Inverse Trig Functions


 Silvia Dawson
 2 years ago
 Views:
Transcription
1 Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that they are useful not only in the calculation of angles given the lengths of the sies of a right triangle, but they also give us solutions to some common integrals. For example, suppose you nee to evaluate the following integral: b a x x for some appropriate values of a an b. You can use the inverse sine function to solve it! In this capsule we o not attempt to erive the formulas that we will use; you shoul look at your textbook for erivations an complete explanations. This material will simply summarize the key results an go through some examples of how to use them. As usual, all angles use here are in raians. Restrictions on the Domains of the Trig Functions A function must be onetoone for it to have an inverse. As we are sure you know, the trig functions are not onetoone an in fact they are perioic (i.e. their values repeat themselves perioically). So in orer to efine inverse functions we nee to restrict the omain of each trig function to a region in which it is onetoone but also attains all of its values. We o this by selecting a specific perio for each function an using this as a omain on which an inverse can be efine. Clearly there are an infinite number of ifferent restrictions we coul chose but the following are choices that are normally use.
2 Stanar Restricte Domains Function Domain Range sin(x) [ π, π ] [, ] cos(x) [0, π] [, ] tan(x) ( π, π ) (, ) cot(x) (0, π) (, ) sec(x) [0, π ) ( π, π] (, ] [, ) csc(x) [ π, 0) (0, π ] (, ] [, ) Definitions of the Inverse Functions When the trig functions are restricte to the omains above they become onetoone functions, so we can efine the inverse functions. For the sine function we use the notation sin (x) or arcsin(x). Both are rea arc sine. Look carefully at where we have place the . Written this way it inicates the inverse of the sine function. If, instea, we write (sin(x)) we mean the fraction sin(x). The other functions are similar. The following table summarizes the omains an ranges of the inverse trig functions. Note that for each inverse trig function we have simply swappe the omain an range for the corresponing trig function. Stanar Restricte Domains Function Domain Range sin (x) [, ] [ π, π ] cos (x) [, ] [0, π] tan (x) (, ) ( π, π ) cot (x) (, ) (0, π) sec (x) (, ] [, ) [0, π ) ( π, π] csc (x) (, ] [, ) [ π, 0) (0, π ] We can now efine the inverse functions more clearly. For the arcsin function we efine y sin (x) if x, y is in [ π, π ], an sin(y) x
3 Note that this is only efine when x is in the interval [, ]. The other inverse functions are similarly efine using the corresponing trig functions. Some Useful Ientities Here are a few ientities that you may fin helpful. cos (x) + cos ( x) π sin (x) + cos (x) π tan ( x) tan (x) Practicing with the Inverse Functions Example : Fin the value of tan(sin ( 5 ). Solution: The best way to solve this sort of problem is to raw a triangle for yourself using the Pythagorian Theorem. 5 θ 6 Here we use θ for the value of sin ( 5 ). Notice that we labele the hypotenuse an the sie opposite θ by using the value of the sin of the angle. We then use the Pythagorian Theorem to get the remaining sie. We now have the information that is neee to fin tan(θ). Since tan(θ) opposite ajacent, the answer is 4 6 Example : Fin the value of sin(cos ( 3 5 )). Solution: Look at the following picture: In this picture we let θ cos ( 3 5 ). Then 0 θ π an cosθ 3 5. Because cos(θ) is negative, θ must be in the secon quarant, i.e. θ π. Using the Pythagorean 3 π θ
4 Theorem an the fact that θ is in the secon quarant we get that sin(θ) Note that although θ oes not lie in the restricte omain we use to efine the arcsin function, the unrestricte sin function is efine in the secon quarant an so we are free to use this fact. Derivatives of Inverse Trig Functions The erivatives of the inverse trig functions are shown in the following table. Function sin (x) cos (x) tan (x) cot (x) sec (x) csc (x) Derivatives Derivative x (sin x) x, x < x (cos x), x < x x (tan x) +x x (cot x) +x x (sec x) x, x > x x (csc x) x x, x > In practice we often are intereste in calculating the erivatives when the variable x is replace by a function u(x). This requires the use of the chain rule. For example, x (sin u) u u x The other functions are hanle in a similar way. u x u, u < Example : Fin the erivative of y cos (x 3 ) for x 3 < Solution: Note that x 3 < if an only if x <, so the erivative is efine whenever x <. 4
