Section 63 DoubleAngle and HalfAngle Identities


 Baldric Merritt
 2 years ago
 Views:
Transcription
1 63 DoubleAngle and HalfAngle Identities 47 Section 63 DoubleAngle and HalfAngle Identities DoubleAngle Identities HalfAngle Identities This section develops another important set of identities called doubleangle and halfangle identities. We can derive these identities directly from the sum and difference identities given in Section 6. Even though the names use the word angle, the new identities hold for real numbers as well. DoubleAngle Identities Start with the sum identity for sine, sin ( y) sin cos y cos sin y and replace y with to obtain sin ( ) sin cos cos sin On simplification, this gives sin sin cos Doubleangle identity for sine () If we start with the sum identity for cosine, cos ( y) cos cos y sin sin y and replace y with, we obtain cos ( ) cos cos sin sin On simplification, this gives cos cos sin First doubleangle identity for cosine () Now, using the Pythagorean identity sin cos (3) in the form cos sin (4) and substituting it into equation (), we get cos sin sin On simplification, this gives cos sin Second doubleangle identity for cosine (5)
2 47 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Or, if we use equation (3) in the form sin cos and substitute it into equation (), we get cos cos ( cos ) On simplification, this gives cos cos Third doubleangle identity for cosine (6) Doubleangle identities can be established for the tangent function in the same way by starting with the sum formula for tangent (a good eercise for you). We list the doubleangle identities below for convenient reference. DOUBLEANGLE IDENTITIES sin sin cos cos cos sin sin cos tan tan tan cot cot cot tan The identities in the second row can be solved for sin and cos to obtain the identities sin cos cos cos These are useful in calculus to transform a power form to a nonpower form. Eplore/Discuss (A) Discuss how you would show that, in general, sin sin cos cos tan tan (B) Graph y sin and y sin in the same viewing window. Conclusion? Repeat the process for the other two statements in part A. Identity Verification Verify the identity cos tan. tan
3 63 DoubleAngle and HalfAngle Identities 473 Verification We start with the right side: tan tan sin cos sin cos cos sin cos sin cos sin cos Quotient identities Algebra Pythagorean identity Doubleangle identity Key Algebraic Steps in Eample a b b a b a b b a b b a b a Solution Verify the identity sin Finding Eact Values tan tan. Find the eact values, without using a calculator, of sin and cos if tan 3 4 and is a quadrant IV angle. First draw the reference triangle for and find any unknown sides: r 4 3 r (3) 4 5 sin 3 5 cos 4 5 Now use doubleangle identities for sine and cosine: sin sin cos ( 3 5 )( 4 5 ) 4 5 cos cos ( 4 5 ) 7 5 Find the eact values, without using a calculator, of cos and tan if sin and is a quadrant II angle. 4 5
4 474 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS HalfAngle Identities Halfangle identities are simply doubleangle identities stated in an alternate form. Let s start with the doubleangle identity for cosine in the form cos m sin m Now replace m with / and solve for sin (/) [if m is twice m, then m is half of m think about this]: cos sin sin cos sin cos Halfangle identity for sine (7) where the choice of the sign is determined by the quadrant in which / lies. To obtain a halfangle identity for cosine, start with the doubleangle identity for cosine in the form cos m cos m and let m / to obtain cos cos Halfangle identity for cosine (8) where the sign is determined by the quadrant in which / lies. To obtain a halfangle identity for tangent, use the quotient identity and the halfangle formulas for sine and cosine: Thus, tan sin cos cos cos cos cos tan cos cos Halfangle identity for tangent (9) where the sign is determined by the quadrant in which / lies. Simpler versions of equation (9) can be obtained as follows: tan cos cos cos cos cos cos (0)
5 cos ( cos ) sin ( cos ) sin ( cos ) sin cos 63 DoubleAngle and HalfAngle Identities 475 sin sin and ( cos ) cos, since cos is never negative. All absolute value signs can be dropped, since it can be shown that tan (/) and sin always have the same sign (a good eercise for you). Thus, tan sin cos Halfangle identity for tangent () By multiplying the numerator and the denominator in the radicand in equation (0) by cos and reasoning as before, we also can obtain tan cos sin Halfangle identity for tangent () We now list all the halfangle identities for convenient reference. HALFANGLE IDENTITIES sin cos cos cos tan cos cos sin cos cos sin where the sign is determined by the quadrant in which / lies. Eplore/Discuss (A) Discuss how you would show that, in general, sin cos tan sin cos tan (B) Graph y sin and y sin in the same viewing window. Conclusion? Repeat the process for the other two statements in part A.
