Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 167.
|
|
- Holly Chase
- 6 years ago
- Views:
Transcription
1 lgebra Chapter 8: nalytical Trigonometry 8- Inverse Trigonometric Functions Chapter 8: nalytical Trigonometry Inverse Trigonometric Function: - use when we are given a particular trigonometric ratio and we are asked to solve for the original angle measure. - it is sometimes referred to as arc-sine sin, arc-cosine cos, or arc-tangent tan. Note: sin x sin x sin x sin x sin x sin x csc x Examples: For angles between 0 and, sin sin 6 6 cos cos 6 tan tan 6 To access Inverse Trigonometric Function on most calculators: nd IN IN or nd CO CO Graph of Inverse ine Function and its Domain and Range: or Recall that a function, fx, has to pass the vertical line test. Hence, for an inverse function, f x has to pass the horizontal line test one to one. y sin x nd TN TN y sin x Calculator is in Radian Mode y sin x revised Domain: x Range: y 6 Note that y sin x does NOT pass the horizontal line test between [, ]. However, if we take the interval at [ ], it will pass the horizontal line test. Hence, we can graph y sin x for y:[ ].,, y sin x Therefore, the output from a calculator of a sin x input, where x, is always between[ ], y sin x Domain: x Range: y sinsin x x for x sin sin x x for x Copyrighted by Gabriel Tang B.Ed., B.c. Page 67.
2 Chapter 8: nalytical Trigonometry lgebra Graph of Inverse Cosine Function and its Domain and Range: Recall that a function, fx, has to pass the vertical line test. Hence, for an inverse function, f x has to pass the horizontal line test one to one. y cos x y cos x y cos x revised Domain: 0 x Range: y Note that y cos x does NOT pass the horizontal line test between [, ]. However, if we take the interval at [0, ], it will pass the horizontal line test. Hence, we can graph y cos x for y: [0, ]. Therefore, the output from a calculator of a cos x input, where x, is always between [0, ] y cos x y cos x Domain: x Range: 0 x coscos x x for x cos cos x x for 0 x Graph of Inverse Tangent Function and its Domain and Range: Recall that a function, fx, has to pass the vertical line test. Hence, for an inverse function, f x has to pass the horizontal line test one to one. y tan x y tan x y tan x revised Domain: x Range: y R Note that y tan x does NOT pass the horizontal line test between [, ]. However, if we take the. interval at [ ], it will pass the horizontal line test. Hence, we can graph y tan x for y:[ ],, y tan x Therefore, the output from a calculator of a sin x input, where x, is always between[ ] y tan x Domain: x R Range: y, tantan x x tan tan x x for x R for x Page 68. Copyrighted by Gabriel Tang B.Ed., B.c.
3 lgebra Chapter 8: nalytical Trigonometry Example : Find the exact value of each expression, if it is defined. a. sin sin b. tan tan c. cos Example : Find the exact value of each expression, if it is defined. Calculator gives the a. sin sin b. sin same decimal value. sin Hence, sin interval allowed. sin e. tan From the unit circle, at, the y- value is. Verify: sin and is within the range of y sin x, which is y. sinsin x x for x and is within [, ] coscos x x for x and is within [, ] Verify: tan and range of y tan x, which is y. Copyrighted by Gabriel Tang B.Ed., B.c. Page 69. No olution tan sinsin x x for x and is within [, ] Calculator gives the c. cos cos d. cos 7 cos same decimal value. 6 Hence, cos interval allowed. cos 6 tantan x x for x R. cos cos x x for 0 x 7 and 6 is not within [0, ] Calculator gives the tan f. tan same decimal value. tan Hence, interval allowed. tan 6 tan tan tan x x for x and is within [, ] g. sin cos ½ h. tan sin 6 sin tan ½ 0.66 rad The range of y tan x is The domain of y cos x [, ], and 0.66 rad is [, ], and ½ is within is within this interval. this interval. The domain of y cos x is [, ]. is outside this interval. Calculator gives the same decimal value. Hence, interval allowed. Calculator does not give the same decimal value. 7 Hence, 6 is not within the interval allowed. Calculator gives the same decimal value. Hence, interval allowed.