5 x (cos (x 3 )) (x 3 ) x (x3 ) (3x ) (x 3 ) 3x x 6 Example : Fin the erivative of y tan ( 3x). Solution: x (tan ( 3x)) + ( 3x) x ( 3x) + ( 3x) 3x 3 3 3x ( + 3x) Exercise : For each of the following, fin the erivative of the given function with respect to the inepenent variable. (a) y tan t 4 (b) z t cot ( + t ) (c) x sin t 4 () s t t + cos t (e) y sin x (f) z cot ( y ) y 5
6 Solutions: (a) y tan t 4 y t t tan (t 4 ) + (t 4 ) t (t4 ) 4t3 + t 8 (b) z t cot ( + t ) z t t t cot ( + t ) cot ( + t ) + t cot ( + t ) + ( + t ) (t) t t 4 + t + (c) x sin t 4 x t t sin t 4 ( t 4 ) t ( t 4 ) ( t 4 ) ( t4 ) ( 4t 3 ) + t 4 t 4 ( t3 ) t t 4 ( t3 ) t t 4 6
7 () s t t + cos t s t t t t + t cos t ( t ) t ( t ) ( t) ( + t ) t t + ( t ) t t ( t )( t ) + t ( t )( t ) ( t ) + t ( t ) ( t )( t ) t ( t ) 3 t ( t ) ( t ) ( t ) (e) y sin x y x x sin x ( x) x x x x x( x) 7
8 (f) z cot ( y ) y z y y cot ( y ) y y +( y ) ( y ) +y ( y ) y ( y ) y y ( y ) y ( y ) ( y ) +y ( y ) +y ( y +y ) y +y 4 +y (+y ) y +y 4 ( y ) y ( y) ( y ) ( y ) y ( y) Solving Integrals The formulas liste above for the erivatives lea us to some nice ways to solve some common integrals. The following is a list of useful ones. These formulas hol for any constant a 0 u a u sin ( u a ) + C for u < a u a +u a tan ( u a ) + C for all u u u u a a sec u a + C for u > a > 0 Exercise : Verify each of the equations above by taking the erivative of the right han sie. We now want to use these formulas to solve some common integrals. Example : Evaluate the integral x 9 6x Solution: Let a 3 an u 4x. Then 6x (4x) u an u 4x. We get the following for 6x < 9: 8
9 x 9 6x 4 u a u 4 sin ( u a ) + C 4 sin ( 4x 3 ) + C 4 sin ( 4 3 x) + C Exercise 3: Evaluate the following integrals. (a) (b) (c) () (e) (f) x 5 4x y 36+4y z z 5+z 4 sin x x 0 cos x x 5+4x x 7 x 5 x+4x Solutions: (a) x u 5 4x. For this problem use the formula a 5, u x an u x, giving you (b) y. Use the formula u 36+4y an u y. This gives us y 36+4y a +u a u x 5 4x sin u a + C with u a u sin ( x 5 ) + C a tan ( u a ) + C with a 6, u y u ( a +u )( 6 ) tan ( y 6 ) + C tan ( y 3 ) + C (c) z z. In orer to make the calculations a bit simpler, it is useful to multiply 5+z 4 the numerator an enominator by in orer to get the term 4z 4 instea of z 4 in the enominator. This gives us z z z z. 5+z 4 0+4z 4 Now let u z, u 4z z an a 0 an we have z z 4 z z 5+z 4 u 0+4z 4 ( 0) +u 0 tan ( z 0 ) + C () sin x x 0 cos x. Let u cos x, u sin x x an a 0. Then sin x x 0 cos x ( ) sin x x 0 cos x sin ( cos x 0 ) + C 9
10 (e) form x 5+4x x. We want to transform this expression into something with the u a. To o this we nee to complete the square of the expression in the u enominator as follows: 5 + 4x x x x x x 9 (x 4x + 4) (3) (x ) This gives us x 5+4x x x (3) (x ) sin ( x 3 ) + C (f) 7 x. Again we nee to complete the square. This time we want to transform 5 x+4x the expression into something with the form u. We rewrite the enominator as a +u follows: 5 x + 4x x + 4x (4) + (x 3) Now, letting u x 3 an u x we get 7 x 5 x+4x 7 x 7 (4) +(x 3) 8 tan ( x 3 4 ) + C 0
arcsine (inverse sine) function
Inverse Trigonometric Functions c 00 Donal Kreier an Dwight Lahr We will introuce inverse functions for the sine, cosine, an tangent. In efining them, we will point out the issues that must be consiere
More informationLecture 17: Implicit differentiation
Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we
More informationInverse Trig Functions
Inverse Trig Functions Trig functions are not onetoone, so we can not formally get an inverse. To efine the notion of inverse trig functions we restrict the omains to obtain onetoone functions: [ Restrict
More informationHere the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and
Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric
More information6.6 The Inverse Trigonometric Functions. Outline
6.6 The Inverse Trigonometric Functions Tom Lewis Fall Semester 2015 Outline The inverse sine function The inverse cosine function The inverse tangent function The other inverse trig functions Miscellaneous
More informationLecture 13: Differentiation Derivatives of Trigonometric Functions
Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the
More informationTrigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics.
