Lesson 28 Working with Special Triangles
|
|
|
- Austen Hodge
- 5 years ago
- Views:
Transcription
1 Lesson 28 Working with Special Triangles Pre-Calculus 3/3/14 Pre-Calculus 1
2 Review Where We ve Been We have a new understanding of angles as we have now placed angles in a circle on a coordinate plane We also have an understanding of how to calculate the trig ratios of the angles in standard position 3/3/14 Pre-Calculus 2
3 Lesson Objectives Know the trig ratios of all multiples of 30, 45, 60, 90 angles Understand the concepts behind the trig ratios of special angles in all four quadrants Solve simple trig equations involving special trig ratios Tabulate the trig ratios to begin graphing trig functions 3/3/14 Pre-Calculus 3
4 (A) Review Special Triangles Review triangle sin(45 ) = sin(π/4) = cos(45 ) = cos(π/4) = tan(45 ) = tan(π/4) = csc(45 ) = csc(π/4) = sec(45 ) = sec(π/4) = cot(45 ) = cot(π/4) = 3/3/14 Pre-Calculus 4
5 (A) Review Special Triangles Review triangle 3/3/14 Pre-Calculus 5
6 (A) Review Special Triangles Review triangle 30 π/6 rad Review triangle 60 π/3 rad sin(30 ) = sin(π/6) = cos(30 ) = cos(π/6) = tan(30 ) = cot(π/6) = csc(30 ) = csc(π/6) = sec(30 ) = sec(π/6) = cot(30 ) = cot(π/6) = sin(60 ) = sin(π/3) = cos(60 ) = cos(π/3) = tan(60 ) = tan(π/3) = csc(60 ) = csc(π/3) = sec(60 ) = sec(π/3) = cot(60 ) = cot(π/3) = 3/3/14 Pre-Calculus 6
7 (A) Review Special Triangles triangle 3/3/14 Pre-Calculus 7
8 (B) Trig Ratios of First Quadrant Angles We have already reviewed first quadrant angles in that we have discussed the sine and cosine (as well as other ratios) of 30, 45, and 60 angles What about the quadrantal angles of 0 and 90? 3/3/14 Pre-Calculus 8
9 (B) Trig Ratios of First Quadrant Angles Quadrantal Angles Let s go back to the x,y,r definitions of sine and cosine ratios and use ordered pairs of angles whose terminal arms lie on the positive x axis (0 angle) and the positive y axis (90 angle) sin(0 ) = cos (0 ) = tan(0 ) = sin(90 ) = sin(π/2) = cos(90 ) = cos(π/2) = tan(90 ) = tan(π/2) = 3/3/14 Pre-Calculus 9
10 (B) Trig Ratios of First Quadrant Angles Quadrantal Angles Let s go back to the x,y,r definitions of sine and cosine ratios and use ordered pairs of angles whose terminal arms lie on the positive x axis (0 angle) and the positive y axis (90 angle) sin(0 ) = 0/1 = 0 cos (0 ) = 1/1 = 1 tan(0 ) = 0/1 = 0 sin(90 ) = sin(π/2) =1/1 = 1 cos(90 ) = cos(π/2) =0/1 = 0 tan(90 ) = tan(π/2) =1/0 = undefined 3/3/14 Pre-Calculus 10
11 (B) Trig Ratios of First Quadrant Angles - Summary 3/3/14 Pre-Calculus 11
12 (C) Trig Ratios of Second Quadrant Angles Now let s apply the same ideas & concepts to considering special second quadrant angles of 120 (2π/3), 135 (3π/4), 150 (5π/6) and 180 (π) 3/3/14 Pre-Calculus 12
13 (C) Trig Ratios of Second Quadrant Angles Now let s apply the same ideas & concepts to considering special second quadrant angles of 120, 135, 150 and 180 θ 120 Sin(θ) Cos(θ) Tan(θ) (2π/3) 150 (5π/6) 3/3/14 Pre-Calculus 13
14 (C) Trig Ratios of Second Quadrant Angles Now let s apply the same ideas & concepts to considering special second quadrant angles of 120, 135, 150 and 180 θ 135 Sin(θ) Cos(θ) Tan(θ) (3π/4) 180 (π) 3/3/14 Pre-Calculus 14
15 (D) Trig Ratios of Third Quadrant Angles Now let s apply the same ideas & concepts to considering special second quadrant angles of 210 (7π/6), 225 (5π/4), 240 (4π/3) and 270 (3π/2) θ 210 Sin(θ) Cos(θ) Tan(θ) (7π/6) 240 (4π/3) 3/3/14 Pre-Calculus 15
16 (D) Trig Ratios of Third Quadrant Angles Now let s apply the same ideas & concepts to considering special second quadrant angles of 210 (7π/6), 225 (5π/4), 240 (4π/3) and 270 (3π/2) θ 225 Sin(θ) Cos(θ) Tan(θ) (5π/4) 270 (3π/2) 3/3/14 Pre-Calculus 16
17 (D) Trig Ratios of Fourth Quadrant Angles Now let s apply the same ideas & concepts to considering special second quadrant angles of 300 (5π/3), 315 (7π/4), 330 (11π/4) and 360 (2π) θ Sin(θ) Cos(θ) Tan(θ) 300 (5π/3) 330 (11π/6) 3/3/14 Pre-Calculus 17
