DOCSLIB.ORG

Section 5.1 Trigonometric Functions of Real Numbers in Calculus and In

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Section 5.1 Trigonometric Functions of Real Numbers in Calculus and In Section 5.1 Trigonometric Functions of Real Numbers In calculus and in the sciences many of the applications of the trigonometric functions require that the inputs be real numbers, rather than angles. By making this small but crucial change in our viewpoint, we can define the trigonometric functions in such a way that the inputs are real numbers. The definitions of the trig functions, and the identities that we have already met (and will meet later) will remain the same, and will be valid whether the inputs are angles or real numbers. Definition of the Trigonometric Functions of Real Numbers Let Pxy(, ) be a point on the unit circle xy22 1 whose arc length from the point (1,0) is t . Then the six trigonometric functions of the real number t is defined as: 1 sin(ty ) csc( t ) ( y 0) y 1 cos(tx ) sec( t ) ( x 0) x yx tan(tty ) cot( ) ( 0) xy 1 Here are some identities you need to know: Reciprocal Identities Pythagorean Identities sin(t) sin22 (tt ) cos ( ) 1 tan(t) cos(t) 1tan()sec()22tt cos(t) 1cot()csc()22tt cot(t) sin(t) 1 sec(tt ) , cos( ) 0 cos(t ) 1 csc(tt ) , sin( ) 0 sin(t ) 1 cot(tt ) , tan( ) 0 tan(t ) 2 1 Example 1: If tan(t ) and sin(t ) 0, find sec(t ) . 5 1 Example 2: Given that cos(t) and tan(t ) 0 ; find sin(t ) and tan(t ). 4 3 Opposite Angle Identities The positive direction from the point (1,0) is the counterclockwise direction and the negative direction is the clockwise direction. sin(tt ) sin( ) cos(tt ) cos( ) tan(tt ) tan( ) cot(tt ) cot( ) sec(tt ) sec( ) csc(tt ) csc( ) These identities tell us that: Sine, tangent, cotangent and cosecant are ODD functions; Cosine and Secant are EVEN functions. These identities hold true both for angles and real numbers. 4 Example 3: Use the opposite-angle identities to find: 2 a. sin 3 5 b. cos 6 3 c. tan cot 44 5 12 Example 4: Given: sin(x ) , find the value of: sin(x ) cos(x ) . 13 6 POPPER FOR SECTION 5.1 Question#1: Which of the following statements is/are true? I. 1tan()sec()22tt II. cos(x ) cos(x ) III. sin(x )csc(x ) 1 A) I only B) II only C) III only D) I and III only E) I, II and III F) None of these 7 Periodic functions: A function is periodic if there is some number p such that f (x p) f (x) . The smallest such number is called the “period” of this function. Periodicity of Sine and Cosine Functions The circumference of the unit circle is 2 . Thus, if we start at a point P and travel for 2 units, we come back to that point P. The period for sine and cosine functions is 2 . sin(tt 2 ) sin( ) cos(tt 2 ) cos( ) More generally: sin(tk 2 ) sin( t ) cos(tk 2 ) cos( t ) for all real numbers t and all integers k. 8 Periodicity of Tangent and Cotangent Functions The tangent and cotangent functions are also periodic functions. However, these functions repeat themselves after each . sin(tt ) sin( ) tan(tt ) tan( ) cos(tt ) cos( ) So tan(t ) tan(t) more generally tan(t k ) tan(t) cot(t ) cot(t) cot(t k ) cot(t) for all real numbers t and all integers k. 9 Remark: The period for the sine and cosine functions is 2 while the period for the tangent and cotangent functions is . Since secant and cosecant functions are the reciprocal functions, they will follow the same periodicity rules as sine and cosine. sec(t 2k) sec(t) csc(t 2k) csc(t) for all real numbers t and all integers k. 10 We will use the identities and periodicity to evaluate trig functions of real numbers. Periodicity sin(t 2k ) sin(t) tan(t k ) tan(t) cos(t 2k ) cos(t) cot(t k ) cot(t) sec(t 2k) sec(t) csc(t 2k) csc(t) (for all real numbers t and all integers k.) 15 Example 4: Evaluate tan 4 25 Example 5: Evaluate cos 6 11 19 Example 6: Evaluate sin 3 20 Example 7: Evaluate sin 3 12 19 21 cos tan 34 Example 8: Evaluate 25 cos(8 )sin 6 14 17 tan 21 4 Example 9: Evaluate cot 4 17 cos(11 )sin 2 15 POPPER FOR SECTION 5.1 41 Question#2: Evaluate: cos(19 ) sin tan 64 A) 1 B) 5/2 C) 1/2 D) -1 E) -1/2 F) None of these 16 Simplifying Trigonometric Expressions Using Identities Example 1: Simplify: cot(tt )sec( ) 17 Example 2: Simplify: cos(t) cos(t) tan2 (t) 18 sin 6 csc 2 Example 3: Simplify: 1tan2 5 19 5 Example 4: Given cos(x ) and 0 x , evaluate the following 13 2 expression: sinxx sin 2 Remark: Recall the fact about “complementary angles”: sinx cos(x ) 2 cosx sin(x ) 2 tanx cot(x ) 2 20 4 Example 5: Given sin(x ) and 0 x , evaluate the following 5 2 expression: sin xxx 11 cos 9 tan 17 21.
