Upload File

introduction

39
Introduction In the real world, if you needed to verify the size of a television, you could get some measuring tools and hold them up to the television to determine that the TV was advertised at the correct size. Imagine, however, that you are a fact checker for The Guinness Book of World Records. It is your job to verify that the tallest building in the world is in fact Burj Khalifa, located in Dubai. Could you use measuring tools to determine the size of a building so large? It would be extremely difficult and impractical to do so. 1 2.2.1: Calculating Sine, Cosine, and Tangent

Upload: phiala

Post on 26-Feb-2016

26 views

Category:

Documents


0 download

Report

Tags:

DESCRIPTION

Introduction - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: Introduction

IntroductionIn the real world, if you needed to verify the size of a television, you could get some measuring tools and hold them up to the television to determine that the TV was advertised at the correct size. Imagine, however, that you are a fact checker for The Guinness Book of World Records. It is your job to verify that the tallest building in the world is in fact Burj Khalifa, located in Dubai. Could you use measuring tools to determine the size of a building so large? It would be extremely difficult and impractical to do so.

1

2.2.1: Calculating Sine, Cosine, and Tangent

Page 2: Introduction

Introduction, continuedYou can use measuring tools for direct measurements of distance, but you can use trigonometry to find indirect measurements. First, though, you must be able to calculate the three basic trigonometric functions that will be applied to finding those larger distances. Specifically, we are going study and practice calculating the sine, cosine, and tangent functions of right triangles as preparation for measuring indirect distances.

2

2.2.1: Calculating Sine, Cosine, and Tangent

Page 3: Introduction

Key Concepts• The three basic trigonometric ratios are ratios of the

side lengths of a right triangle with respect to one of its acute angles.

• As you learned previously:

• Given the angle

• Given the angle

• Given the angle

3

2.2.1: Calculating Sine, Cosine, and Tangent

Page 4: Introduction

Key Concepts, continued• Notice that the trigonometric ratios contain three

unknowns: the angle measure and two side lengths. • Given an acute angle of a right triangle and the

measure of one of its side lengths, use sine, cosine, or tangent to find another side.

• If you know the value of a trigonometric function, you can use the inverse trigonometric function to find the measure of the angle.

4

2.2.1: Calculating Sine, Cosine, and Tangent

Page 5: Introduction

Key Concepts, continued• The inverse trigonometric functions are arcsine,

arccosine, and arctangent.• Arcsine is the inverse of the sine function, written

• Arccosine is the inverse of the cosine function, written

• Arctangent is the inverse of the tangent function, written

5

2.2.1: Calculating Sine, Cosine, and Tangent

Page 6: Introduction

Key Concepts, continued• There are two different types of notation for these

functions:

• if

• if

• if

6

2.2.1: Calculating Sine, Cosine, and Tangent

Page 7: Introduction

Key Concepts, continued• Note that “–1” is not an exponent; it is simply the

notation for an inverse trigonometric function. Because this notation can lead to confusion, the “arc-” notation is frequently used instead.

7

2.2.1: Calculating Sine, Cosine, and Tangent

Page 8: Introduction

Key Concepts, continued• To calculate inverse trigonometric functions on your

graphing calculator:

On a TI-83/84:Step 1: Press [2ND][SIN]. Step 2: Type in the ratio. Step 3: Press [ENTER].

8

2.2.1: Calculating Sine, Cosine, and Tangent

Page 9: Introduction

Key Concepts, continuedOn a TI-Nspire:

Step 1: In the calculate window from the home screen, press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "sin–1.”

Step 2: Type in the ratio. Step 3: Press [enter].

• Use the inverses of the trigonometric functions to find the acute angle measures given two sides of a right triangle.

9

2.2.1: Calculating Sine, Cosine, and Tangent

Page 10: Introduction

Common Errors/Misconceptions• not using the given angle as the guide in determining

which side is opposite or adjacent • forgetting to ensure the calculator is in degree mode

before completing the operations • using the trigonometric function instead of the inverse

trigonometric function when calculating the acute angle measure

• confusing inverse notation with reciprocal notation

10

2.2.1: Calculating Sine, Cosine, and Tangent

Page 11: Introduction

Guided PracticeExample 2A trucker drives 1,027 feet up a hill that has a constant slope. When the trucker reaches the top of the hill, he has traveled a horizontal distance of 990 feet. At what angle did the trucker drive to reach the top? Round your answer to the nearest degree.