MATH19730 Part 1 Trigonometry Section2 Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics. An angle can be measured in degrees
More informationInverse Trigonometric Functions
Section 3 Inverse Trigonometric Functions 00 Kiryl Tsishchanka Inverse Trigonometric Functions DEFINITION: The inverse sine function, denoted by sin x or arcsin x), is defined to be the inverse of the
More informationThe Inverse Trigonometric Functions
The Inverse Trigonometric Functions These notes amplify on the book s treatment of inverse trigonometric functions an supply some neee practice problems. Please see pages 543 544 for the graphs of sin
More informationM147 Practice Problems for Exam 2
M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work
More information4.7 Solving Problems with Inverse Trig Functions
4. Solving Problems with Inverse Trig Functions 4.. Inverse trig functions create right triangles An inverse trig function has an angle (y or θ) as its output. That angle satisfies a certain trig expression
More informationSept 20, 2011 MATH 140: Calculus I Tutorial 2. ln(x 2 1) = 3 x 2 1 = e 3 x = e 3 + 1
Sept, MATH 4: Calculus I Tutorial Solving Quadratics, Dividing Polynomials Problem Solve for x: ln(x ) =. ln(x ) = x = e x = e + Problem Solve for x: e x e x + =. Let y = e x. Then we have a quadratic
More informationTrigonometry CheatSheet
Trigonometry CheatSheet 1 How to use this ocument This ocument is not meant to be a list of formulas to be learne by heart. The first few formulas are very basic (they escen from the efinition an/or Pythagoras
More information2 2 [ 1, 1]. Thus there exists a function sin 1 or arcsin. sin x arcsin x(= sin 1 x) arcsin(x) = y sin(y) = x. arcsin(sin x) = x if π 2 x π 2
7.4 Inverse Trigonometric Functions From looking at the graphs of the trig functions, we see that they fail the horizontal line test spectacularly. However, if you restrict their domain, you can find an
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More information4. Graphing and Inverse Functions
4. Graphing and Inverse Functions 4. Basic Graphs 4. Amplitude, Reflection, and Period 4.3 Vertical Translation and Phase Shifts 4.4 The Other Trigonometric Functions 4.5 Finding an Equation From Its Graph
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationInverse Circular Function and Trigonometric Equation
Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse
More informationChapter 5 The Trigonometric Functions
P a g e 40 Chapter 5 The Trigonometric Functions Section 5.1 Angles Initial side Terminal side Standard position of an angle Positive angle Negative angle Coterminal Angles Acute angle Obtuse angle Complementary
More information2 HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;
More informationWORKBOOK. MATH 30. PRECALCULUS MATHEMATICS.
WORKBOOK. MATH 30. PRECALCULUS MATHEMATICS. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributor: U.N.Iyer Department of Mathematics and Computer Science, CP 315, Bronx Community College, University
More informationAngles and Their Measure
Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationTrigonometry. Week 1 Right Triangle Trigonometry
Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge
More informationSolutions to modified 2 nd Midterm
Math 125 Solutions to moifie 2 n Miterm 1. For each of the functions f(x) given below, fin f (x)). (a) 4 points f(x) = x 5 + 5x 4 + 4x 2 + 9 Solution: f (x) = 5x 4 + 20x 3 + 8x (b) 4 points f(x) = x 8
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationHyperbolic functions (CheatSheet)
Hyperbolic functions (CheatSheet) 1 Intro For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus course, but as a matter of fact they have the same ignity
More informationM3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.