6 476 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 3 Finding Eact Values Compute the eact value of sin 65 without a calculator using a halfangle identity. Solution sin 65 sin 330 cos 330 (3/) 3 Use halfangle identity for sine with a positive radical, since sin 65 is positive. 3 4 Solution Compute the eact value of tan 05 without a calculator using a halfangle identity. Finding Eact Values Find the eact values of cos (/) and cot (/) without using a calculator if sin 3 5, 3/. Draw a reference triangle in the third quadrant, and find cos. Then use appropriate halfangle identities. a a 5 (3) 4 cos (a, 3) If 3/, then 3 4 Divide each member of 3/ by. Thus, / is an angle in the second quadrant where cosine and cotangent are negative, and cos cos (4 5 ) 0 or 0 0 cot tan (/) sin cos 3 5 ( 4 5 ) 3
7 63 DoubleAngle and HalfAngle Identities Verification 5 Find the eact values of sin (/) and tan (/) without using a calculator if cot 4 3, /. Identity Verification Verify the identity: sin sin cos sin cos tan tan cos tan tan cos tan tan sin tan Verify the identity cos tan sin tan Halfangle identity for sine Square both sides. Algebra Algebra tan sin. tan Quotient identity Answers to Matched Problems tan tan sin cos cos sin cos sin cos sin cos sin cos sin. sin cos cos sin cos. cos cos 5, tan 4 3 sin (/) 30/0, tan (/) 3 cos tan 7 tan cos tan tan cos tan sin tan tan EXERCISE 63 A 5. sin cos, (Choose the correct sign.) In Problems 6, verify each identity for the values indicated.. cos cos sin, 30. sin sin cos, tan cot tan, 3 4. tan tan tan, 6 6. cos cos, (Choose the correct sign.) In Problems 7 0, find the eact value without a calculator using doubleangle and halfangle identities. 7. sin.5 8. tan cos tan 5
8 478 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS In Problems 4, graph y and y in the same viewing window for. Use TRACE to compare the two graphs.. y cos, y cos sin. y sin, y sin cos B Verify the identities in Problems (sin cos ) sin 6. sin (tan )( cos ) 7. sin ( cos ) 8. cos (cos ) 9. cos tan sin 0. sin t (sin t cos t). sin. cos 3. cot tan 4. cot tan 5. cot 6. sin cos 7. cos u tan u 8. tan u sec sec csc tan tan sec cot tan 3. cos tan (/) 3. cos tan (/) cot tan Compute the eact values of sin, cos, and tan using the information given in Problems and appropriate identities. Do not use a calculator. 33. sin 3 5, / In Problems 37 40, compute the eact values of sin (/), cos (/), and tan (/) using the information given and appropriate identities. Do not use a calculator y tan, y y tan, y sin cos tan tan tan 5, / 0 cot 5, / 0 sin 3, 3/ cos 4, 3/ 39. cot 3 4, / 40. tan 3 4, / cos cos cot tan cot cos sin cos u sin u tan u tan u 4 cos 5, / Suppose you are tutoring a student who is having difficulties in finding the eact values of sin and cos from the information given in Problems 4 and 4. Assuming you have worked through each problem and have identified the key steps in the solution process, proceed with your tutoring by guiding the student through the solution process using the following questions. Record the epected correct responses from the student. (A) The angle is in what quadrant and how do you know? (B) How can you find sin and cos? Find each. (C) What identities relate sin and cos with either sin or cos? (D) How would you use the identities in part C to find sin and cos eactly, including the correct sign? (E) What are the eact values for sin and cos? 4. Find the eact values of sin and cos, given tan, Find the eact values of sin and cos, given sec, Verify each of the following identities for the value of indicated in Problems Compute values to five significant digits using a calculator. (A) tan tan (B) cos cos tan (Choose the correct sign.) In Problems 47 50, graph y and y in the same viewing window for, and state the intervals for which the equation y y is an identity C y cos (/), y cos y cos (/), y cos y sin (/), y y sin (/), y Verify the identities in Problems cos 3 4 cos 3 3 cos 5. sin 3 3 sin 4 sin 3 cos cos 53. cos 4 8 cos 4 8 cos 54. sin 4 (cos )(4 sin 8 sin 3 )