4 Chapter 8: nalytical Trigonometry lgebra Example : Evaluate cos hyp θ adj x sin opp by sketching a triangle. Recall OH CH TO opp sin θ hyp Example : Rewrite sintan x as an algebraic expression in x adj cos θ hyp opp tan θ adj When we simplify sin, we are solving for the angle θ. Therefore, we can let θ sin cos sin cos θ adj x cos θ hyp Using Pythagorean Theorem: x + x 69 x x cos sin We can let θ tan opp x x, which can be rewrite as tan θ x. adj. Now drawing and labelling the triangle, we can find an expression for the hypotenuse. hyp x + hyp x + Going back to the original expression, sintan opp x x sin θ hyp x + hyp? θ adj sintan x x x + opp x 8- ssignment: pg #,,,, 7, 9,,,, 7 Honours #9,,,, 8 Page 70. Copyrighted by Gabriel Tang B.Ed., B.c.
5 lgebra Chapter 8: nalytical Trigonometry 8- Trigonometric Equations ome Basic Trigonometric Definitions and Identities proven equations tan θ cot θ sinθ cosθ tanθ cosθ sinθ Using regular factoring technique common factor and factoring trinomials, simplifying rational expressions, substituting with basic trigonometric definitions listed above, and referring to the unit circle, we can solve for the solution of various type of trigonometric equations. Example : Find the solutions for 0 x <. csc θ sinθ c. cos x + cos x + 0 d. cscx sec θ cos θ + sin θ Copyrighted by Gabriel Tang B.Ed., B.c. Page 7. cosθ a. tan x 0 b. sin x cos x sin x, tan x sin x cos x sin x 0 sin x cos x 0 x or sin x 0 cos x 0 cos x Let a cos x a + a + 0 a + a + 0 a + 0 a + 0 a ½ a cos x ½ cos x,,, 0, x 0 or Let a x a x csc a sin a, 0 x No solutions sin a ½ 7 9 a 6, 6,,,, x 6, 6,,,, , 0 0 x < solve x in rev 0 a < 0 a < 6 solve a in rev 6, x or x 7 9 x,,,,, e. sin x f. sin x 9 cos x round to the th decimal place 0 sin x 9cos x [ubstitute identity: x Let a 0 x < solve x in rev 0 cos x 9cos x sin x cos x] a x 0 a < 0 a < solve a in ½ rev, y sin a sin a ½ sin a ± sin a ± ± a, x 6, x, x,,, 0 cos x 9cos x 0 cos x 9cos x + cos x + 9cos x 0 cos x cos x + 0, cos x 0 cos x + 0 cos x cos x cos x x No olution x.69 rad,.97 rad, y
6 Chapter 8: nalytical Trigonometry lgebra Example : Using a graphing calculator, determine the solutions of sin x x. In Radian Mode, enter each side of the equation as Y and Y in Y screen ZOOM elect ZTrig Run Intersect twice from nd TRCE x ±.0 Example : The height of a tidal wave above the average sea level is related to time by the function, dt. sin 0.6t. +., where d represents depth in metres, above sea level and t is the time in hours t 0 means midnight. When within one whole day is the tidal wave at a depth of at least m for a ship to dock safely? mplitude a. m Vert. Disp. c. Maximum. m +. m.9 m Minimum. m. m 0.9 m ince we are given the output m to find the input time, we need to enter two equations and find the intersecting points. In Radian Mode, enter the equations as Y and Y in Y screen Water depth will be above m at 0.6 hrs < t < 8.7 hrs and. hrs < t < 0.96 hrs : M < t < 8: M and : PM < t < 8:6 PM 8- ssignment: pg #,, 7, 9,,,,, b, b, 7, 79, 8; Honour # Page 7. Copyrighted by Gabriel Tang B.Ed., B.c.