11

2.2.1: Calculating Sine, Cosine, and Tangent

Page 12: Introduction

Guided Practice: Example 2, continued1. Determine which trigonometric function

to use by identifying the given information. Given an angle of w°, the horizontal distance, 990 feet, is adjacent to the angle.

The distance traveled by the trucker is the hypotenuse since it is opposite the right angle of the triangle.

12

2.2.1: Calculating Sine, Cosine, and Tangent

Page 13: Introduction

Guided Practice: Example 2, continued

Cosine is the trigonometric function that uses

adjacent and hypotenuse,

so we will use it to calculate the angle the truck drove to reach the bottom of the road. 13

2.2.1: Calculating Sine, Cosine, and Tangent

Page 14: Introduction

Guided Practice: Example 2, continuedSet up an equation using the cosine function and

the given measurements.

Therefore,

Solve for w.

14

2.2.1: Calculating Sine, Cosine, and Tangent

Page 15: Introduction

Guided Practice: Example 2, continuedSolve for w by using the inverse cosine since we

are finding an angle instead of a side length.

15

2.2.1: Calculating Sine, Cosine, and Tangent

Page 16: Introduction

Guided Practice: Example 2, continued2. Use a calculator to calculate the value

of w. On a TI-83/84: First, make sure your calculator is in DEGREE mode:

Step 1: Press [MODE]. Step 2: Arrow down twice to RADIAN. Step 3: Arrow right to DEGREE. Step 4: Press [ENTER]. The word “DEGREE”

should be highlighted inside a black rectangle.

16

2.2.1: Calculating Sine, Cosine, and Tangent

Page 17: Introduction

Guided Practice: Example 2, continuedStep 5: Press [2ND]. Step 6: Press [MODE] to QUIT. Note: You will not have to change to Degree mode again unless you have changed your calculator to Radian mode.

Next, perform the calculation.

Step 1: Press [2ND][COS][990][÷][1027][)].Step 2: Press [ENTER].

w = 15.426, or 15°.

17

2.2.1: Calculating Sine, Cosine, and Tangent

Page 18: Introduction

Guided Practice: Example 2, continuedOn a TI-Nspire: First, make sure the calculator is in Degree mode:

Step 1: Choose 5: Settings & Status, then 2: Settings, and 2: Graphs and Geometry.

Step 2: Move to the Geometry Angle field and choose “Degree”.

Step 3: Press [tab] to “ok” and press [enter].

18

2.2.1: Calculating Sine, Cosine, and Tangent

Page 19: Introduction

Guided Practice: Example 2, continuedThen, if necessary, set the Scratchpad in Degree mode:

Step 1: In the calculate window from the home screen, press [doc].

Step 2: Select 7: Settings and Status, then 2: Settings, and 1: General.

Step 3: Move to the Angle field and choose “Degree”.

Step 4: Press [tab] to “Make Default” and press [enter] twice to apply this as the new default setting.

19

2.2.1: Calculating Sine, Cosine, and Tangent

Page 20: Introduction

Guided Practice: Example 2, continuedNext, perform the calculation.

Step 1: In the calculate window from the home screen, press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "cos–1." Enter 990, then press [÷] and enter 1027.

Step 2: Press [enter].

w = 15.426, or 15°.The trucker drove at an angle of 15° to the top of the hill.

20

2.2.1: Calculating Sine, Cosine, and Tangent

✔

Page 21: Introduction

Guided Practice: Example 2, continued

21

2.2.1: Calculating Sine, Cosine, and Tangent

http://www.walch.com/ei/00185
Page 22: Introduction

22

2.2.1: Calculating Sine, Cosine, and Tangent

Guided PracticeExample 4Solve the right triangle. Round sides to the nearest thousandth.