9 63 DoubleAngle and HalfAngle Identities 479 In Problems 55 60, find the eact value of each without using a calculator. 3 5 )] 55. cos [ cos ( 3 5 )] 56. sin [ cos ( 57. tan [ cos ( 4 5 )] 58. tan [ tan ( 59. cos [ cos ( 3 5 )] 60. sin [ tan ( 4 3 )] 3 4 )] (B) Using the resulting equation in part A, determine the angle that will produce the maimum distance d for a given initial speed v 0. This result is an important consideration for shotputters, javelin throwers, and discus throwers. In Problems 6 66, graph f() in a graphing utility, find a simpler function g() that has the same graph as f(), and verify the identity f() g(). [Assume g() k A T(B) where T() is one of the si trigonometric functions.] 6. f() csc cot 6. f() csc cot cos cos 63. f() 64. f() sin cos 65. f() 66. f() cot cot sin cos APPLICATIONS 70. Geometry. In part (a) of the figure, M and N are the midpoints of the sides of a square. Find the eact value of cos. [Hint: The solution uses the Pythagorean theorem, the definition of sine and cosine, a halfangle identity, and some auiliary lines as drawn in part (b) of the figure.] M M 67. Indirect Measurement. Find the eact value of in the figure; then find and to three decimal places. [Hint: Use cos cos.] s N s / / N 68. Indirect Measurement. Find the eact value of in the figure; then find and to three decimal places. [Hint: Use tan ( tan )/( tan ).] 69. Sports Physics. The theoretical distance d that a shotputter, discus thrower, or javelin thrower can achieve on a given throw is found in physics to be given approimately by d 4 feet v 0 sin cos 3 feet per second per second 8 m where v 0 is the initial speed of the object thrown (in feet per second) and is the angle above the horizontal at which the object leaves the hand (see the figure). (A) Write the formula in terms of sin by using a suitable identity. feet 7 m s (a) s (b) 7. Area. An nsided regular polygon is inscribed in a circle of radius R. (A) Show that the area of the nsided polygon is given by A n nr sin n [Hint: (Area of a triangle) ( )(base)(altitude). Also, a doubleangle identity is useful.] (B) For a circle of radius, complete Table, to five decimal places, using the formula in part A: T A B L E n 0 00,000 0,000 A n (C) What number does A n seem to approach as n increases without bound? (What is the area of a circle of radius?) (D) Will A n eactly equal the area of the circumscribed circle for some sufficiently large n? How close can A n be made to get to the area of the circumscribed circle? [In calculus, the area of the circumscribed circle is called the limit of A n as n increases without bound. In symbols, for a circle of radius, we would write lim A n n. The limit concept is the cornerstone on which calculus is constructed.]
Trigonometric Identities and Conditional Equations C
Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving realworld problems and in the development of mathematics. Whatever their use, it is often of value
More informationAnalytic Trigonometry
Name Chapter 5 Analytic Trigonometry Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate trigonometric functions and
More informationSection 9.4 Trigonometric Functions of any Angle
Section 9. Trigonometric Functions of any Angle So far we have only really looked at trigonometric functions of acute (less than 90º) angles. We would like to be able to find the trigonometric functions
More informationSection 8.1: The Inverse Sine, Cosine, and Tangent Functions
Section 8.1: The Inverse Sine, Cosine, and Tangent Functions The function y = sin x doesn t pass the horizontal line test, so it doesn t have an inverse for every real number. But if we restrict the function
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationTrigonometric Identities and Equations
LIALMC07_0768.QXP /6/0 0:7 AM Page 605 7 Trigonometric Identities and Equations In 8 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now
More informationChapter 5: Trigonometric Functions of Real Numbers
Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the
More informationInverse Circular Function and Trigonometric Equation
Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse
More informationM3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.