SM 2. Date: Section: Objective: The Pythagorean Theorem: In a triangle, or
SM 2 Date: Section: Objective: The Pythagorean Theorem: In a triangle, or. It doesn t matter which leg is a and which leg is b. The hypotenuse is the side across from the right angle. To find the length
More information1.6 Applying Trig Functions to Angles of Rotation
wwwck1org Chapter 1 Right Triangles and an Introduction to Trigonometry 16 Applying Trig Functions to Angles of Rotation Learning Objectives Find the values of the six trigonometric functions for angles
More informationTrigonometry I. Exam 0
Trigonometry I Trigonometry Copyright I Standards 006, Test Barry Practice Mabillard. Exam 0 www.math0s.com 1. The minimum and the maximum of a trigonometric function are shown in the diagram. a) Write
More informationA lg e b ra II. Trig o n o m e tric F u n c tio
1 A lg e b ra II Trig o n o m e tric F u n c tio 2015-12-17 www.njctl.org 2 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector
More informationAppendix D Trigonometry
Math 151 c Lynch 1 of 8 Appendix D Trigonometry Definition. Angles can be measure in either degree or radians with one complete revolution 360 or 2 rad. Then Example 1. rad = 180 (a) Convert 3 4 into degrees.
More informationMATH 1113 Exam 3 Review. Fall 2017
MATH 1113 Exam 3 Review Fall 2017 Topics Covered Section 4.1: Angles and Their Measure Section 4.2: Trigonometric Functions Defined on the Unit Circle Section 4.3: Right Triangle Geometry Section 4.4:
More informationAlgebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions
Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions 2015-12-17 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Co-terminal angles Arc
More informationPre Calculus Worksheet: Fundamental Identities Day 1
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationUNIT 5 TRIGONOMETRY Lesson 5.4: Calculating Sine, Cosine, and Tangent. Instruction. Guided Practice 5.4. Example 1
Lesson : Calculating Sine, Cosine, and Tangent Guided Practice Example 1 Leo is building a concrete pathway 150 feet long across a rectangular courtyard, as shown in the following figure. What is the length
More informationFunction f. Function f -1
Page 1 REVIEW (1.7) What is an inverse function? Do all functions have inverses? An inverse function, f -1, is a kind of undoing function. If the initial function, f, takes the element a to the element
More information1. Let be a point on the terminal side of θ. Find the 6 trig functions of θ. (Answers need not be rationalized). b. P 1,3. ( ) c. P 10, 6.
Q. Right Angle Trigonometry Trigonometry is an integral part of AP calculus. Students must know the basic trig function definitions in terms of opposite, adjacent and hypotenuse as well as the definitions
More informationTrigonometry is concerned with the connection between the sides and angles in any right angled triangle.
Trigonometry Obj: I can to use trigonometry to find unknown sides and unknown angles in a triangle. Trigonometry is concerned with the connection between the sides and angles in any right angled triangle.
More informationUnit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses.
Unit 2: Trigonometry This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Pythagorean Theorem Recall that, for any right angled triangle, the square
More informationLesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231
1 Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231 What is Trigonometry? 2 It is defined as the study of triangles and the relationships between their sides and the angles between these sides.
More informationAlgebra II Trigonometric Functions
Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc
More information1. The circle below is referred to as a unit circle. Why is this the circle s name?
Right Triangles and Coordinates on the Unit Circle Learning Task: 1. The circle below is referred to as a unit circle. Why is this the circle s name? Part I 2. Using a protractor, measure a 30 o angle
More informationA trigonometric ratio is a,
ALGEBRA II Chapter 13 Notes The word trigonometry is derived from the ancient Greek language and means measurement of triangles. Section 13.1 Right-Triangle Trigonometry Objectives: 1. Find the trigonometric
More informationPrecalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant Practice Problems. Questions
Questions 1. Describe the graph of the function in terms of basic trigonometric functions. Locate the vertical asymptotes and sketch two periods of the function. y = 3 tan(x/2) 2. Solve the equation csc
More informationThe diagram above shows a sketch of the curve C with parametric equations
1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate
More informationUnit 3 Trigonometry. 3.4 Graph and analyze the trigonometric functions sine, cosine, and tangent to solve problems.
