Inverse Trigonometric Functions
|
|
- Elaine Daniels
- 5 years ago
- Views:
Transcription
1 Inverse Trigonometric Functions. Inverse of a function f eists, if function is one-one and onto, i.e., bijective.. Trignometric functions are many-one functions but these become one-one, onto, if we restrict the domain of trignometric functions. We can say that inverse of trigonometric functions are defined within restricted domains of corresponding trigonometric functions.. Inverse of sin is denoted by sin (arc sin function). We also write as sin. Similary, other inverse trigonometric functions are given by cos, tan,sec,cot and. Note that sin sin and (sin ) sin. Also sin (sin ). 5. Table for domain and range of inverse trigonometrical functions : cosec. Functions Domain Range (Principal value Branch) y = sin y y = cos 0 y y = tan < < < y < y = cosec or y, y 0 y = sec or 0 y, y y = cot < < 0 < y < Properties of Inverse Trigonometric Functions. (i) sin (sin θ ) = θ, for all θ [, ] cos (cos θ ) = θ, for all θ [0, ] tan (tan θ ) = θ, for all θ [, ] (iv) cosec (cosec θ ) = θ, for all θ [, ], θ 0 (v) sec (sec θ ) = θ, for all θ [ θ, ], θ /
2 (vi) cot (cot ) θ = θ, for all θ [0, ]. (i) sin(sin ) =, for all [, ] cos(cos ) =, for all [, ] tan(tan ) =, for all R (iv) cosec(cosec ) =, for all (, ] [, ] (v) sec(sec ) =, for all (, ] [, ] (vi). (i) cot(cot ) sin =, for all R = cosec, for all (, ] [, ] = cos sec tan > + <, for all (, ] [, ] cot, for 0 cot, for 0. (i) sin ( ) = sin ( ), for all [, ] cos ( ) cos, = for all [, ] tan ( ) = tan,, for all R (iv) cosec ( ) = cosec,, for all (, ] [, ] (v) sec ( ) = sec,, for all (, ] [, ] (vi) cot ( ) = cot, for all R 5. (i) sin cos / + =, for all [, ] tan cot / + =, for all R sec + cosec = /, for all (, ] [, ] 6. (i) sin + sin y = { } y y sin +, if, y and + y or if y < 0 and + y > sin sin y { y y } = sin, if, y and + y or if y < 0 and + y > 7. (i) cos + cos y = { } y y cos, if, y and + y 0
3 cos cos y = { } y y cos, if, y and + y 0 y 8. (i) tan tan y tan + + = y, if y < y tan tan y tan, = + y if y > 9. (i) sin = sin ( ), if cos = cos ( ), if 0 tan = tan, if < < 0. (i) sin = sin ( ), if cos = cos ( ), if tan = tan, if < <. (i) + tan = sin, tan = cos, + if if 0 <. (i) sin = cos = tan = cot = sec cosec = cos = sin = tan = cot sec cosec = =
4 tan = sin cos = + + = cot sec cosec + = + = Important substitution to simplify trigonometrical epressions involving inverse trigonometric functions : Epression Substitution a a + = a tan θ or = a cot θ = a sin θ or = a cosθ a = a secθ or = a cosec θ a + a or a a + = a cos θ Important Facts (i) If no branch of an inverse trigonometric function is mentioned, we mean the principal value branch of that function. sin sin or (sin ) and same holds true for other trigonometric functions also. If sin y sin = then and y are the elements of domain and range of principal value branch of respectively. i.e., [, ] and y,, Similar fact is also applicable for other inverse trigonometric functions. Question for Practice Very Short Answer Type Questions ( Mark). Evaluate : sin sin. Using principal value, evaluate : cos cos + sin sin
5 . Show that sin ( ) = sin.. Solve for = > + :tan tan, What is the principal value of tan ( )? 6. What is the principal value of 7. Using principal value evaluate cos? sin sin Write the principal value of 9. Write the principal value of cos cos 6. 7 tan tan. 0. What is principal value of sin.. What is domain of the function sin. Write principal value of sec ( ).?. Write the principal value of cos sin.. Find the principal value of tan sec ( ). 5. Write the value of cot(tan a + cot a). 6. Write the principal value of tan () + cos. 7. Write the value of tan tan Write the principal vlaue of tan ( ) cot ( ). 9. Write the value of tan sin cos. 0. Write the principal value of 9 tan tan. 8
6 . Write the value of sin sin. 5 Very Short Answer Type Questions ( Mark). Prove that : tan + cos + tan cos = b b a a b. a. Solve for 8 : tan ( + ) + tan ( ) = tan.. Prove that : tan + tan = tan Prove that : 5. Prove that : 6 sin + cos + tan. 5 6 tan + tan + tan + tan = Solve for 7. Solve for + = : tan tan. :tan tan. + + = + 8. Prove that : tan tan tan + = Prove that : tan + tan + tan = Prove that : cot (7) + cot (8) + cot (8) = cot ().. Prove that : + = + < < tan tan,0.. Solve for = > + :tan tan 0, 0.. Prove that :. Prove that : 5 6 sin + sin + sin. 5 5 tan + tan = cos 9 5.
