Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Trigonometry LESSON ONE - Degrees and Radians Lesson Notes"

Transcription

1 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles: Draw = 120 Draw = -120 c) reference angle: Find the reference angle of = 10.

2 Trigonometry = 7 6 d) co-terminal angles: Draw the first positive co-terminal angle of 60. e) principal angle: Find the principal angle of = 20. f) general form of co-terminal angles: Find the first four positive co-terminal angles of =. Find the first four negative co-terminal angles of =.

3 = 7 6 Trigonometry Example 2 Three Angle Types: Degrees, Radians, and Revolutions. a) Define degrees, radians, and revolutions. Angle Types and Conversion Multipliers i) Degrees: Draw = 1 ii) Radians: Draw = 1 rad iii) Revolutions: Draw = 1 rev

4 Trigonometry = 7 6 b) Use conversion multipliers to answer the questions and fill in the reference chart. Round all decimals to the nearest hundredth. Conversion Multiplier Reference Chart i) 2 = rad degree radian revolution degree ii) 2 = rev radian iii) 2.6 = revolution iv) 2.6 = rev v) 0.7 rev = vi) 0.7 rev = rad c) Contrast the decimal approximation of a radian with the exact value of a radian. i) Decimal Approximation: = rad ii) Exact Value: = rad

5 = 7 6 Trigonometry Example Convert each angle to the requested form. Round all decimals to the nearest hundredth. a) convert 17 to an approximate radian decimal. Angle Conversion Practice b) convert 210 to an exact-value radian. c) convert 120 to an exact-value revolution. d) convert 2. to degrees. e) convert to degrees. 2 f) write as an approximate radian decimal. 2 g) convert to an exact-value revolution. 2 h) convert 0. rev to degrees. i) convert rev to radians.

6 Trigonometry = 7 6 Example The diagram shows commonly used degrees. Find exact-value radians that correspond to each degree. When complete, memorize the diagram. Commonly Used Degrees and Radians a) Method One: Find all exact-value radians using a conversion multiplier. b) Method Two: Use a shortcut. (Counting Radians) 90 = = 10 = 1 = = = 0 = 0 = = = = 210 = 22 = 20 0 = 1 = 00 = = 270

7 = 7 6 Trigonometry Example a) 210 Draw each of the following angles in standard position. State the reference angle. Reference Angles b) -260 c). d) - e) 12 7

8 Trigonometry = 7 6 Example 6 a) 90 Draw each of the following angles in standard position. State the principal and reference angles. Principal and Reference Angles b) -8 c) 9 d) - 10

9 = 7 6 Trigonometry For each angle, find all co-terminal Example 7 Co-terminal Angles angles within the stated domain. a) 60, Domain: -60 < 1080 b) 9, Domain: < 720 c) 11.78, Domain: -2 < d) 8, Domain: 1 2 < 7

10 Trigonometry = 7 6 Example 8 For each angle, use estimation to find the principal angle. a) 189 b) 7.2 Principal Angle of a Large Angle 912 c) d) 1 9 6

11 = 7 6 Trigonometry Example 9 a) principal angle = 00 (find co-terminal angle rotations counter-clockwise) Use the general form of co-terminal angles to find the specified angle. General Form of Co-terminal Angles 2 b) principal angle = (find co-terminal angle 1 rotations clockwise) c) How many rotations are required to find the principal angle of 00? State the principal angle. d) How many rotations are required to find 2 the principal angle of? State the principal angle.

12 Trigonometry = 7 6 Example 10 Six Trigonometric Ratios In addition to the three primary trigonometric ratios (sin, cos, and tan), there are three reciprocal ratios (csc, sec, and cot). Given a triangle with side lengths of x and y, and a hypotenuse of length r, the six trigonometric ratios are as follows: sin = y r csc = 1 sin = r y r y cos = x r sec = 1 cos = r x x tan = y x cot = 1 tan = x y a) If the point P(-, 12) exists on the terminal arm of an angle in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle. b) If the point P(2, -) exists on the terminal arm of an angle in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle.

13 = 7 6 Trigonometry Example 11 Determine the sign of each trigonometric ratio in each quadrant. Signs of Trigonometric Ratios a) sin b) cos c) tan d) csc e) sec f) cot g) How do the quadrant signs of the reciprocal trigonometric ratios (csc, sec, and cot) compare to the quadrant signs of the primary trigonometric ratios (sin, cos, and tan)?