Load more

Recommended publications
  • Refresher Course Content
    Calculus I Refresher Course Content 4. Exponential and Logarithmic Functions Section 4.1: Exponential Functions Section 4.2: Logarithmic Functions Section 4.3: Properties of Logarithms Section 4.4: Exponential and Logarithmic Equations Section 4.5: Exponential Growth and Decay; Modeling Data 5. Trigonometric Functions Section 5.1: Angles and Radian Measure Section 5.2: Right Triangle Trigonometry Section 5.3: Trigonometric Functions of Any Angle Section 5.4: Trigonometric Functions of Real Numbers; Periodic Functions Section 5.5: Graphs of Sine and Cosine Functions Section 5.6: Graphs of Other Trigonometric Functions Section 5.7: Inverse Trigonometric Functions Section 5.8: Applications of Trigonometric Functions 6. Analytic Trigonometry Section 6.1: Verifying Trigonometric Identities Section 6.2: Sum and Difference Formulas Section 6.3: Double-Angle, Power-Reducing, and Half-Angle Formulas Section 6.4: Product-to-Sum and Sum-to-Product Formulas Section 6.5: Trigonometric Equations 7. Additional Topics in Trigonometry Section 7.1: The Law of Sines Section 7.2: The Law of Cosines Section 7.3: Polar Coordinates Section 7.4: Graphs of Polar Equations Section 7.5: Complex Numbers in Polar Form; DeMoivre’s Theorem Section 7.6: Vectors Section 7.7: The Dot Product 8. Systems of Equations and Inequalities (partially included) Section 8.1: Systems of Linear Equations in Two Variables Section 8.2: Systems of Linear Equations in Three Variables Section 8.3: Partial Fractions Section 8.4: Systems of Nonlinear Equations in Two Variables Section 8.5: Systems of Inequalities Section 8.6: Linear Programming 10. Conic Sections and Analytic Geometry (partially included) Section 10.1: The Ellipse Section 10.2: The Hyperbola Section 10.3: The Parabola Section 10.4: Rotation of Axes Section 10.5: Parametric Equations Section 10.6: Conic Sections in Polar Coordinates 11.
