 16 Find sin A and cos A. 2) 2) 9 15 6 Find tan A and cot A.
More information1 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 informationMathematics 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 informationSolutions 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 informationMath 36 "Fall 08" 5.2 "Sum and Di erence Identities" * Find exact values of functions of rational multiples of by using sum and di erence identities.
Math 36 "Fall 08" 5.2 "Sum and Di erence Identities" Skills Objectives: * Find exact values of functions of rational multiples of by using sum and di erence identities. * Develop new identities from the
More information13-1 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 informationMath 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 informationUnit 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 informationAlgebra 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 informationName: 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 informationof 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 information13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. ANSWER: 2. If, find cos θ.
Find the exact value of each expression if 0 < θ < 90 1. If cot θ = 2, find tan θ. 8. CCSS PERSEVERANCE When unpolarized light passes through polarized sunglass lenses, the intensity of the light is cut
More informationChapter 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 informationMath 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 information2. 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 informationName 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 informationTrigonometry 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 informationChapter 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 informationPrecalculus ~ Review Sheet
Period: Date: Precalculus ~ Review Sheet 4.4-4.5 Multiple Choice 1. The screen below shows the graph of a sound recorded on an oscilloscope. What is the period and the amplitude? (Each unit on the t-axis
More informationSHORT 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 informationSection 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 informationMath 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 informationAlgebra 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 informationTrig/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 informationMULTIPLE 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 informationArkansas 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 informationWARM 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 informationSolving Linear & Graphing Inequalities
Solving Linear & Graphing Inequalities Sep 7 11:06 PM 1 Open circle on the graph means that the inequality will be greater than or less than. > or < Closed circle on the graph means that the inequality
More informationF.TF.A.2: Reciprocal Trigonometric Relationships
Regents Exam Questions www.jmap.org Name: If sin x =, a 0, which statement must be true? a ) csc x = a csc x = a ) sec x = a sec x = a 5 The expression sec 2 x + csc 2 x is equivalent to ) sin x ) cos
More informationCopyright 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 informationSECTION 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= 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 informationMath 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 informationPREREQUISITE/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 information6.4 & 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 informationMAC 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 information4.3. 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 information13-3The The Unit Unit Circle
13-3The The Unit Unit Circle Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Find the measure of the reference angle for each given angle. 1. 120 60 2. 225 45 3. 150 30 4. 315 45 Find the exact value
More informationChapter 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 informationGRAPHING 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 informationDate 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 informationIn 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 informationPerfect Squares that are Written as Fractions or Decimals
Math 9: Unit 1 Lesson 2 Perfect Squares that are Written as Fractions or Decimals Part 1: Fractions There are two ways to determine the square root of a perfect square that is written as a fraction: 1.
More informationMath 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 informationSection 7.1 Graphs of Sine and Cosine
Section 7.1 Graphs of Sine and Cosine OBJECTIVE 1: Understanding the Graph of the Sine Function and its Properties In Chapter 7, we will use a rectangular coordinate system for a different purpose. We
More informationPreCalc: Chapter 6 Test Review
Name: Class: Date: ID: A PreCalc: Chapter 6 Test Review Short Answer 1. Draw the angle. 135 2. Draw the angle. 3. Convert the angle to a decimal in degrees. Round the answer to two decimal places. 8. If
More informationMATH 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MATH 1113 Exam III PRACTICE TEST FALL 2015 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact values of the indicated trigonometric
More information13-1 Practice. Trigonometric Identities. Find the exact value of each expression if 0 < θ < 90. 1, find sin θ. 1. If cos θ = 1, find cot θ.
1-1 Practice Trigonometric Identities Find the exact value of each expression if 0 < θ < 90. 1. If cos θ = 5 1, find sin θ.. If cot θ = 1, find sin θ.. If tan θ = 4, find sec θ. 4. If tan θ =, find cot
More informationVerifying Trigonometric Identities
25 PART I: Solutions to Odd-Numbered Exercises and Practice Tests a 27. sina =- ==> a = c. sin A = 20 sin 28 ~ 9.39 c B = 90 -A = 62 b cosa=- ==~ b=c.cosa~ 7.66 c 29. a = ~/c 2 - b 2 = -~/2.542-6.22 ~
More informationTrigonometry 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 information7.1 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 informationθ = = 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