14 Trigonometry = 7 6 Example 12 Given the following conditions, find the quadrant(s) where the angle could potentially exist. What Quadrant(s) is the Angle in? a) i) sin < 0 ii) cos > 0 iii) tan > 0 b) i) sin > 0 and cos > 0 ii) sec > 0 and tan < 0 iii) csc < 0 and cot > 0 c) i) sin < 0 and csc = 1 ii) and csc < 0 iii) sec > 0 and tan = 1 2

15 = 7 6 Trigonometry Example 1 Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a radian. Exact Values of Trigonometric Ratios a) b)

16 Trigonometry = 7 6 Example 1 Given one trigonometric ratio, find the exact Exact Values of values of the other five trigonometric ratios. Trigonometric Ratios State the reference angle and the standard position angle, to the nearest hundredth of a degree. a) b)

17 = 7 6 Trigonometry Example 1 Calculating with a calculator. Calculator Concerns a) When you solve a trigonometric equation in your calculator, the answer you get for can seem unexpected. Complete the following chart to learn how the calculator processes your attempt to solve for. If the angle could exist in either quadrant or... The calculator always picks quadrant I or II I or III I or IV II or III II or IV III or IV b) Given the point P(, ), Mark tries to find the reference angle using a sine ratio, Jordan tries to find it using a cosine ratio, and Dylan tries to find it using a tangent ratio. Why does each person get a different result from their calculator? P(, ) Mark s Calculation of (using sine) sin = Jordan s Calculation of (using cosine) cos = Dylan s Calculation of (using tan) tan = = 6.87 = 1.1 = -6.87

18 Trigonometry = 7 6 Example 16 Arc Length The formula for arc length is a = r, where a is the arc length, is the central angle in radians, and r is the radius of the circle. The radius and arc length must have the same units. r a) Derive the formula for arc length, a = r. a b) Solve for a, to the nearest hundredth. c) Solve for. (express your answer as a degree, to the nearest hundredth.) 6 cm cm 1 cm a d) Solve for r, to the nearest hundredth. e) Solve for n. (express your answer as an exact-value radian.) 1.2 cm cm r 2 6 cm n

19 = 7 6 Trigonometry Example 17 Area of a circle sector. r 2 a) Derive the formula for the area of a circle sector, A =. 2 Sector Area r In parts (b - e), find the area of each shaded region. b) c) cm 7 6 cm 20 d) e) 9 cm cm cm

20 Trigonometry = 7 6 Example 18 The formula for angular speed is, where ω (Greek: Omega) is the angular speed, is the change in angle, and T is the change in time. Calculate the requested quantity in each scenario. Round all decimals to the nearest hundredth. a) A bicycle wheel makes 100 complete revolutions in 1 minute. Calculate the angular speed in degrees per second. b) A Ferris wheel rotates 1020 in. minutes. Calculate the angular speed in radians per second.

21 = 7 6 Trigonometry c) The moon orbits Earth once every 27 days. Calculate the angular speed in revolutions per second. If the average distance from the Earth to the moon is 8 00 km, how far does the moon travel in one second? d) A cooling fan rotates with an angular speed of 200 rpm. What is the speed in rps? e) A bike is ridden at a speed of 20 km/h, and each wheel has a diameter of 68 cm. Calculate the angular speed of one of the bicycle wheels and express the answer using revolutions per second.

22 Trigonometry = 7 6 Example 19 A satellite orbiting Earth 0 km above the surface makes one complete revolution every 90 minutes. The radius of Earth is approximately 670 km. a) Calculate the angular speed of the satellite. Express your answer as an exact value, in radians/second. 0 km 670 km b) How many kilometres does the satellite travel in one minute? Round your answer to the nearest hundredth of a kilometre.

Section 3.1 Radian Measure

Section 3.1 Radian Measure Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle

More information

Mathematics Book Two. Trigonometry One Trigonometry Two Permutations and Combinations

Mathematics Book Two. Trigonometry One Trigonometry Two Permutations and Combinations Mathematics 0- Book Two Trigonometry One Trigonometry Two Permutations and Combinations A workbook and animated series by Barry Mabillard Copyright 04 This page has been left blank for correct workbook

More information

Trigonometry Chapter 3 Lecture Notes

Trigonometry Chapter 3 Lecture Notes Ch Notes Morrison Trigonometry Chapter Lecture Notes Section. Radian Measure I. Radian Measure A. Terminology When a central angle (θ) intercepts the circumference of a circle, the length of the piece

More information

Chapter 6 Trigonometric Functions of Angles

Chapter 6 Trigonometric Functions of Angles 6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.