M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides
More informationWEEK #5: Trig Functions, Optimization
WEEK #5: Trig Functions, Optimization Goals: Trigonometric functions and their derivatives Optimization Textbook reading for Week #5: Read Sections 1.8, 2.10, 3.3 Trigonometric Functions From Section 1.8
More informationf(x) = undefined otherwise We have Domain(f) = [ π, π ] and Range(f) = [ 1, 1].
Lecture 6 : Inverse Trigonometric Functions Inverse Sine Function (arcsin = sin ) The trigonometric function sin is not onetoone functions, hence in orer to create an inverse, we must restrict its omain.
More information6.3 Inverse Trigonometric Functions
Chapter 6 Periodic Functions 863 6.3 Inverse Trigonometric Functions In this section, you will: Learning Objectives 6.3.1 Understand and use the inverse sine, cosine, and tangent functions. 6.3. Find the
More informationInverse Trig Functions
Inverse Trig Functions Previously in Math 30 DEFINITION 24 A function g is the inverse of the function f if g( f ()) = for all in the omain of f 2 f (g()) = for all in the omain of g In this situation
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More informationAbout Trigonometry. Triangles
About Trigonometry TABLE OF CONTENTS About Trigonometry... 1 What is TRIGONOMETRY?... 1 Triangles... 1 Background... 1 Trigonometry with Triangles... 1 Circles... 2 Trigonometry with Circles... 2 Rules/Conversion...
More informationMath 115 Spring 2014 Written Homework 10SOLUTIONS Due Friday, April 25
Math 115 Spring 014 Written Homework 10SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain
More informationUsing implicit di erentiation for good: Inverse functions.
Using implicit di erentiation for good: Inverse functions. Warmup: Calculate dy dx if 1. e y = xy Take d dx So of both sides to find ey dy dx = x dy dx + y. y = e y dy dx x dy dx = dy dx (ey x), implying
More informationElliptic Functions sn, cn, dn, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota
Elliptic Functions sn, cn, n, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota Backgroun: Jacobi iscovere that rather than stuying elliptic integrals themselves, it is simpler to think of them as inverses
More informationSection 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
More informationInverse Trigonometric Functions
SECTION 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the exact value of an inverse trigonometric function. Use a calculator to approximate
More informationStudent Academic Learning Services Page 1 of 6 Trigonometry
Student Academic Learning Services Page 1 of 6 Trigonometry Purpose Trigonometry is used to understand the dimensions of triangles. Using the functions and ratios of trigonometry, the lengths and angles
More informationEvaluating trigonometric functions
MATH 1110 0090906 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationy = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions
MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y
More informationINVERSE TRIGONOMETRIC FUNCTIONS. Colin Cox
INVERSE TRIGONOMETRIC FUNCTIONS Colin Cox WHAT IS AN INVERSE TRIG FUNCTION? Used to solve for the angle when you know two sides of a right triangle. For example if a ramp is resting against a trailer,
More information3. Right Triangle Trigonometry
. Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities . Reference Angle Reference Angle Reference angle
More informationSection 0.4 Inverse Trigonometric Functions
Section 0.4 Inverse Trigonometric Functions Short Recall from Trigonometry Definition: A function f is periodic of period T if f(x + T ) = f(x) for all x such that x and x+t are in the domain of f. The
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationSection 8.1: The Inverse Sine, Cosine, and Tangent Functions
Section 8.1: The Inverse Sine, Cosine, and Tangent Functions The function y = sin x doesn t pass the horizontal line test, so it doesn t have an inverse for every real number. But if we restrict the function
More informationMath 2201 Unit 3: Acute Triangle Trigonometry. Ch. 3 Notes
Rea Learning Goals, p. 17 text. Math 01 Unit 3: ute Triangle Trigonometry h. 3 Notes 3.1 Exploring Siengle Relationships in ute Triangles (0.5 lass) Rea Goal p. 130 text. Outomes: 1. Define an aute triangle.