M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More information1. Introduction identity algbriac factoring identities
1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as
More informationSection 59 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 59 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationMA Lesson 19 Summer 2016 Angles and Trigonometric Functions
DEFINITIONS: An angle is defined as the set of points determined by two rays, or halflines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common
More informationTrigonometry. Week 1 Right Triangle Trigonometry
Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge
More information5.2. Trigonometric Functions Of Real Numbers. Copyright Cengage Learning. All rights reserved.
5.2 Trigonometric Functions Of Real Numbers Copyright Cengage Learning. All rights reserved. Objectives The Trigonometric Functions Values of the Trigonometric Functions Fundamental Identities 2 Trigonometric
More informationSUM AND DIFFERENCE FORMULAS
SUM AND DIFFERENCE FORMULAS Introduction We have several identities that we are concentrating on in this section: o Difference Identities for Cosine o Sum Identities for Cosine o Cofunction Identities
More informationPreCalculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.
PreCalculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationTrigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:
First, a couple of things to help out: Page 1 of 24 Use periodic properties of the trigonometric functions to find the exact value of the expression. 1. cos 2. sin cos sin 2cos 4sin 3. cot cot 2 cot Sin
More informationInverse Trigonometric Functions
SECTION 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the exact value of an inverse trigonometric function. Use a calculator to approximate
More information4.1 Radian and Degree Measure
Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position
More informationTrigonometry Lesson Objectives
Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the
More informationMPE Review Section II: Trigonometry
MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. ) 56
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More information39 Verifying Trigonometric Identities
39 Verifying Trigonometric Identities In this section, you will learn how to use trigonometric identities to simplify trigonometric expressions. Equations such as (x 2)(x + 2) x 2 4 or x2 x x + are referred
More informationTrigonometric Identities
Trigonometric Identities Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout dpresents some of the most useful trigonometric identities. It also explains how to derive new
More information9.1 Trigonometric Identities
9.1 Trigonometric Identities r y x θ x y θ r sin (θ) = y and sin (θ) = y r r so, sin (θ) =  sin (θ) and cos (θ) = x and cos (θ) = x r r so, cos (θ) = cos (θ) And, Tan (θ) = sin (θ) =  sin (θ)
More informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are welldefined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
More informationLesson Two. Pre Calculus Math 40S: Explained! 152
Lesson Two Pre Calculus Math 40S: Eplained! www.math40s.com 5 PART I MULTIPLICATION & DIVISION IDENTITLES Algebraic proofs of trigonometric identities In this lesson, we will look at various strategies
More information11 Trigonometric Functions of Acute Angles
Arkansas Tech University MATH 10: Trigonometry Dr. Marcel B. Finan 11 Trigonometric Functions of Acute Angles In this section you will learn (1) how to find the trigonometric functions using right triangles,
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationWe d like to explore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( x)
Inverse Trigonometric Functions: We d like to eplore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( ) sin. Recall the graph: MATH 30 Lecture 0 of 0 Well, we can
More informationpp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 11: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 12: Algebraic Expressions
More informationFURTHER TRIGONOMETRIC IDENTITIES AND EQUATIONS
Mathematics Revision Guides Further Trigonometric Identities and Equations Page of 7 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C4 Edexcel: C3 OCR: C3 OCR MEI: C4 FURTHER TRIGONOMETRIC
More informationy = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions
MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y
More information18 Verifying Trigonometric Identities
Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 18 Verifying Trigonometric Identities In this section, you will learn how to use trigonometric identities to simplify trigonometric
More informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationUnit 8 Inverse Trig & Polar Form of Complex Nums.