7 sin sin 5. Prove that : cot + + =, 0,. + sin sin 6. Solve for : tan (cos ) = tan (cosec ). 7. Solve for + = :cos tan Prove that : + tan = cos, (0,). 56 Prove that : cos + sin = sin Prove that following : a a b + b + = b a tan cos tan cos. 0. Prove that following : Prove the following : tan + tan = tan. + cos[tan {sin(cot )}] =. + sin sin. Prove the following : cot + + =, 0,. + sin sin Find the value of y tan tan +. y + y. Prove the following : = 8 sin sin. Solve the following equation for. tan = tan, > Prove that : + tan = cos,. + +
8 cos r. Prove that : tan r r =,,. sin 8 5 Prove that : sin + sin = cos Prove the following : cos sin + cot = Prove that : sin = sin + cos Solve for : tan (sin ) = tan (sec ),. y 7. Find the value of the following : tan sin cos,, y 0 and y. + < > < + + y Prove that : tan + tan + tan = Show that : tan sin =. Solve the following equation : cos(tan ) = sin cot. 9. If cos(tan ) = sin cot, then find the value of. If y = cot ( cos ) tan ( cos ), then prove that sin y = tan. ANSWERS Very Short Answer Questions :
9 Short Answer Questions :. 8.. [, ] ± , y y 8. = 9. =±
Preview from Notesale.co.uk Page 2 of 42
. CONCEPTS & FORMULAS. INTRODUCTION Radian The angle subtended at centre of a circle by an arc of length equal to the radius of the circle is radian r o = o radian r r o radian = o = 6 Positive & Negative
More informationSection 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
More informationTHE COMPOUND ANGLE IDENTITIES
TRIGONOMETRY THE COMPOUND ANGLE IDENTITIES Question 1 Prove the validity of each of the following trigonometric identities. a) sin x + cos x 4 4 b) cos x + + 3 sin x + 2cos x 3 3 c) cos 2x + + cos 2x cos
More informationUsing 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
More informationPart r A A A 1 Mark Part r B B B 2 Marks Mark P t ar t t C C 5 Mar M ks Part r E 4 Marks Mark Tot To a t l
Part Part P t Part Part Total A B C E 1 Mark 2 Marks 5 Marks M k 4 Marks CIRCLES 12 Marks approximately Definition ; A circle is defined as the locus of a point which moves such that its distance from
More information4-3 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
More informationFundamental Trigonometric Identities
Fundamental Trigonometric Identities MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: recognize and write the fundamental trigonometric
More informationMATH section 3.1 Maximum and Minimum Values Page 1 of 7
MATH section. Maimum and Minimum Values Page of 7 Definition : Let c be a number in the domain D of a function f. Then c ) is the Absolute maimum value of f on D if ) c f() for all in D. Absolute minimum
More informationNOTES ON INVERSE TRIGONOMETRIC FUNCTIONS
NOTES ON INVERSE TRIGONOMETRIC FUNCTIONS MATH 5 (S). Definitions of Inverse Trigonometric Functions () y = sin or y = arcsin is the inverse function of y = sin on [, ]. The omain of y = sin = arcsin is
More information4.3 Inverse Trigonometric Properties
www.ck1.org Chapter. Inverse Trigonometric Functions. Inverse Trigonometric Properties Learning Objectives Relate the concept of inverse functions to trigonometric functions. Reduce the composite function
More informationMath 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
More informationare its positions as it is moving in anti-clockwise direction through angles 1, 2, 3 &
T: Introduction: The word trigonometry is derived from Greek words trigon meaning a triangle and metron meaning measurement. In this branch of mathematics, we study relationship of sides and angles of