1 General Outcome: Develop trigonometric reasoning. Specific Outcomes: Unit 3 Trigonometry 3.1 Demonstrate an understanding of angles in standard position, expressed in degrees and radians. 3. Develop
More informationRight Triangle Trigonometry Definitions (Instructor Notes)
Right Triangle Trigonometry Definitions (Instructor Notes) This activity is designed for a 50 min. class. Materials: Triangles Print out the last 10 pages of this document. It helps to use different colors
More informationTrigonometry Ratios. For each of the right triangles below, the labelled angle is equal to 40. Why then are these triangles similar to each other?
Name: Trigonometry Ratios A) An Activity with Similar Triangles Date: For each of the right triangles below, the labelled angle is equal to 40. Why then are these triangles similar to each other? Page
More informationTrigonometric Functions of Any Angle
Trigonometric Functions of Any Angle MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of any angle,
More informationDAY 1 - GEOMETRY FLASHBACK
DAY 1 - GEOMETRY FLASHBACK Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse sin θ = opp. hyp. cos θ = adj. hyp. tan θ = opp. adj. Tangent Opposite Adjacent a 2 + b 2 = c 2 csc θ = hyp. opp. sec θ =
More informationSecondary Math 3- Honors. 7-4 Inverse Trigonometric Functions
Secondary Math 3- Honors 7-4 Inverse Trigonometric Functions Warm Up Fill in the Unit What You Will Learn How to restrict the domain of trigonometric functions so that the inverse can be constructed. How
More information8.6 Other Trigonometric Functions
8.6 Other Trigonometric Functions I have already discussed all the trigonometric functions and their relationship to the sine and cosine functions and the x and y coordinates on the unit circle, but let
More informationMath 144 Activity #2 Right Triangle Trig and the Unit Circle
1 p 1 Right Triangle Trigonometry Math 1 Activity #2 Right Triangle Trig and the Unit Circle We use right triangles to study trigonometry. In right triangles, we have found many relationships between the
More informationTriangle Trigonometry
Honors Finite/Brief: Trigonometry review notes packet Triangle Trigonometry Right Triangles All triangles (including non-right triangles) Law of Sines: a b c sin A sin B sin C Law of Cosines: a b c bccos
More informationC. HECKMAN TEST 2A SOLUTIONS 170
C HECKMN TEST SOLUTIONS 170 (1) [15 points] The angle θ is in Quadrant IV and tan θ = Find the exact values of 5 sin θ, cos θ, tan θ, cot θ, sec θ, and csc θ Solution: point that the terminal side of the
More informationA Quick Review of Trigonometry
A Quick Review of Trigonometry As a starting point, we consider a ray with vertex located at the origin whose head is pointing in the direction of the positive real numbers. By rotating the given ray (initial
More information4-6 Inverse Trigonometric Functions
Find the exact value of each expression, if it exists. 29. The inverse property applies, because lies on the interval [ 1, 1]. Therefore, =. 31. The inverse property applies, because lies on the interval
More informationName Trigonometric Functions 4.2H
TE-31 Name Trigonometric Functions 4.H Ready, Set, Go! Ready Topic: Even and odd functions The graphs of even and odd functions make it easy to identify the type of function. Even functions have a line
More informationMA 154 PRACTICE QUESTIONS FOR THE FINAL 11/ The angles with measures listed are all coterminal except: 5π B. A. 4
. If θ is in the second quadrant and sinθ =.6, find cosθ..7.... The angles with measures listed are all coterminal except: E. 6. The radian measure of an angle of is: 7. Use a calculator to find the sec
More information9.1 Use Trigonometry with Right Triangles
9.1 Use Trigonometry with Right Triangles Use the Pythagorean Theorem to find missing lengths in right triangles. Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle
More informationTrigonometry and the Unit Circle. Chapter 4
Trigonometry and the Unit Circle Chapter 4 Topics Demonstrate an understanding of angles in standard position, expressed in degrees and radians. Develop and apply the equation of the unit circle. Solve
More informationYou found and graphed the inverses of relations and functions. (Lesson 1-7)
You found and graphed the inverses of relations and functions. (Lesson 1-7) LEQ: How do we evaluate and graph inverse trigonometric functions & find compositions of trigonometric functions? arcsine function
More informationYoungstown State University Trigonometry Final Exam Review (Math 1511)
Youngstown State University Trigonometry Final Exam Review (Math 1511) 1. Convert each angle measure to decimal degree form. (Round your answers to thousandths place). a) 75 54 30" b) 145 18". Convert
More informationUse the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before.