18 (D) Trig Ratios of Fourth Quadrant Angles Now let s apply the same ideas & concepts to considering special second quadrant angles of 300 (5π/3), 315 (7π/4), 330 (11π/4) and 360 θ Sin(θ) Cos(θ) Tan(θ) (2π) 315 (7π/4) 360 (2π) 3/3/14 Pre-Calculus 18
19 (G) Summary (As a Table of Values) sin cos sin cos 3/3/14 Pre-Calculus 19
20 (G) Summary As a Unit Circle The Unit Circle is a tool used in understanding sines and cosines of angles found in right triangles. It is so named because its radius is exactly one unit in length, usually just called "one". The circle's center is at the origin, and its circumference comprises the set of all points that are exactly one unit from the origin while lying in the plane. 3/3/14 Pre-Calculus 20
21 (G) Summary As a Unit Circle 3/3/14 Pre-Calculus 21
22 (H) Examples Complete the worksheet: trigonometry104.htm trigonometry108.htm 3/3/14 Pre-Calculus 22
23 (H) Trig Equations Simplify the following: 3/3/14 Pre-Calculus 23
24 (H) Trig Equations Simplify or solve 3/3/14 Pre-Calculus 24
Sect 7.4 Trigonometric Functions of Any Angles
Sect 7.4 Trigonometric Functions of Any Angles Objective #: Extending the definition to find the trigonometric function of any angle. Before we can extend the definition our trigonometric functions, we
Calculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)
Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h)
Analytic Trigonometry
Chapter 5 Analytic Trigonometry Course Number Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate trigonometric functions
Math Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically
1 MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram
I.e., the range of f(x) = arctan(x) is all real numbers y such that π 2 < y < π 2
Inverse Trigonometric Functions: The inverse sine function, denoted by fx = arcsinx or fx = sin 1 x is defined by: y = sin 1 x if and only if siny = x and π y π I.e., the range of fx = arcsinx is all real
More with Angles Reference Angles
More with Angles Reference Angles A reference angle is the angle formed by the terminal side of an angle θ, and the (closest) x axis. A reference angle, θ', is always 0 o
Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach
Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t
Section 6.2 Trigonometric Functions: Unit Circle Approach
Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal
Math Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
Unit Circle. Return to. Contents
Unit Circle Return to Table of Contents 32 The Unit Circle The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant
Solution. Using the point-slope form of the equation we have the answer immediately: y = 4 5 (x ( 2)) + 9 = 4 (x +2)+9
Chapter Review. Lines Eample. Find the equation of the line that goes through the point ( 2, 9) and has slope 4/5. Using the point-slope form of the equation we have the answer immediately: y = 4 5 ( (
(Section 4.7: Inverse Trig Functions) 4.82 PART F: EVALUATING INVERSE TRIG FUNCTIONS. Think:
PART F: EVALUATING INVERSE TRIG FUNCTIONS Think: (Section 4.7: Inverse Trig Functions) 4.82 A trig function such as sin takes in angles (i.e., real numbers in its domain) as inputs and spits out outputs
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
Chapter 5 Notes. 5.1 Using Fundamental Identities
Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx
Chapter 5: Trigonometric Functions of Angles Homework Solutions
Chapter : Trigonometric Functions of Angles Homework Solutions Section.1 1. D = ( ( 1)) + ( ( )) = + 8 = 100 = 10. D + ( ( )) + ( ( )) = + = 1. (x + ) + (y ) =. (x ) + (y + 7) = r To find the radius, we
Functions & Trigonometry Final Review #3. 3. Please find 2 coterminal angels (one positive and one negative) in the same measure as the given angle.