More informationMATHEMATICS. MINIMUM LEVEL MATERIAL for CLASS XII Project Planned By. Honourable Shri D. Manivannan Deputy Commissioner,KVS RO Hyderabad
MATHEMATICS MINIMUM LEVEL MATERIAL for CLASS XII 06 7 Project Planned By Honourable Shri D. Manivannan Deputy Commissioner,KVS RO Hyderabad Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist
More informationINVERSE TRIGONOMETRIC FUNCTIONS. Mathematics, in general, is fundamentally the science of self-evident things. FELIX KLEIN
. Introduction Mathematics, in general, is fundamentally the science of self-evident things. FELIX KLEIN In Chapter, we have studied that the inverse of a function f, denoted by f, eists if f is one-one
More information2 nd ORDER O.D.E.s SUBSTITUTIONS
nd ORDER O.D.E.s SUBSTITUTIONS Question 1 (***+) d y y 8y + 16y = d d d, y 0, Find the general solution of the above differential equation by using the transformation equation t = y. Give the answer in
More informationC3 Exam Workshop 2. Workbook. 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2
C3 Exam Workshop 2 Workbook 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2 π. Give the value of α to 3 decimal places. (b) Hence write down the minimum value of 7 cos
More informationINVERSE TRIGONOMETRIC FUNCTIONS
Inverse Trigonometric MODULE - IV 8 INVERSE TRIGONOMETRIC FUNCTIONS In the previous lesson, you have studied the definition of a function and different kinds of functions. We have defined inverse function.
More information2.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
More informationSection 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
More informationINVERSE TRIGONOMETRIC FUNCTIONS Notes
Inverse Trigonometric s MODULE - VII INVERSE TRIGONOMETRIC FUNCTIONS In the previous lesson, you have studied the definition of a function and different kinds of functions. We have defined inverse function.
More information5.3 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.
More informationThe function is a periodic function. That means that the functions repeats its values in regular intervals, which we call the period.
Section 5.4 - Inverse Trigonometric Functions The Inverse Sine Function Consider the graph of the sine function f ( x) sin( x). The function is a periodic function. That means that the functions repeats
More informationTrigonometric Identities Exam Questions
Trigonometric Identities Exam Questions Name: ANSWERS January 01 January 017 Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible
More informationAs we know, the three basic trigonometric functions are as follows: Figure 1
Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an
More informationDISCOVERING THE PYTHAGOREAN IDENTITIES LEARNING TASK:
Name: Class Period: DISCOVERING THE PYTHAGOREAN IDENTITIES LEARNING TASK: An identity is an equation that is valid for all values of the variable for which the epressions in the equation are defined. You
More informationSect 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
More informationMath 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
More information6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities
Chapter 6: Trigonometric Identities 1 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages 290 298 6.3 Proving Identities Pages 309 315 Measure of
More informationDifferential Equaitons Equations
Welcome to Multivariable Calculus / Dierential Equaitons Equations The Attached Packet is or all students who are planning to take Multibariable Multivariable Calculus/ Dierential Equations in the all.