    [Show full text]
  • Trigonometric Functions
    Trigonometric Functions This worksheet covers the basic characteristics of the sine, cosine, tangent, cotangent, secant, and cosecant trigonometric functions. Sine Function: f(x) = sin (x) • Graph • Domain: all real numbers • Range: [-1 , 1] • Period = 2π • x intercepts: x = kπ , where k is an integer. • y intercepts: y = 0 • Maximum points: (π/2 + 2kπ, 1), where k is an integer. • Minimum points: (3π/2 + 2kπ, -1), where k is an integer. • Symmetry: since sin (–x) = –sin (x) then sin(x) is an odd function and its graph is symmetric with respect to the origin (0, 0). • Intervals of increase/decrease: over one period and from 0 to 2π, sin (x) is increasing on the intervals (0, π/2) and (3π/2 , 2π), and decreasing on the interval (π/2 , 3π/2). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Cosine Function: f(x) = cos (x) • Graph • Domain: all real numbers • Range: [–1 , 1] • Period = 2π • x intercepts: x = π/2 + k π , where k is an integer. • y intercepts: y = 1 • Maximum points: (2 k π , 1) , where k is an integer. • Minimum points: (π + 2 k π , –1) , where k is an integer. • Symmetry: since cos(–x) = cos(x) then cos (x) is an even function and its graph is symmetric with respect to the y axis. • Intervals of increase/decrease: over one period and from 0 to 2π, cos (x) is decreasing on (0 , π) increasing on (π , 2π). Tutoring and Learning Centre, George Brown College 2014 www.georgebrown.ca/tlc Trigonometric Functions Tangent Function : f(x) = tan (x) • Graph • Domain: all real numbers except π/2 + k π, k is an integer.
    [Show full text]
  • Lecture 5: Complex Logarithm and Trigonometric Functions
    LECTURE 5: COMPLEX LOGARITHM AND TRIGONOMETRIC FUNCTIONS Let C∗ = C \{0}. Recall that exp : C → C∗ is surjective (onto), that is, given w ∈ C∗ with w = ρ(cos φ + i sin φ), ρ = |w|, φ = Arg w we have ez = w where z = ln ρ + iφ (ln stands for the real log) Since exponential is not injective (one one) it does not make sense to talk about the inverse of this function. However, we also know that exp : H → C∗ is bijective. So, what is the inverse of this function? Well, that is the logarithm. We start with a general definition Definition 1. For z ∈ C∗ we define log z = ln |z| + i argz. Here ln |z| stands for the real logarithm of |z|. Since argz = Argz + 2kπ, k ∈ Z it follows that log z is not well defined as a function (it is multivalued), which is something we find difficult to handle. It is time for another definition. Definition 2. For z ∈ C∗ the principal value of the logarithm is defined as Log z = ln |z| + i Argz. Thus the connection between the two definitions is Log z + 2kπ = log z for some k ∈ Z. Also note that Log : C∗ → H is well defined (now it is single valued). Remark: We have the following observations to make, (1) If z 6= 0 then eLog z = eln |z|+i Argz = z (What about Log (ez)?). (2) Suppose x is a positive real number then Log x = ln x + i Argx = ln x (for positive real numbers we do not get anything new).
    [Show full text]
  • An Appreciation of Euler's Formula
    Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 17 An Appreciation of Euler's Formula Caleb Larson North Dakota State University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Larson, Caleb (2017) "An Appreciation of Euler's Formula," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 17. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/17 Rose- Hulman Undergraduate Mathematics Journal an appreciation of euler's formula Caleb Larson a Volume 18, No. 1, Spring 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 [email protected] a scholar.rose-hulman.edu/rhumj North Dakota State University Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 1, Spring 2017 an appreciation of euler's formula Caleb Larson Abstract. For many mathematicians, a certain characteristic about an area of mathematics will lure him/her to study that area further. That characteristic might be an interesting conclusion, an intricate implication, or an appreciation of the impact that the area has upon mathematics. The particular area that we will be exploring is Euler's Formula, eix = cos x + i sin x, and as a result, Euler's Identity, eiπ + 1 = 0. Throughout this paper, we will develop an appreciation for Euler's Formula as it combines the seemingly unrelated exponential functions, imaginary numbers, and trigonometric functions into a single formula. To appreciate and further understand Euler's Formula, we will give attention to the individual aspects of the formula, and develop the necessary tools to prove it.