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More informationTrigonometry LESSON TWO  The Unit Circle Lesson Notes
(cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More information20. Product rule, Quotient rule
20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that
More informationInverse trig functions create right triangles. Elementary Functions. Some Worked Problems on Inverse Trig Functions
Inverse trig functions create right triangles An inverse trig function has an angle (y or ) as its output That angle satisfies a certain trig expression and so we can draw a right triangle that represents
More informationIntroduction to Integration Part 1: AntiDifferentiation
Mathematics Learning Centre Introuction to Integration Part : AntiDifferentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More informationInverse Trigonometric Functions
Inverse Trigonometric Functions I. Four Facts About Functions and Their Inverse Functions:. A function must be onetoone (an horizontal line intersects it at most once) in order to have an inverse function..
More informationInverse Trig Functions
MATH 7 Inverse Trig Functions Dr. Neal, WKU A function y = f (x) is one to one if it is always the case that different x values are assigned to different y values. For example, y = x + 4 is one to one,
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More informationHow to Avoid the Inverse Secant (and Even the Secant Itself)
How to Avoi the Inverse Secant (an Even the Secant Itself) S A Fulling Stephen A Fulling (fulling@mathtamue) is Professor of Mathematics an of Physics at Teas A&M University (College Station, TX 7783)
More informationPractical Lab 2 The Diffraction Grating
Practical Lab 2 The Diffraction Grating OBJECTIVES: 1) Observe the interference pattern prouce when laser light passes through multipleslit grating (a iffraction grating). 2) Graphically verify the wavelength
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationLines. We have learned that the graph of a linear equation. y = mx +b
Section 0. Lines We have learne that the graph of a linear equation = m +b is a nonvertical line with slope m an intercept (0, b). We can also look at the angle that such a line makes with the ais. This
More informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More information( ) = sin x so that it is a onetoone function
(Section 4.7: Inverse Trig Functions) 4.72 SECTION 4.7: INVERSE TRIG FUNCTIONS You may want to review Section 1.8 on inverse functions. PART A : GRAPH OF sin 1 x (or arcsin x) Warning: Remember that f
More informationThe graph of. horizontal line between1 and 1 that the sine function is not 11 and therefore does not have an inverse.
Inverse Trigonometric Functions The graph of If we look at the graph of we can see that if you draw a horizontal line between1 and 1 that the sine function is not 11 and therefore does not have an inverse.
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More information11 Trigonometric Functions of Acute Angles
Arkansas Tech University MATH 10: Trigonometry Dr. Marcel B. Finan 11 Trigonometric Functions of Acute Angles In this section you will learn (1) how to find the trigonometric functions using right triangles,
More informationInverse Trigonometric Functions  Trigonometric Equations
Inverse Trigonometric Functions  Trigonometric Equations Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout defines the inverse of the sine, cosine and tangent functions.
More informationTrigonometric Substitution Created by Tynan Lazarus November 3, 2015
. Trig Identities Trigonometric Substitution November 3, 0 tan(θ) = sin(θ) cos(θ) sec(θ) = cos(θ) cot(θ) = cos(θ) sin(θ) csc(θ) = sin(θ). Trig Integrals sin(θ) dθ = cos(θ) + C sec (θ) dθ = tan(θ) + C sec(θ)
More informationChapter 6 Trigonometric Functions of Angles
6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.