HARTFIELD PRECALCULUS UNIT 8 NOTES PAGE 1 Unit 8 Inverse Trig & Polar Form of Complex Nums. This is a SCIENTIFIC OR GRAPHING CALCULATORS ALLOWED unit. () Inverse Functions (3) Invertibility of Trigonometric
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will also
More informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
More informationThe Natural Logarithmic Function: Integration. Log Rule for Integration
6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. EXPLORATION Integrating Rational Functions Earl in Chapter, ou learned rules that allowed ou to integrate
More informationAlgebra 2/ Trigonometry Extended Scope and Sequence (revised )
Algebra 2/ Trigonometry Extended Scope and Sequence (revised 2012 2013) Unit 1: Operations with Radicals and Complex Numbers 9 days 1. Operations with radicals (p.88, 94, 98, 101) a. Simplifying radicals
More informationTRIGONOMETRIC EQUATIONS
CHAPTER 3 CHAPTER TABLE OF CONTENTS 3 FirstDegree Trigonometric Equations 3 Using Factoring to Solve Trigonometric Equations 33 Using the Quadratic Formula to Solve Trigonometric Equations 34 Using
More information3. EVALUATION OF TRIGONOMETRIC FUNCTIONS
. EVALUATIN F TIGNMETIC FUNCTINS In this section, we obtain values of the trigonometric functions for quadrantal angles, we introduce the idea of reference angles, and we discuss the use of a calculator
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More informationCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS ShiahSen Wang The graphs are prepared by ChienLun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationVerifying Trigonometric Identities. Introduction. is true for all real numbers x. So, it is an identity. Verifying Trigonometric Identities
333202_0502.qxd 382 2/5/05 Chapter 5 5.2 9:0 AM Page 382 Analytic Trigonometry Verifying Trigonometric Identities What you should learn Verify trigonometric identities. Why you should learn it You can
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More informationYou can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure
Solving a Right Triangle A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve
More informationLesson 6.1 Exercises, pages
Lesson 6. Eercises, pages 7 80 A. Use technolog to determine the value of each trigonometric ratio to the nearest thousandth. a) sin b) cos ( 6 ) c) cot 7 d) csc 8 0.89 0. tan 7 sin 8 0..0. Sketch each
More informationSECTION 14 Absolute Value in Equations and Inequalities
14 Absolute Value in Equations and Inequalities 37 SECTION 14 Absolute Value in Equations and Inequalities Absolute Value and Distance Absolute Value in Equations and Inequalities Absolute Value and
More informationTrigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011
Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles
More information6.3 Polar Coordinates
6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard
More informationAP Calculus AB First Semester Final Exam Practice Test Content covers chapters 13 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1 Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
More informationApplications of Trigonometry
chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced
More informationRight Triangle Trigonometry
Right Triangle Trigonometry MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of acute angles, use
More informationSimilar Right Triangles
9.1 Similar Right Triangles Goals p Solve problems involving similar right triangles formed b the altitude drawn to the hpotenuse of a right triangle. p Use a geometric mean to solve problems. THEOREM
More informationMORE TRIG IDENTITIES MEMORIZE!
MORE TRIG IDENTITIES MEMORIZE! SUM IDENTITIES Memorize: sin( u + v)= sinu cosv + cosu sinv Think: Sum of the mixedup products (Multiplication and addition are commutative, but start with the sinu cosv
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
More informationInverse Trigonometric Functions
Inverse Trigonometric Functions I. Four Facts About Functions and Their Inverse Functions:. A function must be onetoone (an horizontal line intersects it at most once) in order to have an inverse function..
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More information13 Absolute Value in Equations and Inequalities
SECTION 1 3 Absolute Value in Equations and Inequalities 103 13 Absolute Value in Equations and Inequalities Z Relating Absolute Value and Distance Z Solving Absolute Value Equations and Inequalities
More informationEnrichment The Physics of Soccer Recall from Lesson 71 that the formula for the maximum height h h v 0 2 sin 2
71 The Physics of Soccer Recall from Lesson 71 that the formula for the maximum height h h v 0 2 sin 2 of a projectile is 2g, where is the measure of the angle of elevation in degrees, v 0 is the initial
More informationSection 7.5 Unit Circle Approach; Properties of the Trigonometric Functions. cosθ a. = cosθ a.
Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions A unit circle is a circle with radius = 1 whose center is at the origin. Since we know that the formula for the circumference
More informationTRIGONOMETRIC IDENTITIES AND EQUATIONS
Chapter 7 Unit Trigonometry (Chapters 5 8) TRIGONOMETRIC IDENTITIES ND EQUTIONS CHPTER OBJECTIVES Use reciprocal, quotient, Pythagorean, symmetry, and oppositeangle identities. (Lesson 7) Verify trigonometric
More information1. The Six Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1.2 The Rectangular Coordinate System 1.3 Definition I: Trigonometric
1. The Si Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1. The Rectangular Coordinate Sstem 1.3 Definition I: Trigonometric Functions 1.4 Introduction to Identities 1.5 More on Identities
More informationAbout Trigonometry. Triangles
About Trigonometry TABLE OF CONTENTS About Trigonometry... 1 What is TRIGONOMETRY?... 1 Triangles... 1 Background... 1 Trigonometry with Triangles... 1 Circles... 2 Trigonometry with Circles... 2 Rules/Conversion...
More informationList of trigonometric identities
List of trigonometric identities From Wikipedia, the free encyclopedia In mathematics, trigonometric identities are equalities involving trigonometric functions that are true for all values of the occurring
More information6.6 The Inverse Trigonometric Functions. Outline
6.6 The Inverse Trigonometric Functions Tom Lewis Fall Semester 2015 Outline The inverse sine function The inverse cosine function The inverse tangent function The other inverse trig functions Miscellaneous
More informationIntegration Involving Trigonometric Functions and Trigonometric Substitution
Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving
More information5.2 Unit Circle: Sine and Cosine Functions
Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and
More informationy = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)
5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If
More informationClass Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson
Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationMATH 150 TOPIC 9 RIGHT TRIANGLE TRIGONOMETRY. 9a. Right Triangle Definitions of the Trigonometric Functions
Math 50 T9aRight Triangle Trigonometry Review Page MTH 50 TOPIC 9 RIGHT TRINGLE TRIGONOMETRY 9a. Right Triangle Definitions of the Trigonometric Functions 9a. Practice Problems 9b. 5 5 90 and 0 60 90
More informationALGEBRA 2/ TRIGONOMETRY
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA 2/ TRIGONOMETRY Friday, June 14, 2013 1:15 4:15 p.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 28.......................
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationInverse Trigonometric Functions. Inverse Sine Function 4.71 FIGURE. Definition of Inverse Sine Function. The inverse sine function is defined by
0_007.qd /7/05 :0 AM Page Section.7.7 Inverse Trigonometric Functions Inverse Trigonometric Functions What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse
More informationSection 105 Parametric Equations
88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative
More informationTrigonometric Functions
Trigonometric Functions MATH 10, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: identify a unit circle and describe its relationship to real
More informationMath 115 Spring 2014 Written Homework 10SOLUTIONS Due Friday, April 25
Math 115 Spring 014 Written Homework 10SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain
More informationOverview. Essential Questions. Precalculus, Quarter 2, Unit 2.5 Proving Trigonometric Identities. Number of instruction days: 5 7 (1 day = 53 minutes)
Precalculus, Quarter, Unit.5 Proving Trigonometric Identities Overview Number of instruction days: 5 7 (1 day = 53 minutes) Content to Be Learned Verify proofs of Pythagorean identities. Apply Pythagorean,
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More information3 Unit Circle Trigonometry
0606_CH0_78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, June 14, 013 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationSection 64 Product Sum and Sum Product Identities
480 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Section 64 Product Sum and Sum Product Identities Product Sum Identities Sum Product Identities Our work with identities is concluded by developing
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationFinal Exam PracticeProblems
Name: Class: Date: ID: A Final Exam PracticeProblems Problem 1. Consider the function f(x) = 2( x 1) 2 3. a) Determine the equation of each function. i) f(x) ii) f( x) iii) f( x) b) Graph all four functions
More informationMidChapter Quiz: Lessons 41 through 44
Find the exact values of the six trigonometric functions of θ. Find the value of x. Round to the nearest tenth if necessary. 1. The length of the side opposite is 24, the length of the side adjacent to
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationSection 33 Approximating Real Zeros of Polynomials
 Approimating Real Zeros of Polynomials 9 Section  Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More information