Page 23: Introduction

Guided Practice: Example 4, continued1. Find the measures of and .

Solving the right triangle means to find all the missing angle measures and all the missing side lengths. The given angle is 64.5° and 17 is the length of the adjacent side. With this information, we could either use cosine or tangent since both functions’ ratios include the adjacent side of a right triangle. Start by using the tangent function to find .

23

2.2.1: Calculating Sine, Cosine, and Tangent

Page 24: Introduction

Guided Practice: Example 4, continued

Recall that

24

2.2.1: Calculating Sine, Cosine, and Tangent

Page 25: Introduction

Guided Practice: Example 4, continuedOn a TI-83/84:

Step 1: Press [17][TAN][64.5][)]. Step 2: Press [ENTER]. x = 35.641

25

2.2.1: Calculating Sine, Cosine, and Tangent

Page 26: Introduction

Guided Practice: Example 4, continuedOn a TI-Nspire:

Step 1: In the calculate window from the home screen, enter 17, then press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "tan," then enter 64.5.

Step 2: Press [enter]. x = 35.641

The measure of AC = 35.641.

26

2.2.1: Calculating Sine, Cosine, and Tangent

Page 27: Introduction

Guided Practice: Example 4, continuedTo find the measure of , either acute angle may be used as an angle of interest. Since two side lengths are known, the Pythagorean Theorem may be used as well.

Note: It is more precise to use the given values instead of approximated values.

27

2.2.1: Calculating Sine, Cosine, and Tangent

Page 28: Introduction

Guided Practice: Example 4, continued2. Use the cosine function based on the

given information.

Recall that

28

2.2.1: Calculating Sine, Cosine, and Tangent

Page 29: Introduction

Guided Practice: Example 4, continued

29

2.2.1: Calculating Sine, Cosine, and Tangent

Page 30: Introduction

Guided Practice: Example 4, continuedOn a TI-83/84: Check to make sure your calculator is in Degree mode first. Refer to the directions in the previous example.

Step 1: Press [17][÷][COS][64.5][)]. Step 2: Press [ENTER]. y = 39.488

30

2.2.1: Calculating Sine, Cosine, and Tangent

Page 31: Introduction

Guided Practice: Example 4, continuedOn a TI-Nspire: Check to make sure your calculator is in Degree mode first. Refer to the directions in the previous example.

Step 1: In the calculate window from the home screen, enter 17, then press [÷][trig]. Use the keypad to select "cos," and then enter 64.5.

Step 2: Press [enter]. y = 39.488

The measure of AB = 39.488.

31

2.2.1: Calculating Sine, Cosine, and Tangent

Page 32: Introduction

Guided Practice: Example 4, continued3. Use the Pythagorean Theorem to check

your trigonometry calculations.

AC = 35.641 and AB = 39.488.

32

2.2.1: Calculating Sine, Cosine, and Tangent

The answer checks out.

Page 33: Introduction

Guided Practice: Example 4, continued4. Find the value of ∠A.

33

2.2.1: Calculating Sine, Cosine, and Tangent

Page 34: Introduction

Guided Practice: Example 4, continuedUsing trigonometry, you could choose any of the three functions since you have solved for all three side lengths. In an attempt to be as precise as possible, let’s choose the given side length and one of the approximate side lengths.

34

2.2.1: Calculating Sine, Cosine, and Tangent

Page 35: Introduction

Guided Practice: Example 4, continued5. Use the inverse trigonometric function

since you are solving for an angle measure.

35

2.2.1: Calculating Sine, Cosine, and Tangent

Page 36: Introduction

Guided Practice: Example 4, continuedOn a TI-83/84:

Step 1: Press [2ND][SIN][17][÷][39.488][)]. Step 2: Press [ENTER]. z = 25.500°

36

2.2.1: Calculating Sine, Cosine, and Tangent

Page 37: Introduction

Guided Practice: Example 4, continuedOn a TI-Nspire:

Step 1: In the calculate window from the home screen, press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "sin–1," and then enter 17, press [÷], and enter 39.488.