More informationChapter 5: Trigonometric Functions of Real Numbers
Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationSection 8.1 The Inverse Sine, Cosine, and Tangent Functions
Section 8.1 The Inverse Sine, Cosine, and Tangent Functions You learned about inverse unctions in both college algebra and precalculus. The main characteristic o inverse unctions is that composing one
More information6.5 Trigonometric formulas
100 CHAPTER 6. TRIGONOMETRIC FUNCTIONS 6.5 Trigonometric formulas There are a few very important formulas in trigonometry, which you will need to know as a preparation for Calculus. These formulas are
More informationSection 5.4 More Trigonometric Graphs. Graphs of the Tangent, Cotangent, Secant, and Cosecant Function
Section 5. More Trigonometric Graphs Graphs of the Tangent, Cotangent, Secant, and Cosecant Function 1 REMARK: Many curves have a U shape near zero. For example, notice that the functions secx and x +
More informationChapter 11. Techniques of Integration
Chapter Techniques of Integration Chapter 6 introduced the integral. There it was defined numerically, as the limit of approximating Riemann sums. Evaluating integrals by applying this basic definition
More informationTrigonometric Identities and Equations
Chapter 4 Trigonometric Identities and Equations Trigonometric identities describe equalities between related trigonometric expressions while trigonometric equations ask us to determine the specific values
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More information18 Verifying Trigonometric Identities
Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 18 Verifying Trigonometric Identities In this section, you will learn how to use trigonometric identities to simplify trigonometric
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationPlane Trigonometry  Fall 1996 Test File
Plane Trigonometry  Fall 1996 Test File Test #1 1.) Fill in the blanks in the two tables with the EXACT values of the given trigonometric functions. The total point value for the tables is 10 points.
More informationList of trigonometric identities
List of trigonometric identities From Wikipedia, the free encyclopedia In mathematics, trigonometric identities are equalities involving trigonometric functions that are true for all values of the occurring
More informationEuler s Formula and Trig Identities
Euler s Formula and Trig Identities Steven W. Nydick May 5, 0 Introduction Many of the identities from trigonometry can be demonstrated relatively easily using Euler s formula, rules of exponents, basic
More informationFunctions  Inverse Trigonometry
10.9 Functions  Inverse Trigonometry We used a special function, one of the trig functions, to take an angle of a triangle and find the side length. Here we will do the opposite, take the side lengths
More informationSection 2.7 OnetoOne Functions and Their Inverses
Section. OnetoOne Functions and Their Inverses OnetoOne Functions HORIZONTAL LINE TEST: A function is onetoone if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More informationBlue Pelican Calculus First Semester
Blue Pelican Calculus First Semester Teacher Version 1.01 Copyright 20112013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More information9.1 Trigonometric Identities
9.1 Trigonometric Identities r y x θ x y θ r sin (θ) = y and sin (θ) = y r r so, sin (θ) =  sin (θ) and cos (θ) = x and cos (θ) = x r r so, cos (θ) = cos (θ) And, Tan (θ) = sin (θ) =  sin (θ)
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationMAC 1114: Trigonometry Notes
MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant
More information#11: Inverse trigonometric relations
#11: Inverse trigonometric relations November 14, 2008 a shady character Problem 1. A shadylooking character slinks up next to you and whispers, Pssst. Hey kid. I ll tell you a real number x, and I ll
More informationMATH 109 TOPIC 8 TRIGONOMETRIC IDENTITIES. I. Using Algebra in Trigonometric Forms
Math 109 T8Trigonometric Identities Page 1 MATH 109 TOPIC 8 TRIGONOMETRIC IDENTITIES I. Using Algebra in Trigonometric Forms Practice Problems II. Verifying Identities Practice Problems I. Using Algebra
More informationSection 8 Inverse Trigonometric Functions
Section 8 Inverse Trigonometric Functions Inverse Sine Function Recall that for every function y = f (x), one may de ne its INVERSE FUNCTION y = f 1 (x) as the unique solution of x = f (y). In other words,
More informationFinding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6.
1 Math 3 Final Review Guide This is a summary list of many concepts covered in the different sections and some examples of types of problems that may appear on the Final Exam. The list is not exhaustive,
More informationCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS ShiahSen Wang The graphs are prepared by ChienLun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole
More informationUnit 8 Inverse Trig & Polar Form of Complex Nums.
HARTFIELD PRECALCULUS UNIT 8 NOTES PAGE 1 Unit 8 Inverse Trig & Polar Form of Complex Nums. This is a SCIENTIFIC OR GRAPHING CALCULATORS ALLOWED unit. () Inverse Functions (3) Invertibility of Trigonometric
More informationnparameter families of curves
1 nparameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,
More informationUnit 6 Introduction to Trigonometry Inverse Trig Functions (Unit 6.6)
Unit 6 Introduction to Trigonometry Inverse Trig Functions (Unit 6.6) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Understand
More informationWe d like to explore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( x)
Inverse Trigonometric Functions: We d like to eplore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( ) sin. Recall the graph: MATH 30 Lecture 0 of 0 Well, we can
More information