1. Please convert the following angles to degrees. a. 5 3 revolutions CCW = b. 5π 9 = c. 9 12π 4 revolutions CW = d. 5 = 2. Please convert the following angles to radians. a. c. 3 5 revolutions CCW = b.
Inverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.
Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise
Pre- Calculus Mathematics Trigonometric Identities and Equations
Pre- Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
sin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
Lesson 33 - Trigonometric Identities. Pre-Calculus
Lesson 33 - Trigonometric Identities Pre-Calculus 1 (A) Review of Equations An equation is an algebraic statement that is true for only several values of the variable The linear equation 5 = 2x 3 is only
Using the Definitions of the Trigonometric Functions
1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities February 1, 2013 Mrs. Poland Objectives Objective
Practice Test - Chapter 4
Find the value of x. Round to the nearest tenth, if necessary. Find the measure of angle θ. Round to the nearest degree, if necessary. 1. An acute angle measure and the length of the hypotenuse are given,
Trig Identities, Solving Trig Equations Answer Section
Trig Identities, Solving Trig Equations Answer Section MULTIPLE CHOICE. ANS: B PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas. ANS: A PTS: REF: Knowledge and Understanding OBJ:
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS MR. FORTIER 1. Trig Functions of Any Angle We now extend the definitions of the six basic trig functions beyond triangles so that we do not have to restrict
Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
4-3 Trigonometric Functions on the Unit Circle
Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on
( 3 ) = (r) cos (390 ) =
MATH 7A Test 4 SAMPLE This test is in two parts. On part one, you may not use a calculator; on part two, a (non-graphing) calculator is necessary. When you complete part one, you turn it in and get part
Mth 133 Trigonometry Review Problems for the Final Examination
Mth 1 Trigonometry Review Problems for the Final Examination Thomas W. Judson Stephen F. Austin State University Fall 017 Final Exam Details The final exam for MTH 1 will is comprehensive and will cover
Chapter 5 Trigonometric Functions of Angles
Chapter 5 Trigonometric Functions of Angles Section 3 Points on Circles Using Sine and Cosine Signs Signs I Signs (+, +) I Signs II (+, +) I Signs II (, +) (+, +) I Signs II (, +) (+, +) I III Signs II
CK- 12 Algebra II with Trigonometry Concepts 1
1.1 Pythagorean Theorem and its Converse 1. 194. 6. 5 4. c = 10 5. 4 10 6. 6 5 7. Yes 8. No 9. No 10. Yes 11. No 1. No 1 1 1. ( b+ a)( a+ b) ( a + ab+ b ) 1 1 1 14. ab + c ( ab + c ) 15. Students must
Notes on Radian Measure
MAT 170 Pre-Calculus Notes on Radian Measure Radian Angles Terri L. Miller Spring 009 revised April 17, 009 1. Radian Measure Recall that a unit circle is the circle centered at the origin with a radius
4 The Trigonometric Functions
Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater
2.Draw each angle in standard position. Name the quadrant in which the angle lies. 2. Which point(s) lies on the unit circle? Explain how you know.
Chapter Review Section.1 Extra Practice 1.Draw each angle in standard position. In what quadrant does each angle lie? a) 1 b) 70 c) 110 d) 00.Draw each angle in standard position. Name the quadrant in
TRIGONOMETRY REVIEW (PART 1)
TRIGONOMETRY REVIEW (PART ) MATH 52, SECTION 55 (VIPUL NAIK) Difficulty level: Easy to moderate, given that you are already familiar with trigonometry. Covered in class?: Probably not (for the most part).
(c) cos Arctan ( 3) ( ) PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER
PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER Work the following on notebook paper ecept for the graphs. Do not use our calculator unless the problem tells ou to use it. Give three decimal places