More informationLecture 5: Inverse Trigonometric Functions
Lecture 5: Inverse Trigonometric Functions 5 The inverse sine function The function f(x = sin(x is not one-to-one on (,, but is on [ π, π Moreover, f still has range [, when restricte to this interval
More information16 Inverse Trigonometric Functions
6 Inverse Trigonometric Functions Concepts: Restricting the Domain of the Trigonometric Functions The Inverse Sine Function The Inverse Cosine Function The Inverse Tangent Function Using the Inverse Trigonometric
More informationHonors 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
More informationMath Calculus II Homework # Due Date Solutions
Math 35 - Calculus II Homework # - 007.08.3 Due Date - 007.09.07 Solutions Part : Problems from sections 7.3 and 7.4. Section 7.3: 9. + d We will use the substitution cot(θ, d csc (θ. This gives + + cot
More informationPre- 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
More informationRecapitulation of Mathematics
Unit I Recapitulation of Mathematics Basics of Differentiation Rolle s an Lagrange s Theorem Tangent an Normal Inefinite an Definite Integral Engineering Mathematics I Basics of Differentiation CHAPTER
More informationFundamentals of Mathematics (MATH 1510)
Fundamentals of Mathematics () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University March 14-18, 2016 Outline 1 2 s An angle AOB consists of two rays R 1 and R
More informationCalculus 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
More informationTRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)
TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents
More informationMethods of Integration
Methods of Integration Essential Formulas k d = k +C sind = cos +C n d = n+ n + +C cosd = sin +C e d = e +C tand = ln sec +C d = ln +C cotd = ln sin +C + d = tan +C lnd = ln +C secd = ln sec + tan +C cscd
More information6.5 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
More informationMATH 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
More information(ii) y = ln 1 ] t 3 t x x2 9
Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside
More informationCK- 12 Algebra II with Trigonometry Concepts 1
14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.
More information1. 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 ( ) = (
More informationMore 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
More informationSPM Past Year Questions : AM Form 5 Chapter 5 Trigonometric Functions
SPM 1993 SPM PAST YEAR QUESTIONS ADDITIONAL MATHEMATICS FORM 5 CHAPTER 5 : TRIGONOMETRIC FUNCTIONS 1. Solve the equation sec x = 3 tan x for 0 x 360. [5 marks]. Given that tan θ = 1, without using a calculator,
More informationADDITONAL MATHEMATICS
ADDITONAL MATHEMATICS 00 0 CLASSIFIED TRIGONOMETRY Compiled & Edited B Dr. Eltaeb Abdul Rhman www.drtaeb.tk First Edition 0 5 Show that cosθ + + cosθ = cosec θ. [3] 0606//M/J/ 5 (i) 6 5 4 3 0 3 4 45 90
More information3.5 Derivatives of Trig Functions
3.5 Derivatives of Trig Functions Problem 1 (a) Suppose we re given the right triangle below. Epress sin( ) and cos( ) in terms of the sides of the triangle. sin( ) = B C = B and cos( ) = A C = A (b) Suppose
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.1 Trigonometric Identities Copyright Cengage Learning. All rights reserved. Objectives Simplifying Trigonometric Expressions Proving
More informationEXAM. Practice for Second Exam. Math , Fall Nov 4, 2003 ANSWERS
EXAM Practice for Second Eam Math 135-006, Fall 003 Nov 4, 003 ANSWERS i Problem 1. In each part, find the integral. A. d (4 ) 3/ Make the substitution sin(θ). d cos(θ) dθ. We also have Then, we have d/dθ
More informationHALF SYLLABUS TEST. Topics : Ch 01 to Ch 08
Ma Marks : Topics : Ch to Ch 8 HALF SYLLABUS TEST Time : Minutes General instructions : (i) All questions are compulsory (ii) Please check that this question paper contains 9 questions (iii) Questions
More informationsin 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
More informationTrigonometry 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,
More informationChapter 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
More informationMTH 112: Elementary Functions
1/19 MTH 11: Elementary Functions Section 6.6 6.6:Inverse Trigonometric functions /19 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line crosses the graph of a 1
More informationFrom 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.