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationReview Notes for the Calculus I/Precalculus Placement Test
Review Notes for the Calculus I/Precalculus Placement Test Part 9 -. Degree and radian angle measures a. Relationship between degrees and radians degree 80 radian radian 80 degree Example Convert each
More informationPre-calculus Chapter 4 Part 1 NAME: P.
Pre-calculus NAME: P. Date Day Lesson Assigned Due 2/12 Tuesday 4.3 Pg. 284: Vocab: 1-3. Ex: 1, 2, 7-13, 27-32, 43, 44, 47 a-c, 57, 58, 63-66 (degrees only), 69, 72, 74, 75, 78, 79, 81, 82, 86, 90, 94,
More informationReview of Trigonometry
Worksheet 8 Properties of Trigonometric Functions Section Review of Trigonometry This section reviews some of the material covered in Worksheets 8, and The reader should be familiar with the trig ratios,
More informationObjectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using
Ch 13 - RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric
More informationThe Sine and Cosine Functions
Concepts: Graphs of Tangent, Cotangent, Secant, and Cosecant. We obtain the graphs of the other trig functions by thinking about how they relate to the sin x and cos x. The Sine and Cosine Functions Page
More informationWarm Up: please factor completely
Warm Up: please factor completely 1. 2. 3. 4. 5. 6. vocabulary KEY STANDARDS ADDRESSED: MA3A2. Students will use the circle to define the trigonometric functions. a. Define and understand angles measured
More informationFinding Angles and Solving Right Triangles NEW SKILLS: WORKING WITH INVERSE TRIGONOMETRIC RATIOS. Calculate each angle to the nearest degree.
324 MathWorks 10 Workbook 7.5 Finding Angles and Solving Right Triangles NEW SKILLS: WORKING WITH INVERSE TRIGONOMETRIC RATIOS The trigonometric ratios discussed in this chapter are unaffected by the size
More informationChapter 4/5 Part 1- Trigonometry in Radians
Chapter 4/5 Part - Trigonometry in Radians Lesson Package MHF4U Chapter 4/5 Part Outline Unit Goal: By the end of this unit, you will be able to demonstrate an understanding of meaning and application
More informationSec 4.1 Trigonometric Identities Basic Identities. Name: Reciprocal Identities:
Sec 4. Trigonometric Identities Basic Identities Name: Reciprocal Identities: Quotient Identities: sin csc cos sec csc sin sec cos sin tan cos cos cot sin tan cot cot tan Using the Reciprocal and Quotient
More informationAP Calculus Summer Review Packet
AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** Email anytime with questions: danna.seigle@henry.k1.ga.us Complex Fractions
More informationMathematics for Computer Graphics. Trigonometry
Mathematics for Computer Graphics Trigonometry Trigonometry...????? The word trigonometry is derived from the ancient Greek language and means measurement of triangles. trigonon triangle + metron measure
More informationMATHEMATICS FOR ENGINEERING TRIGONOMETRY
MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL SOME MORE RULES OF TRIGONOMETRY This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves
More informationMATH EXAM 1 - SPRING 2018 SOLUTION
MATH 140 - EXAM 1 - SPRING 018 SOLUTION 8 February 018 Instructor: Tom Cuchta Instructions: Show all work, clearly and in order, if you want to get full credit. If you claim something is true you must
More informationChapter 9: Right Triangle Trigonometry
Haberman MTH 11 Section I: The Trigonometric Functions Chapter 9: Right Triangle Trigonometry As we studied in Intro to the Trigonometric Functions: Part 1, if we put the same angle in the center of two
More informationSNAP Centre Workshop. Introduction to Trigonometry
SNAP Centre Workshop Introduction to Trigonometry 62 Right Triangle Review A right triangle is any triangle that contains a 90 degree angle. There are six pieces of information we can know about a given
More informationSolving Trigonometric Equations
OpenStax-CNX module: m49398 1 Solving Trigonometric Equations OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you
More informationTrigonometry Curriculum Guide Scranton School District Scranton, PA
Trigonometry Scranton School District Scranton, PA Trigonometry Prerequisite: Algebra II, Geometry, Algebra I Intended Audience: This course is designed for the student who has successfully completed Algebra
More information2.0 Trigonometry Review Date: Pythagorean Theorem: where c is always the.