    [Show full text]
  • Appendix B. Random Number Tables
    Appendix B. Random Number Tables Reproduced from Million Random Digits, used with permission of the Rand Corporation, Copyright, 1955, The Free Press. The publication is available for free on the Internet at http://www.rand.org/publications/classics/randomdigits. All of the sampling plans presented in this handbook are based on the assumption that the packages constituting the sample are chosen at random from the inspection lot. Randomness in this instance means that every package in the lot has an equal chance of being selected as part of the sample. It does not matter what other packages have already been chosen, what the package net contents are, or where the package is located in the lot. To obtain a random sample, two steps are necessary. First it is necessary to identify each package in the lot of packages with a specific number whether on the shelf, in the warehouse, or coming off the packaging line. Then it is necessary to obtain a series of random numbers. These random numbers indicate exactly which packages in the lot shall be taken for the sample. The Random Number Table The random number tables in Appendix B are composed of the digits from 0 through 9, with approximately equal frequency of occurrence. This appendix consists of 8 pages. On each page digits are printed in blocks of five columns and blocks of five rows. The printing of the table in blocks is intended only to make it easier to locate specific columns and rows. Random Starting Place Starting Page. The Random Digit pages numbered B-2 through B-8.
    [Show full text]
  • Effects of a Prescribed Fire on Oak Woodland Stand Structure1
    Effects of a Prescribed Fire on Oak Woodland Stand Structure1 Danny L. Fry2 Abstract Fire damage and tree characteristics of mixed deciduous oak woodlands were recorded after a prescription burn in the summer of 1999 on Mt. Hamilton Range, Santa Clara County, California. Trees were tagged and monitored to determine the effects of fire intensity on damage, recovery and survivorship. Fire-caused mortality was low; 2-year post-burn survey indicates that only three oaks have died from the low intensity ground fire. Using ANOVA, there was an overall significant difference for percent tree crown scorched and bole char height between plots, but not between tree-size classes. Using logistic regression, tree diameter and aspect predicted crown resprouting. Crown damage was also a significant predictor of resprouting with the likelihood increasing with percent scorched. Both valley and blue oaks produced crown resprouts on trees with 100 percent of their crown scorched. Although overall tree damage was low, crown resprouts developed on 80 percent of the trees and were found as shortly as two weeks after the fire. Stand structural characteristics have not been altered substantially by the event. Long term monitoring of fire effects will provide information on what changes fire causes to stand structure, its possible usefulness as a management tool, and how it should be applied to the landscape to achieve management objectives. Introduction Numerous studies have focused on the effects of human land use practices on oak woodland stand structure and regeneration. Studies examining stand structure in oak woodlands have shown either persistence or strong recruitment following fire (McClaran and Bartolome 1989, Mensing 1992).
    [Show full text]
  • Call Numbers
    Call numbers: It is our recommendation that libraries NOT put J, +, E, Ref, etc. in the call number field in front of the Dewey or other call number. Use the Home Location field to indicate the collection for the item. It is difficult if not impossible to sort lists if the call number fields aren’t entered systematically. Dewey Call Numbers for Non-Fiction Each library follows its own practice for how long a Dewey number they use and what letters are used for the author’s name. Some libraries use a number (Cutter number from a table) after the letters for the author’s name. Other just use letters for the author’s name. Call Numbers for Fiction For fiction, the call number is usually the author’s Last Name, First Name. (Use a comma between last and first name.) It is usually already in the call number field when you barcode. Call Numbers for Paperbacks Each library follows its own practice. Just be consistent for easier handling of the weeding lists. WCTS libraries should follow the format used by WCTS for the items processed by them for your library. Most call numbers are either the author’s name or just the first letters of the author’s last name. Please DO catalog your paperbacks so they can be shared with other libraries. Call Numbers for Magazines To get the call numbers to display in the correct order by date, the call number needs to begin with the date of the issue in a number format, followed by the issue in alphanumeric format.