Step 2: Press [enter].z = 25.500°

37

2.2.1: Calculating Sine, Cosine, and Tangent

Page 38: Introduction

Guided Practice: Example 4, continuedCheck your angle measure by using the Triangle Sum Theorem.

∠A is 25.5°.

38

2.2.1: Calculating Sine, Cosine, and Tangent

✔The answer checks out.

Page 39: Introduction

Guided Practice: Example 4, continued

39

2.2.1: Calculating Sine, Cosine, and Tangent

http://www.walch.com/ei/00186

Introduction Chapitre 1 · Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction to Databases: Introduction

Introduction Introduction (ctnd.)

INTRODUCTIONS (LEADS) Academic introduction Creative introduction Action introduction Dialogue introduction Overarching societal statements Personal introduction

1. INTRODUCTION INTRODUCTION - Deusto

1. INTRODUCTION • INTRODUCTION • EINFÜHRUNG • INLEIDING • INTRODUCCIÓN … · 2016-12-16 · 1. INTRODUCTION • INTRODUCTION • EINFÜHRUNG • INLEIDING • INTRODUCCIÓN

Introduction to Logic Introduction

1. INTRODUCTION 1.1 Introduction

Introduction - University of Cambridge · Introduction. Introduction

Introduction & Strategy Module Introduction & Administrative

Systems Analysis I introduction I introduction. 2Systems Analysis - Introduction Contents Introduction Introduction 3 classic phases 3 classic phases

Introduction and Welcome€¦ · Introduction and Welcome Introduction 4

1. Introduction 10 - therackonline.com€¦ · 1. Introduction 10 - therackonline.com ... Introduction)

voip-iptel-introduction-SRK1.ppt voip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-ip

introduction | EinlEitung | introduction | introducción · introduction | EinlEitung | introduction | introducción En Description Operation Maintenance Safety instructions dE Benennung

Optimisation des Requêtes. Introduction Introduction

An Introduction to NSAn Introduction to NS2An Introduction

CrowPiwithMineCraftPiEdition · 2 CrowPiwithMineCraftPiEdition-Introduction-MinecraftPiEdition-Introduction-Whatyouwillneed-Introduction-RunningMinecraft-Introduction

Introduction to introduction to introduction to … Optimization

Introduction | Introduction | Millennium Serials · 2018. 4. 17. · MARC Display | Print | Introduction | Millennium Serials INTRODUCTION Introduction This tutorial introduces the

Introduction Introduction Introduction Introduction

The Sacrament of Baptism Introduction Video Introduction Video The Sacrament of Baptism Introduction Video Introduction Video

Introduction: Definition Introduction: Philosophy

Topics Introduction 1 / 3 Introduction 2 / 3 Introduction 3 / 3

An Introduction for An Introduction for Introduction for ......An Introduction for An Introduction for Introduction for TeachersTeachersTeachers . exChange, Blackburn Cathedral’s

URBAN STREAM REHABILITATION. INTRODUCTION, OBJECTIVES & IMPACTS INTRODUCTION, OBJECTIVES & IMPACTS INTRODUCTION, OBJECTIVES & IMPACTS INTRODUCTION, OBJECTIVES

Introduction to Transducers, Introduction to Transducers ... on... · Introduction to Transducers, Introduction to Transducers, Sensors and ActuatorsSensors and Actuators Introduction

Introduction - daks2k3a4ib2z.cloudfront.net€¦ · Introduction Establishedin1974MarbleHillFireplacesofferoneofthelargestandmostcomprehensiveselectionsof …

R as a Web Service Introduction Introduction Introduction

Module: Introduction...Module: Introduction Page: Introduction CC0.1 Introduction Please give a general description and introduction to your organization. Providing electrical, hydraulic

 · Introduction . 6 . Contents . Introduction

1. Introduction 1. Introduction 1. Introduction 1 ... · Introduction 1. Introduction 1. Introduction 1. Introduction 6061B0242401. The Phone Screen From the Phone screen, you can

PRODUCT INTRODUCTION Company Introduction

Industry introduction Industry introduction Company introduction Company introduction Production introduction Display time time content

Introducción / Introduction / Introduction