2.0 Trigonometry Review Date: Key Ideas: The three angles in a triangle sum to. Pythagorean Theorem: where c is always the. In trigonometry problems, all vertices (corners or angles) of the triangle are
More informationAlgebra II. Slide 1 / 92. Slide 2 / 92. Slide 3 / 92. Trigonometry of the Triangle. Trig Functions
Slide 1 / 92 Algebra II Slide 2 / 92 Trigonometry of the Triangle 2015-04-21 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 92 Trigonometry of the Right Triangle Inverse
More informationUnit 2 Intro to Angles and Trigonometry
HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 1 Unit 2 Intro to Angles and Trigonometry This is a BASIC CALCULATORS ONLY unit. (2) Definition of an Angle (3) Angle Measurements & Notation (4) Conversions of
More informationName Student Activity
Open the TI-Nspire document Proofs_of_Identities.tns. An identity is an equation that is true for all values of the variables for which both sides of the equation are defined. In this activity, you will
More information4.1 Angles and Angle Measure. 1, multiply by
4.1 Angles and Angle Measure Angles can be measured in degrees or radians. Angle measures without units are considered to be in radians. Radian: One radian is the measure of the central angle subtended
More informationAdding vectors. Let s consider some vectors to be added.
Vectors Some physical quantities have both size and direction. These physical quantities are represented with vectors. A common example of a physical quantity that is represented with a vector is a force.
More informationIB SL Review Questions
I SL Review Questions. Solve the equation 3 cos x = 5 sin x, for x in the interval 0 x 360, giving your answers to the nearest degree.. Given that sin θ =, cos θ = 3 and 0 < θ < 360, find the value of
More information: Find the values of the six trigonometric functions for θ. Special Right Triangles:
ALGEBRA 2 CHAPTER 13 NOTES Section 13-1 Right Triangle Trig Understand and use trigonometric relationships of acute angles in triangles. 12.F.TF.3 CC.9- Determine side lengths of right triangles by using
More informationCh. 2 Trigonometry Notes
First Name: Last Name: Block: Ch. Trigonometry Notes.0 PRE-REQUISITES: SOLVING RIGHT TRIANGLES.1 ANGLES IN STANDARD POSITION 6 Ch..1 HW: p. 83 #1,, 4, 5, 7, 9, 10, 8. - TRIGONOMETRIC FUNCTIONS OF AN ANGLE
More informationWarm-Up Up Exercises. Use this diagram for Exercises If PR = 12 and m R = 19, find p. ANSWER If m P = 58 and r = 5, find p.