Math Analysis Chapter 5 Notes: Analytic Trigonometric
Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot
A. Incorrect! For a point to lie on the unit circle, the sum of the squares of its coordinates must be equal to 1.
Algebra - Problem Drill 19: Basic Trigonometry - Right Triangle No. 1 of 10 1. Which of the following points lies on the unit circle? (A) 1, 1 (B) 1, (C) (D) (E), 3, 3, For a point to lie on the unit circle,
Since 1 revolution = 1 = = Since 1 revolution = 1 = =
Fry Texas A&M University Math 150 Chapter 8A Fall 2015! 207 Since 1 revolution = 1 = = Since 1 revolution = 1 = = Convert to revolutions (or back to degrees and/or radians) a) 45! = b) 120! = c) 450! =
PreCalculus First Semester Exam Review
PreCalculus First Semester Eam Review Name You may turn in this eam review for % bonus on your eam if all work is shown (correctly) and answers are correct. Please show work NEATLY and bo in or circle
MATH 127 SAMPLE FINAL EXAM I II III TOTAL
MATH 17 SAMPLE FINAL EXAM Name: Section: Do not write on this page below this line Part I II III TOTAL Score Part I. Multiple choice answer exercises with exactly one correct answer. Each correct answer
Practice Problems for MTH 112 Exam 2 Prof. Townsend Fall 2013
Practice Problems for MTH 11 Exam Prof. Townsend Fall 013 The problem list is similar to problems found on the indicated pages. means I checked my work on my TI-Nspire software Pages 04-05 Combine the
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. and θ is in quadrant IV. 1)
Chapter 5-6 Review Math 116 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the fundamental identities to find the value of the trigonometric
Mathematics Trigonometry: Unit Circle
a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and
Trigonometric Ratios. θ + k 360
Trigonometric Ratios These notes are intended as a summary of section 6.1 (p. 466 474) in your workbook. You should also read the section for more complete explanations and additional examples. Coterminal
Summer Packet Greetings Future AP Calculus Scholar,
Summer Packet 2017 Greetings Future AP Calculus Scholar, I am excited about the work that we will do together during the 2016-17 school year. I do not yet know what your math capability is, but I can assure
MTH 122: Section 204. Plane Trigonometry. Test 1
MTH 122: Section 204. Plane Trigonometry. Test 1 Section A: No use of calculator is allowed. Show your work and clearly identify your answer. 1. a). Complete the following table. α 0 π/6 π/4 π/3 π/2 π
2 Trigonometric functions
Theodore Voronov. Mathematics 1G1. Autumn 014 Trigonometric functions Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics..1
Chapter 4/5 Part 2- Trig Identities and Equations
Chapter 4/5 Part 2- Trig Identities and Equations Lesson Package MHF4U Chapter 4/5 Part 2 Outline Unit Goal: By the end of this unit, you will be able to solve trig equations and prove trig identities.
Honors Algebra 2 Chapter 14 Page 1
Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using t-bar method. A. Sine and Cosecant. y = sinθ y y y y 0 --- --- 80 --- --- 30 0 0 300 5 35 5 35 60 50 0
Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.
Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs
1. Trigonometry.notebook. September 29, Trigonometry. hypotenuse opposite. Recall: adjacent
Trigonometry Recall: hypotenuse opposite adjacent 1 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2 Example: Determine the value of x. a)
5.3 Properties of Trigonometric Functions Objectives
Objectives. Determine the Domain and Range of the Trigonometric Functions. 2. Determine the Period of the Trigonometric Functions. 3. Determine the Signs of the Trigonometric Functions in a Given Quadrant.
Lesson 22 - Trigonometric Identities
POP QUIZ Lesson - Trigonometric Identities IB Math HL () Solve 5 = x 3 () Solve 0 = x x 6 (3) Solve = /x (4) Solve 4 = x (5) Solve sin(θ) = (6) Solve x x x x (6) Solve x + = (x + ) (7) Solve 4(x ) = (x
Chapter 4 Trigonometric Functions
Chapter 4 Trigonometric Functions Overview: 4.1 Radian and Degree Measure 4.2 Trigonometric Functions: The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of Any Angle 4.5 Graphs
6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
MATH 130 FINAL REVIEW
MATH 130 FINAL REVIEW Problems 1 5 refer to triangle ABC, with C=90º. Solve for the missing information. 1. A = 40, c = 36m. B = 53 30', b = 75mm 3. a = 91 ft, b = 85 ft 4. B = 1, c = 4. ft 5. A = 66 54',
MATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions
Math 09 Ta-Right Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b. 5 5 90 and 0 60 90 Triangles
From now on angles will be drawn with their vertex at the. The angle s initial ray will be along the positive. Think of the angle s
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 1 Chapter 8A Angles and Circles From now on angles will be drawn with their vertex at the The angle s initial ray will be along the positive.
CALCULUS ASSESSMENT REVIEW
CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness
FUNDAMENTAL TRIGONOMETRIC INDENTITIES 1 = cos. sec θ 1 = sec. = cosθ. Odd Functions sin( t) = sint. csc( t) = csct tan( t) = tant
NOTES 8: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 8.1 TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sinθ 1 cscθ cosθ 1 secθ tanθ 1
Crash Course in Trigonometry
Crash Course in Trigonometry Dr. Don Spickler September 5, 003 Contents 1 Trigonometric Functions 1 1.1 Introduction.................................... 1 1. Right Triangle Trigonometry...........................