More informationDifferentiation 9F. tan 3x. Using the result. The first term is a product with. 3sec 3. 2 x and sec x. Using the product rule for the first term: then
Differentiation 9F a tan Using the reslt tan k k sec k sec 4tan Let tan ; then 4 sec and sec tan sec d tan tan The first term is a proct with and v tan and sec Using the proct rle for the first term: sec
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Final Revision CLASS XII CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION.
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Final Revision CLASS XII 2016 17 CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.),
More informationExact Equations. M(x,y) + N(x,y) y = 0, M(x,y) dx + N(x,y) dy = 0. M(x,y) + N(x,y) y = 0
Eact Equations An eact equation is a first order differential equation that can be written in the form M(, + N(,, provided that there eists a function ψ(, such that = M (, and N(, = Note : Often the equation
More informationInverse Relations. 5 are inverses because their input and output are switched. For instance: f x x. x 5. f 4
Inverse Functions Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the
More informationMath 141: Trigonometry Practice Final Exam: Fall 2012
Name: Math 141: Trigonometry Practice Final Eam: Fall 01 Instructions: Show all work. Answers without work will NOT receive full credit. Clearly indicate your final answers. The maimum possible score is
More informationTHE INVERSE TRIGONOMETRIC FUNCTIONS
THE INVERSE TRIGONOMETRIC FUNCTIONS Question 1 (**+) Solve the following trigonometric equation ( x ) π + 3arccos + 1 = 0. 1 x = Question (***) It is given that arcsin x = arccos y. Show, by a clear method,
More informationSome commonly encountered sets and their notations
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their
More informationExercise Set 4.1: Special Right Triangles and Trigonometric Ratios
Eercise Set.1: Special Right Triangles and Trigonometric Ratios Answer the following. 9. 1. If two sides of a triangle are congruent, then the opposite those sides are also congruent. 2. If two angles
More informationTrigonometry and modelling 7E
Trigonometry and modelling 7E sinq +cosq º sinq cosa + cosq sina Comparing sin : cos Comparing cos : sin Divide the equations: sin tan cos Square and add the equations: cos sin (cos sin ) since cos sin
More informationSection 6.1 Angles and Radian Measure Review If you measured the distance around a circle in terms of its radius, what is the unit of measure?
Section 6.1 Angles and Radian Measure Review If you measured the distance around a circle in terms of its radius, what is the unit of measure? In relationship to a circle, if I go half way around the edge
More informationMATH 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',
More informationSolutions for Trigonometric Functions of Any Angle
Solutions for Trigonometric Functions of Any Angle I. Souldatos Answers Problem... Consider the following triangle with AB = and AC =.. Find the hypotenuse.. Find all trigonometric numbers of angle B..
More informationAMB121F 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
More informationChapter 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
More informationMATH 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
More informationTrigonometry 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
More informationSection: I. u 4 du. (9x + 1) + C, 3
EXAM 3 MAT 168 Calculus II Fall 18 Name: Section: I All answers must include either supporting work or an eplanation of your reasoning. MPORTANT: These elements are considered main part of the answer and
More informationSHORT 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
More informationFUNDAMENTAL 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
More informationMPE Review Section II: Trigonometry
MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the
More informationMath 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
More information1. Evaluate the integrals. a. (9 pts) x e x/2 dx. Solution: Using integration by parts, let u = x du = dx and dv = e x/2 dx v = 2e x/2.
MATH 8 Test -SOLUTIONS Spring 4. Evaluate the integrals. a. (9 pts) e / Solution: Using integration y parts, let u = du = and dv = e / v = e /. Then e / = e / e / e / = e / + e / = e / 4e / + c MATH 8
More informationLesson-3 TRIGONOMETRIC RATIOS AND IDENTITIES
Lesson- TRIGONOMETRIC RATIOS AND IDENTITIES Angle in trigonometry In trigonometry, the measure of an angle is the amount of rotation from B the direction of one ray of the angle to the other ray. Angle
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.6 Inverse Functions and Logarithms In this section, we will learn about: Inverse functions and logarithms. INVERSE FUNCTIONS The table gives data from an experiment
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More informationJUST THE MATHS SLIDES NUMBER 3.1. TRIGONOMETRY 1 (Angles & trigonometric functions) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 3.1 TRIGONOMETRY 1 (Angles & trigonometric functions) by A.J.Hobson 3.1.1 Introduction 3.1.2 Angular measure 3.1.3 Trigonometric functions UNIT 3.1 - TRIGONOMETRY 1 - ANGLES
More informationTrigonometric substitutions (8.3).