More information

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, 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 information

Trigonometry LESSON TWO - The Unit Circle Lesson Notes

Trigonometry LESSON TWO - The Unit Circle Lesson Notes (cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each

More information

5.2. Trigonometric Functions Of Real Numbers. Copyright Cengage Learning. All rights reserved.

5.2. Trigonometric Functions Of Real Numbers. Copyright Cengage Learning. All rights reserved. 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 information

Chapter 5: Trigonometric Functions of Real Numbers

Chapter 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 information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150

MULTIPLE 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 information

TEST #3 RADIAN MEASURE AND CIRCULAR FUNCTIONS, PRACTICE. 1) Convert each degree measure to radians. Leave answers as multiples of π. cos 4.

TEST #3 RADIAN MEASURE AND CIRCULAR FUNCTIONS, PRACTICE. 1) Convert each degree measure to radians. Leave answers as multiples of π. cos 4. PRECALCULUS B TEST # RADIAN MEASURE AND CIRCULAR FUNCTIONS, PRACTICE SECTION. Radian Measure ) Convert each degree measure to radians. Leave answers as multiples of. A) 45 B) 60 C) 0 D) 50 E) 75 F) 40

More information

Section 9.4 Trigonometric Functions of any Angle

Section 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 information

y = 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)

y = 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 information

Trigonometric Functions and Triangles

Trigonometric 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 information

Definition III: Circular Functions

Definition III: Circular Functions SECTION 3.3 Definition III: Circular Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Evaluate a trigonometric function using the unit circle. Find the value of a

More information

4.1 Radian and Degree Measure

4.1 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 information

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

D.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 information

5.1-Angles and Their Measure

5.1-Angles and Their Measure 5.1-Angles and Their Measure Objectives: 1. Find the degree or radian measure of co-terminal angles. 2. Convert between degrees minutes and seconds and decimal degrees. 3. Convert between degrees and radians.

More information

Mid-Chapter Quiz: Lessons 4-1 through 4-4

Mid-Chapter Quiz: Lessons 4-1 through 4-4 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 information

Unit 1 - Radian and Degree Measure Classwork

Unit 1 - Radian and Degree Measure Classwork Unit 1 - Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side - starting and ending Position of the ray Standard position origin if the

More information

3. Right Triangle Trigonometry

3. Right Triangle Trigonometry . Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities . Reference Angle Reference Angle Reference angle

More information

Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)

Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.) 1- The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview

More information

Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.

Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle. Pre-Calculus 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 information

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places. SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.

More information

Trigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:

Trigonometry (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 information

Trigonometry Basics. Angle. vertex at the and initial side along the. Standard position. Positive and Negative Angles

Trigonometry Basics. Angle. vertex at the and initial side along the. Standard position. Positive and Negative Angles Name: Date: Pd: Trigonometry Basics Angle terminal side direction of rotation arrow CCW is positive CW is negative vertex initial side Standard position vertex at the and initial side along the Positive

More information

Solutions to Exercises, Section 5.1

Solutions to Exercises, Section 5.1 Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

More information

Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25

Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25 Math 115 Spring 014 Written Homework 10-SOLUTIONS 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 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. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Assume that the cities lie on the same north-south line and that the radius of the earth

More information

Give an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179

Give 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 information

Trigonometry 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 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 information

FLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A.

FLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A. Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is

More information

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole

More information

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Angles 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 information

5.2 Unit Circle: Sine and Cosine Functions

5.2 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 information

PRE-CALCULUS GRADE 12

PRE-CALCULUS GRADE 12 PRE-CALCULUS 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 information

Chapter. Trigonometry. By the end of this chapter, you will. 4, _ π 3, π_, and their multiples less

Chapter. Trigonometry. By the end of this chapter, you will. 4, _ π 3, π_, and their multiples less Chapter Trigonometry You may be surprised to learn that scientists, engineers, designers, and other professionals who use angles in their daily work generally do not measure the angles in degrees. In this

More information

Trigonometry 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 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 information

Lecture Two Trigonometric

Lecture Two Trigonometric Lecture Two Trigonometric Section.1 Degrees, Radians, Angles and Triangles Basic Terminology Two distinct points determine line AB. Line segment AB: portion of the line between A and B. Ray AB: portion

More information

Unit 1 - Radian and Degree Measure Classwork

Unit 1 - Radian and Degree Measure Classwork Unit - Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side - starting and ending Position of the ray Standard position origin if the vertex,

More information

4.1: Angles and Radian Measure

4.1: Angles and Radian Measure 4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is

More information

11 Trigonometric Functions of Acute Angles

11 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 information

1) Convert 13 32' 47" to decimal degrees. Round your answer to four decimal places.