Lesson 25 Solving Linear Trigonometric Equations
Lesson 25 Solving Linear Trigonometric Equations IB Math HL - Santowski EXPLAIN the difference between the following 2 equations: (a) Solve sin(x) = 0.75 (b) Solve sin -1 (0.75) = x Now, use you calculator
Troy High School AP Calculus Summer Packet
Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by
Precalculus: Trigonometry of Circular Functions Practice Problems. Questions. and sin θ > Find csc θ and cot θ if tan θ = 4 3
Questions. Find csc θ and cot θ if tan θ = 4 3 and sin θ > 0. 2. An airplane flying at an altitude of 8000 ft passes directly over a group of hikers who are at 7400 ft. If θ is the angle of elevation from
A-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019
A-Level Mathematics TRIGONOMETRY G. David Boswell - R2S Explore 2019 1. Graphs the functions sin kx, cos kx, tan kx, where k R; In these forms, the value of k determines the periodicity of the trig functions.
Radian measure and trigonometric functions on the reals
Trigonometry Radian measure and trigonometric functions on the reals The units for measuring angles tend to change depending on the context - in a geometry course, you re more likely to be measuring your
Chapter 6. Trigonometric Functions of Angles. 6.1 Angle Measure. 1 radians = 180º. π 1. To convert degrees to radians, multiply by.
Chapter 6. Trigonometric Functions of Angles 6.1 Angle Measure Radian Measure 1 radians = 180º Therefore, o 180 π 1 rad =, or π 1º = 180 rad Angle Measure Conversions π 1. To convert degrees to radians,
Lesson 5.3. Solving Trigonometric Equations
Lesson 5.3 Solving To solve trigonometric equations: Use standard algebraic techniques learned in Algebra II. Look for factoring and collecting like terms. Isolate the trig function in the equation. Use
Unit 3 Trigonometry Note Package. Name:
MAT40S Unit 3 Trigonometry Mr. Morris Lesson Unit 3 Trigonometry Note Package Homework 1: Converting and Arc Extra Practice Sheet 1 Length 2: Unit Circle and Angles Extra Practice Sheet 2 3: Determining
Chapter 1. Functions 1.3. Trigonometric Functions
1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius
Chapter 13: Trigonometry Unit 1
Chapter 13: Trigonometry Unit 1 Lesson 1: Radian Measure Lesson 2: Coterminal Angles Lesson 3: Reference Angles Lesson 4: The Unit Circle Lesson 5: Trig Exact Values Lesson 6: Trig Exact Values, Radian
Calculus with business applications, Lehigh U, Lecture 05 notes Summer
Calculus with business applications, Lehigh U, Lecture 0 notes Summer 0 Trigonometric functions. Trigonometric functions often arise in physical applications with periodic motion. They do not arise often
Trigonometry LESSON SIX - Trigonometric Identities I Lesson Notes
LESSON SIX - Trigonometric Identities I Example Understanding Trigonometric Identities. a) Why are trigonometric identities considered to be a special type of trigonometric equation? Trigonometric Identities
MATH 2412 Sections Fundamental Identities. Reciprocal. Quotient. Pythagorean
MATH 41 Sections 5.1-5.4 Fundamental Identities Reciprocal Quotient Pythagorean 5 Example: If tanθ = and θ is in quadrant II, find the exact values of the other 1 trigonometric functions using only fundamental
Exercise Set 6.2: Double-Angle and Half-Angle Formulas
Exercise Set : Double-Angle and Half-Angle Formulas Answer the following π 1 (a Evaluate sin π (b Evaluate π π (c Is sin = (d Graph f ( x = sin ( x and g ( x = sin ( x on the same set of axes (e Is sin
Numbers Content Points. Reference sheet (1 pt. each) 1-7 Linear Equations (1 pt. each) / Factoring (2 pt. each) /28
Summer Packet 2015 Your summer packet will be a major test grade for the first nine weeks. It is due the first day of school. You must show all necessary solutions. You will be tested on ALL material;
9.1. Click here for answers. Click here for solutions. PARAMETRIC CURVES
SECTION 9. PARAMETRIC CURVES 9. PARAMETRIC CURVES A Click here for answers. S Click here for solutions. 5 (a) Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the
2 sin 2 (x) 1 = 0 2 sin 2 (x) = 1. sin 2 (x) = sin(x) = ± 2. x 3 = sin 1 ( 1/ 2) = π. x 2 = π x 1 = 3π 4. 4 x 4 = π x 3 = 5π 4
Math 147 - Assignment 1 Solutions - Spring 01 - BSU - Jaimos F Skriletz 1 1. Trigonometric Equations Find all solutions to the following trigonometric equations: (a) sin (x) 1 = 0 sin (x) = 1 If sin(x)
MATH 100 REVIEW PACKAGE
SCHOOL OF UNIVERSITY ARTS AND SCIENCES MATH 00 REVIEW PACKAGE Gearing up for calculus and preparing for the Assessment Test that everybody writes on at. You are strongly encouraged not to use a calculator
Algebra II B Review 5
Algebra II B Review 5 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the measure of the angle below. y x 40 ο a. 135º b. 50º c. 310º d. 270º Sketch
Transition to College Math
Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: F-TF.2 Learning Target: Eplain
Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.
Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain
1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if
. Which of the following defines a function f for which f ( ) = f( )? a. f ( ) = + 4 b. f ( ) = sin( ) f ( ) = cos( ) f ( ) = e f ( ) = log. ln(4 ) < 0 if and only if a. < b. < < < < > >. If f ( ) = (
Trigonometric Functions
Trigonometric Functions Here is a review the basic definitions and properties of the trigonometric functions. We provide a list of trig identities at the end.. Definitions The trigonometric functions are
A List of Definitions and Theorems
Metropolitan Community College Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180. Definition 2. One
SET 1. (1) Solve for x: (a) e 2x = 5 3x
() Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x
Precalculus Lesson 6.1: Angles and Their Measure Lesson 6.2: A Unit Circle Approach Part 2
Precalculus Lesson 6.1: Angles and Their Measure Lesson 6.2: A Unit Circle Approach Part 2 Lesson 6.2 Before we look at the unit circle with respect to the trigonometric functions, we need to get some
CHAPTERS 5-7 TRIG. FORMULAS PACKET
CHAPTERS 5-7 TRIG. FORMULAS PACKET PRE-CALCULUS SECTION 5-2 IDENTITIES Reciprocal Identities sin x = ( 1 / csc x ) csc x = ( 1 / sin x ) cos x = ( 1 / sec x ) sec x = ( 1 / cos x ) tan x = ( 1 / cot x
Chapter 5 Analytic Trigonometry
Chapter 5 Analytic Trigonometry Overview: 5.1 Using Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Solving Trig Equations 5.4 Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-sum
The choice of origin, axes, and length is completely arbitrary.
Polar Coordinates There are many ways to mark points in the plane or in 3-dim space for purposes of navigation. In the familiar rectangular coordinate system, a point is chosen as the origin and a perpendicular
Chapter 3. Radian Measure and Circular Functions. Copyright 2005 Pearson Education, Inc.
Chapter 3 Radian Measure and Circular Functions Copyright 2005 Pearson Education, Inc. 3.1 Radian Measure Copyright 2005 Pearson Education, Inc. Measuring Angles Thus far we have measured angles in degrees
Pre-Calc Trig ~1~ NJCTL.org. Unit Circle Class Work Find the exact value of the given expression. 7. Given the terminal point ( 3, 2 10.
Unit Circle Class Work Find the exact value of the given expression. 1. cos π 3. sin 7π 3. sec π 3. tan 5π 6 5. cot 15π 6. csc 9π 7. Given the terminal point ( 3, 10 ) find tanθ 7 7 8. Given the terminal
Special Angles 1 Worksheet MCR3U Jensen
Special Angles 1 Worksheet 1) a) Draw a right triangle that has one angle measuring 30. Label the sides using lengths 3, 2, and 1. b) Identify the adjacent and opposite sides relative to the 30 angle.
NOTES Show all necessary work. You are not allowed to use your unit circle on the test. The test will include a non-calculator portion
Algebra Trig hapter 1 Review Problems omplete the following problems on a separate piece of paper. NOTES Show all necessary work. You are not allowed to use your unit circle on the test. The test will
( ) Trigonometric identities and equations, Mixed exercise 10
Trigonometric identities and equations, Mixed exercise 0 a is in the third quadrant, so cos is ve. The angle made with the horizontal is. So cos cos a cos 0 0 b sin sin ( 80 + 4) sin 4 b is in the fourth
AMB121F Trigonometry Notes
AMB11F Trigonometry Notes Trigonometry is a study of measurements of sides of triangles linked to the angles, and the application of this theory. Let ABC be right-angled so that angles A and B are acute