Review for Eam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Eam covers: 7.4, 7.6, 7.7, 8-IT, 8., 8.2. Solving differential equations
More informationLesson 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
More information1 Functions and Inverses
October, 08 MAT86 Week Justin Ko Functions and Inverses Definition. A function f : D R is a rule that assigns each element in a set D to eactly one element f() in R. The set D is called the domain of f.
More informationLesson 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
More information*n23494b0220* C3 past-paper questions on trigonometry. 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (2)
C3 past-paper questions on trigonometry physicsandmathstutor.com June 005 1. (a) Given that sin θ + cos θ 1, show that 1 + tan θ sec θ. (b) Solve, for 0 θ < 360, the equation tan θ + secθ = 1, giving your
More informationSection 5.4 The Other Trigonometric Functions
Section 5.4 The Other Trigonometric Functions Section 5.4 The Other Trigonometric Functions In the previous section, we defined the e and coe functions as ratios of the sides of a right triangle in a circle.
More informationInverse 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)
More informationAnalytic 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
More informationThese items need to be included in the notebook. Follow the order listed.
* Use the provided sheets. * This notebook should be your best written work. Quality counts in this project. Proper notation and terminology is important. We will follow the order used in class. Anyone
More information2. Pythagorean Theorem:
Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications 1. A triangle in which one angle is a right angle (90 0 ) is called a. The side opposite the right angle
More informationInverse Functions and Trigonometric Equations - Solution Key
Inverse Functions and Trigonometric Equations - Solution Key CK Editor Say Thanks to the Authors Click http://www.ck.org/saythanks (No sign in required To access a customizable version of this book, as
More informationDerivatives of Trigonometric Functions
Derivatives of Trigonometric Functions 9-8-28 In this section, I ll iscuss its an erivatives of trig functions. I ll look at an important it rule first, because I ll use it in computing the erivative of
More informationCLASS XII CBSE MATHEMATICS INTEGRALS
Using Partial Fractions LSS XII SE MTHEMTIS INTEGRLS () cos ( sin)(sin ) () ns: log sin sin () () (SE 8) tan (sin ) c Let sin t cos ( t)( t ) t ( )( ) cosθ (sin θ)(5 cos θ) t,,, t (SE 8 OMP) dθ (SE 7)
More informationChapter 5 Analytic Trigonometry
Chapter 5 Analytic Trigonometry Section 1 Section 2 Section 3 Section 4 Section 5 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas
More informationy= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2.
. (a).5 0.5 y sin x+sin6x 0.5.5 (A) (C) (b) Period (C) []. (a) y x 0 x O x Notes: Award for end points Award for a maximum of.5 Award for a local maximum of 0.5 Award for a minimum of 0.75 Award for the
More informationMTH 112: Elementary Functions
MTH 11: Elementary Functions F. Patricia Medina Department of Mathematics. Oregon State University Section 6.6 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line
More informationChapter 7, Continued
Math 150, Fall 008, c Benjamin Aurispa Chapter 7, Continued 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas Double-Angle Formulas Formula for Sine: Formulas for Cosine: Formula for Tangent: sin
More informationANNUAL EXAMINATION - ANSWER KEY II PUC - MATHEMATICS PART - A
. LCM of and 6 8. -cosec ( ) -. π a a A a a. A A A A 8 8 6 5. 6. sin d ANNUAL EXAMINATION - ANSWER KEY -7 + d + + C II PUC - MATHEMATICS PART - A 7. or more vectors are said to be collinear vectors if
More information