Warm-Up Up Exercises Use this diagram for Exercises 1 4. 1. If PR = 12 and m R = 19, find p. ANSWER 11.3 2. If m P = 58 and r = 5, find p. ANSWER 8.0 Warm-Up Up Exercises Use this diagram for Exercises
More informationChapter 3: Right Triangle Trigonometry
10C Name: Chapter 3: Right Triangle Trigonometry 3.1 The Tangent Ratio Outcome : Develop and apply the tangent ratio to solve problems that involve right triangles. Definitions: Adjacent side: the side
More informationMath 144 Activity #3 Coterminal Angles and Reference Angles
144 p 1 Math 144 Activity #3 Coterminal Angles and Reference Angles For this activity we will be referring to the unit circle. Using the unit circle below, explain how you can find the sine of any given
More informationTrigonometry Summer Assignment
Name: Trigonometry Summer Assignment Due Date: The beginning of class on September 8, 017. The purpose of this assignment is to have you practice the mathematical skills necessary to be successful in Trigonometry.
More informationMath for Geometric Optics
Algebra skills Math for Geometric Optics general rules some common types of equations units problems with several variables (substitution) Geometry basics Trigonometry Pythagorean theorem definitions,
More informationHW. Pg. 334 #1-9, 11, 12 WS. A/ Angles in Standard Position: Terminology: Initial Arm. Terminal Arm. Co-Terminal Angles. Quadrants
MCR 3UI Introduction to Trig Functions Date: Lesson 6.1 A/ Angles in Standard Position: Terminology: Initial Arm HW. Pg. 334 #1-9, 11, 1 WS Terminal Arm Co-Terminal Angles Quadrants Related Acute Angles
More informationMath-3 Lesson 6-1. Trigonometric Ratios for Right Triangles and Extending to Obtuse angles.
Math-3 Lesson 6-1 Trigonometric Ratios for Right Triangles and Extending to Obtuse angles. Right Triangle: has one angle whose measure is. 90 The short sides of the triangle are called legs. The side osite
More informationA lg e b ra II. Trig o n o m e try o f th e Tria n g le
1 A lg e b ra II Trig o n o m e try o f th e Tria n g le 2015-04-21 www.njctl.org 2 Trig Functions click on the topic to go to that section Trigonometry of the Right Triangle Inverse Trig Functions Problem
More informationCCNY Math Review Chapters 5 and 6: Trigonometric functions and graphs
Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions CCNY Math Review Chapters 5 and 6: Trigonometric functions and
More informationUNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios Instruction
Prerequisite Skills This lesson requires the use of the following skills: manipulating the Pythagorean Theorem given any two sides of a right triangle defining the three basic trigonometric ratios (sine,
More informationUnit 3, Lesson 1.3 Special Angles in the Unit Circle
Unit, Lesson Special Angles in the Unit Circle Special angles exist within the unit circle For these special angles, it is possible to calculate the exact coordinates for the point where the terminal side
More informationGanado Unified School District Pre-Calculus 11 th /12 th Grade
Ganado Unified School District Pre-Calculus 11 th /12 th Grade PACING Guide SY 2016-2017 Timeline & Resources Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to highlight
More information4.1: Angles & Angle Measure
4.1: Angles & Angle Measure In Trigonometry, we use degrees to measure angles in triangles. However, degree is not user friendly in many situations (just as % is not user friendly unless we change it into
More informationMath 144 Activity #7 Trigonometric Identities
44 p Math 44 Activity #7 Trigonometric Identities What is a trigonometric identity? Trigonometric identities are equalities that involve trigonometric functions that are true for every single value of
More informationTrigonometry. 9.1 Radian and Degree Measure
Trigonometry 9.1 Radian and Degree Measure Angle Measures I am aware of three ways to measure angles: degrees, radians, and gradians. In all cases, an angle in standard position has its vertex at the origin,
More informationGanado Unified School District Trigonometry/Pre-Calculus 12 th Grade
Ganado Unified School District Trigonometry/Pre-Calculus 12 th Grade PACING Guide SY 2014-2015 Timeline & Resources Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to
More information5.5 Multiple-Angle and Product-to-Sum Formulas
Section 5.5 Multiple-Angle and Product-to-Sum Formulas 87 5.5 Multiple-Angle and Product-to-Sum Formulas Multiple-Angle Formulas In this section, you will study four additional categories of trigonometric
More informationUnit Circle. Project Response Sheet
NAME: PROJECT ACTIVITY: Trigonometry TOPIC Unit Circle GOALS MATERIALS Explore Degree and Radian Measure Explore x- and y- coordinates on the Unit Circle Investigate Odd and Even functions Investigate
More informationChapter 7: Analytic Trigonometry
Chapter 7: Analytic Trigonometry 7. Trigonometric Identities Below are the basic trig identities discussed in previous chapters. Reciprocal csc(x) sec(x) cot(x) sin(x) cos(x) tan(x) Quotient sin(x) cos(x)
More informationTrigonometry Review Day 1
Name Trigonometry Review Day 1 Algebra II Rotations and Angle Terminology II Terminal y I Positive angles rotate in a counterclockwise direction. Reference Ray Negative angles rotate in a clockwise direction.