    [Show full text]
  • Girls' Elite 2 0 2 0 - 2 1 S E a S O N by the Numbers
    GIRLS' ELITE 2 0 2 0 - 2 1 S E A S O N BY THE NUMBERS COMPARING NORMAL SEASON TO 2020-21 NORMAL 2020-21 SEASON SEASON SEASON LENGTH SEASON LENGTH 6.5 Months; Dec - Jun 6.5 Months, Split Season The 2020-21 Season will be split into two segments running from mid-September through mid-February, taking a break for the IHSA season, and then returning May through mid- June. The season length is virtually the exact same amount of time as previous years. TRAINING PROGRAM TRAINING PROGRAM 25 Weeks; 157 Hours 25 Weeks; 156 Hours The training hours for the 2020-21 season are nearly exact to last season's plan. The training hours do not include 16 additional in-house scrimmage hours on the weekends Sep-Dec. Courtney DeBolt-Slinko returns as our Technical Director. 4 new courts this season. STRENGTH PROGRAM STRENGTH PROGRAM 3 Days/Week; 72 Hours 3 Days/Week; 76 Hours Similar to the Training Time, the 2020-21 schedule will actually allow for a 4 additional hours at Oak Strength in our Sparta Science Strength & Conditioning program. These hours are in addition to the volleyball-specific Training Time. Oak Strength is expanding by 8,800 sq. ft. RECRUITING SUPPORT RECRUITING SUPPORT Full Season Enhanced Full Season In response to the recruiting challenges created by the pandemic, we are ADDING livestreaming/recording of scrimmages and scheduled in-person visits from Lauren, Mikaela or Peter. This is in addition to our normal support services throughout the season. TOURNAMENT DATES TOURNAMENT DATES 24-28 Dates; 10-12 Events TBD Dates; TBD Events We are preparing for 15 Dates/6 Events Dec-Feb.
    [Show full text]
  • Using Differentials to Differentiate Trigonometric and Exponential Functions Tevian Dray
    Using Differentials to Differentiate Trigonometric and Exponential Functions Tevian Dray Tevian Dray ([email protected]) received his B.S. in mathematics from MIT in 1976, his Ph.D. in mathematics from Berkeley in 1981, spent several years as a physics postdoc, and is now a professor of mathematics at Oregon State University. A Fellow of the American Physical Society for his early work in general relativity, his current research interests include the octonions as well as science education. He directs the Vector Calculus Bridge Project. (http://www.math.oregonstate.edu/bridge) Differentiating a polynomial is easy. To differentiate u2 with respect to u, start by computing d.u2/ D .u C du/2 − u2 D 2u du C du2; and then dropping the last term, an operation that can be justified in terms of limits. Differential notation, in general, can be regarded as a shorthand for a formal limit argument. Still more informally, one can argue that du is small compared to u, so that the last term can be ignored at the level of approximation needed. After dropping du2 and dividing by du, one obtains the derivative, namely d.u2/=du D 2u. Even if one regards this process as merely a heuristic procedure, it is a good one, as it always gives the correct answer for a polynomial. (Physicists are particularly good at knowing what approximations are appropriate in a given physical context. A physicist might describe du as being much smaller than the scale imposed by the physical situation, but not so small that quantum mechanics matters.) However, this procedure does not suffice for trigonometric functions.
    [Show full text]
  • Fracking by the Numbers
    Fracking by the Numbers The Damage to Our Water, Land and Climate from a Decade of Dirty Drilling Fracking by the Numbers The Damage to Our Water, Land and Climate from a Decade of Dirty Drilling Written by: Elizabeth Ridlington and Kim Norman Frontier Group Rachel Richardson Environment America Research & Policy Center April 2016 Acknowledgments Environment America Research & Policy Center sincerely thanks Amy Mall, Senior Policy Analyst, Land & Wildlife Program, Natural Resources Defense Council; Courtney Bernhardt, Senior Research Analyst, Environmental Integrity Project; and Professor Anthony Ingraffea of Cornell University for their review of drafts of this document, as well as their insights and suggestions. Frontier Group interns Dana Bradley and Danielle Elefritz provided valuable research assistance. Our appreciation goes to Jeff Inglis for data assistance. Thanks also to Tony Dutzik and Gideon Weissman of Frontier Group for editorial help. We also are grateful to the many state agency staff who answered our numerous questions and requests for data. Many of them are listed by name in the methodology. The authors bear responsibility for any factual errors. The recommendations are those of Environment America Research & Policy Center. The views expressed in this report are those of the authors and do not necessarily reflect the views of our funders or those who provided review. 2016 Environment America Research & Policy Center. Some Rights Reserved. This work is licensed under a Creative Commons Attribution Non-Commercial No Derivatives 3.0 Unported License. To view the terms of this license, visit creativecommons.org/licenses/by-nc-nd/3.0. Environment America Research & Policy Center is a 501(c)(3) organization.
    [Show full text]
  • Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
    Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time.
    [Show full text]
  • The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit ....................................
    [Show full text]