1) Convert 13 32' 47 to decimal degrees. Round your answer to four decimal places. PRECLCULUS FINL EXM, PRCTICE UNIT ONE TRIGONOMETRIC FUNCTIONS ) Convert ' 47" to decimal degrees. Round your answer to four decimal places. ) Convert 5.6875 to degrees, minutes, and seconds. Round to the

More information

Trigonometric Functions: The Unit Circle

Trigonometric 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 information

Chapter 5: Trigonometric Functions of Angles

Chapter 5: Trigonometric Functions of Angles Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has

More information

Right Triangle Trigonometry

Right Triangle Trigonometry Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned

More information

Algebra. 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. 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 information

Semester 2, Unit 4: Activity 21

Semester 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 information

About Trigonometry. Triangles

About 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 information

6.1 - Introduction to Periodic Functions

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

More information

Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6.

Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6. 1 Math 3 Final Review Guide This is a summary list of many concepts covered in the different sections and some examples of types of problems that may appear on the Final Exam. The list is not exhaustive,

More information

θ. The angle is denoted in two ways: angle θ

θ. The angle is denoted in two ways: angle θ 1.1 Angles, Degrees and Special Triangles (1 of 24) 1.1 Angles, Degrees and Special Triangles Definitions An angle is formed by two rays with the same end point. The common endpoint is called the vertex

More information

6.1: Angle Measure in degrees

6.1: Angle Measure in degrees 6.1: Angle Measure in degrees How to measure angles Numbers on protractor = angle measure in degrees 1 full rotation = 360 degrees = 360 half rotation = quarter rotation = 1/8 rotation = 1 = Right angle

More information

Plane Trigonometry - Fall 1996 Test File

Plane Trigonometry - Fall 1996 Test File Plane Trigonometry - Fall 1996 Test File Test #1 1.) Fill in the blanks in the two tables with the EXACT values of the given trigonometric functions. The total point value for the tables is 10 points.

More information

vertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector

vertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector vertex initial side terminal side standard position radian coterminal angles linear speed angular speed sector Convert Between DMS and Decimal Degree Form A. Write 329.125 in DMS form. First, convert 0.125

More information

Algebra 2/Trigonometry Practice Test

Algebra 2/Trigonometry Practice Test Algebra 2/Trigonometry Practice Test Part I Answer all 27 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. For each question, write on the separate

More information

5.1 Angles and Their Measure. Objectives

5.1 Angles and Their Measure. Objectives Objectives 1. Convert between decimal degrees and degrees, minutes, seconds measures of angles. 2. Find the length of an arc of a circle. 3. Convert from degrees to radians and from radians to degrees.

More information

SUM AND DIFFERENCE FORMULAS

SUM 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 information

Objectives After completing this section, you should be able to:

Objectives After completing this section, you should be able to: Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding

More information

Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. 42. The sum of the measures of the angles of a triangle is 180. Therefore, The sine of an angle

More information

Unit 6 Trigonometric Identities, Equations, and Applications

Unit 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 information

Math Placement Test Practice Problems

Math 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 information

Section 6.1 Angle Measure

Section 6.1 Angle Measure Section 6.1 Angle Measure An angle AOB consists of two rays R 1 and R 2 with a common vertex O (see the Figures below. We often interpret an angle as a rotation of the ray R 1 onto R 2. In this case, R

More information

Inverse Trigonometric Functions

Inverse 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 information

Final Exam Practice--Problems

Final Exam Practice--Problems Name: Class: Date: ID: A Final Exam Practice--Problems 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 information

ALGEBRA 2/TRIGONOMETRY

ALGEBRA 2/TRIGONOMETRY ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession

More information

Name Class Date 1 E. Finding the Ratio of Arc Length to Radius

Name Class Date 1 E. Finding the Ratio of Arc Length to Radius Name Class Date 0- Right-Angle Trigonometry Connection: Radian Measure Essential question: What is radian, and how are radians related to degrees? CC.9 2.G.C.5 EXPLORE Finding the Ratio of Arc Length to

More information

Friday, January 29, 2016 9:15 a.m. to 12:15 p.m., only

Friday, January 29, 2016 9:15 a.m. to 12:15 p.m., only ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession

More information

Maths Class 11 Chapter 3. Trigonometric Functions

Maths Class 11 Chapter 3. Trigonometric Functions 1 P a g e Angle Maths Class 11 Chapter 3. Trigonometric Functions When a ray OA starting from its initial position OA rotates about its end point 0 and takes the final position OB, we say that angle AOB

More information

1. The Six Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1.2 The Rectangular Coordinate System 1.3 Definition I: Trigonometric