More information3.0 Trigonometry Review
3.0 Trigonometry Review In trigonometry problems, all vertices (corners or angles) of the triangle are labeled with capital letters. The right angle is usually labeled C. Sides are usually labeled with
More informationMAC Learning Objectives. Learning Objectives (Cont.) Module 2 Acute Angles and Right Triangles
MAC 1114 Module 2 Acute Angles and Right Triangles Learning Objectives Upon completing this module, you should be able to: 1. Express the trigonometric ratios in terms of the sides of the triangle given
More information3.1 The Inverse Sine, Cosine, and Tangent Functions
3.1 The Inverse Sine, Cosine, and Tangent Functions Let s look at f(x) = sin x The domain is all real numbers (which will represent angles). The range is the set of real numbers where -1 sin x 1. However,
More informationUnit 7: Trigonometry Part 1
100 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( α ) = d) Cosecant csc( α ) = α Adjacent Opposite b) Cosine cos( α ) = e) Secant sec( α ) = c) Tangent f) Cotangent tan(
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Precalculus CP Final Exam Review - 01 Name Date: / / MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle in degrees to radians. Express
More informationVector Addition. Qty Item Part Number 1 Force Table ME-9447B 1 Mass and Hanger Set ME Carpenter s level 1 String
rev 05/2018 Vector Addition Equipment List Qty Item Part Number 1 Force Table ME-9447B 1 Mass and Hanger Set ME-8979 1 Carpenter s level 1 String Purpose The purpose of this lab is for the student to gain
More informationPART I: NO CALCULATOR (64 points)
Math 10 Trigonometry 11 th edition Lial, Hornsby, Schneider, and Daniels Practice Midterm (Ch. 1-) PART I: NO CALCULATOR (6 points) (.1,.,.,.) Match each graph with one of the basic circular functions
More informationWalt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC
Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be
More informationSemester Exam Review. 1. Give a real life example of a situation that can be modeled with a periodic function.
Trigonometry Semester Exam Review Name: 1. Give a real life example of a situation that can be modeled with a periodic function.. As a child goes up and down on a seesaw, his or her distance form the ground
More informationTrigonometric Ratios and Functions
Algebra 2/Trig Unit 8 Notes Packet Name: Date: Period: # Trigonometric Ratios and Functions (1) Worksheet (Pythagorean Theorem and Special Right Triangles) (2) Worksheet (Special Right Triangles) (3) Page
More informationSection 10.6 Right Triangle Trigonometry
153 Section 10.6 Right Triangle Trigonometry Objective #1: Understanding djacent, Hypotenuse, and Opposite sides of an acute angle in a right triangle. In a right triangle, the otenuse is always the longest
More informationSection 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.
1 Section 7.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint
More informationCommon Core Standards Addressed in this Resource
Common Core Standards Addressed in this Resource N-CN.4 - Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular
More informationto and go find the only place where the tangent of that
Study Guide for PART II of the Spring 14 MAT187 Final Exam. NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will
More informationGanado Unified School District #20 (Pre-Calculus 11th/12th Grade)
Ganado Unified School District #20 (Pre-Calculus 11th/12th Grade) PACING Guide SY 2018-2019 Timeline & Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to highlight a quantity
More informationUnit 6 Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1)
Unit 6 Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Find
More information