1. 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 information

CHAPTER 28 THE CIRCLE AND ITS PROPERTIES

CHAPTER 28 THE CIRCLE AND ITS PROPERTIES CHAPTER 8 THE CIRCLE AND ITS PROPERTIES EXERCISE 118 Page 77 1. Calculate the length of the circumference of a circle of radius 7. cm. Circumference, c = r = (7.) = 45.4 cm. If the diameter of a circle

More information

MAC 1114: Trigonometry Notes

MAC 1114: Trigonometry Notes MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant

More information

Right Triangle Trigonometry

Right 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 information

Graphing Trigonometric Skills

Graphing 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 information

Trigonometry Lesson Objectives

Trigonometry 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 information

Trigonometry. Week 1 Right Triangle Trigonometry

Trigonometry. 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 information

3. EVALUATION OF TRIGONOMETRIC FUNCTIONS

3. 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 information

MPE Review Section II: Trigonometry

MPE 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 information

Chapter 5 The Trigonometric Functions

Chapter 5 The Trigonometric Functions P a g e 40 Chapter 5 The Trigonometric Functions Section 5.1 Angles Initial side Terminal side Standard position of an angle Positive angle Negative angle Coterminal Angles Acute angle Obtuse angle Complementary

More information

Lesson 6.1 Exercises, pages

Lesson 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 information

Trigonometric Functions

Trigonometric 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 information

4-1 Right Triangle Trigonometry

4-1 Right Triangle Trigonometry Find the exact values of the six trigonometric functions of θ. 1. The length of the side opposite θ is 8 is 18., the length of the side adjacent to θ is 14, and the length of the hypotenuse 3. The length

More information

2. Right Triangle Trigonometry

2. Right Triangle Trigonometry 2. Right Triangle Trigonometry 2.1 Definition II: Right Triangle Trigonometry 2.2 Calculators and Trigonometric Functions of an Acute Angle 2.3 Solving Right Triangles 2.4 Applications 2.5 Vectors: A Geometric

More information

4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles

4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles 4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred

More information

ALGEBRA 2/TRIGONOMETRY

ALGEBRA 2/TRIGONOMETRY ALGEBRA 2/TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA 2/TRIGONOMETRY Friday, January 29, 2016 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The

More information

6.3 Polar Coordinates

6.3 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 information

Mathematical Procedures

Mathematical Procedures CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

More information

Section 8.1: The Inverse Sine, Cosine, and Tangent Functions

Section 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 information

1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic

1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic 1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single

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

Trigonometric Functions

Trigonometric Functions Trigonometric Functions 13A Trigonometry and Angles 13-1 Right-Angle Trigonometry 13- Angles of Rotation Lab Explore the Unit Circle 13-3 The Unit Circle 13-4 Inverses of Trigonometric Functions 13B Applying

More information

Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring

Right 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 information

Section 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. 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 information

The Power of Calculator Software! Zoom Math 400 Version 1.00 Upgrade Manual for people who used Zoom Math 300

The Power of Calculator Software! Zoom Math 400 Version 1.00 Upgrade Manual for people who used Zoom Math 300 The Power of Calculator Software! Zoom Math 400 Version 1.00 Upgrade Manual for people who used Zoom Math 300 Zoom Math 400 Version 1.00 Upgrade Manual Revised 7/27/2010 2010 I.Q. Joe, LLC www.zoommath.com

More information

6.1 Basic Right Triangle Trigonometry

6.1 Basic Right Triangle Trigonometry 6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at

More information

With the Tan function, you can calculate the angle of a triangle with one corner of 90 degrees, when the smallest sides of the triangle are given:

With the Tan function, you can calculate the angle of a triangle with one corner of 90 degrees, when the smallest sides of the triangle are given: Page 1 In game development, there are a lot of situations where you need to use the trigonometric functions. The functions are used to calculate an angle of a triangle with one corner of 90 degrees. By

More information

RIGHT TRIANGLE TRIGONOMETRY

RIGHT 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 information

Angles and Their Measure

Angles and Their Measure Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two

More information

MATH 150 TOPIC 9 RIGHT TRIANGLE TRIGONOMETRY. 9a. Right Triangle Definitions of the Trigonometric Functions

MATH 150 TOPIC 9 RIGHT TRIANGLE TRIGONOMETRY. 9a. Right Triangle Definitions of the Trigonometric Functions Math 50 T9a-Right 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 information

Solution Guide for Chapter 6: The Geometry of Right Triangles

Solution Guide for Chapter 6: The Geometry of Right Triangles Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E-. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab

More information