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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. -
Anel NOVO 18.06.08 Sadrzaj:Layout 1.Qxd
Dr. Mustafa Memić GUSINJSKO-PLAVSKA KRAJINA U VRTLOGU HISTORIJE Sarajevo, 2008. IZDAVAČ: Institut za istraživanje zločina protiv čovječnosti i međunarodnog prava Univerziteta u Sarajevu ZA IZDAVAČA: prof. dr. Smail Čekić UREDNICI: dr. Safet Bandžović prof. mr. Muharem Kreso RECENZENTI: akademik Muhamed Filipović mr. Sefer Halilović LEKTOR: Sadžida Džuvić KORICE: Dževdet Nikočević DTP: Anel Ćuhara ŠTAMPARIJA: AMOS GRAF d.o.o. TIRAŽ: 500 PREDGOVOR Ovom knjigom želim objasniti neke od burnih događaja u mom rodnom kraju koji su bitno utjecali na formiranje nacionalne svijesti mojih sunarodnika i na njihov ekonomski i društveno-politički položaj poslije Drugog svjetskog rata. U narodu moga kraja duboko su urezana dva događaja. Jedan je osvajanje Gusinjsko-plavskog kraja od crnogorske vojske 1912, a drugi se odnosi na razdoblje od 1919. do 1945. Njima se objašnjava dolazak jednog puka srpske vojske, koji je poslije proboja Solunskog fronta nastupao vardarskom dolinom i od Skoplja i Kosovske Mitrovice uputio se prema Crnoj Gori. Pritom se prema Podgorici kretao preko Gusinjsko-plavske krajine, nakon čega je došlo do pobune Bošnjaka i Albanaca, te pokušaja uspostavljanja nove vlasti, a zatim do formiranja dviju vasojevićkih brigada - Donja i Gornja vasojevićka - koje su se kao paravojne jedinice pridružile srpskoj vojsci i djelovale pod rukovodstvom centralne Crnogorske uprave u Podgorici. Tom su prilikom u Plavu i Gusinju formirane i dvije vojne jedinice - dva bataljona - najprije kao komitske jedinice, koje su u početku djelovale u sastavu komitskog pokreta u Crnoj Gori. Strahovalo se da se uspostavljanjem njihove vlasti ne nametnu policijske vlasti, koje su tokom 1912-1913. počinile teške zločine (masovno strijeljanje – prema nekim podacima ubijeno je preko 8.000 Bošnjaka i Albanaca, a došlo je i do nasilnog pokrštavanja oko 12.500 ljudi). -
UNDER ORDERS: War Crimes in Kosovo Order Online
UNDER ORDERS: War Crimes in Kosovo Order online Table of Contents Acknowledgments Introduction Glossary 1. Executive Summary The 1999 Offensive The Chain of Command The War Crimes Tribunal Abuses by the KLA Role of the International Community 2. Background Introduction Brief History of the Kosovo Conflict Kosovo in the Socialist Federal Republic of Yugoslavia Kosovo in the 1990s The 1998 Armed Conflict Conclusion 3. Forces of the Conflict Forces of the Federal Republic of Yugoslavia Yugoslav Army Serbian Ministry of Internal Affairs Paramilitaries Chain of Command and Superior Responsibility Stucture and Strategy of the KLA Appendix: Post-War Promotions of Serbian Police and Yugoslav Army Members 4. march–june 1999: An Overview The Geography of Abuses The Killings Death Toll,the Missing and Body Removal Targeted Killings Rape and Sexual Assault Forced Expulsions Arbitrary Arrests and Detentions Destruction of Civilian Property and Mosques Contamination of Water Wells Robbery and Extortion Detentions and Compulsory Labor 1 Human Shields Landmines 5. Drenica Region Izbica Rezala Poklek Staro Cikatovo The April 30 Offensive Vrbovac Stutica Baks The Cirez Mosque The Shavarina Mine Detention and Interrogation in Glogovac Detention and Compusory Labor Glogovac Town Killing of Civilians Detention and Abuse Forced Expulsion 6. Djakovica Municipality Djakovica City Phase One—March 24 to April 2 Phase Two—March 7 to March 13 The Withdrawal Meja Motives: Five Policeman Killed Perpetrators Korenica 7. Istok Municipality Dubrava Prison The Prison The NATO Bombing The Massacre The Exhumations Perpetrators 8. Lipljan Municipality Slovinje Perpetrators 9. Orahovac Municipality Pusto Selo 10. Pec Municipality Pec City The “Cleansing” Looting and Burning A Final Killing Rape Cuska Background The Killings The Attacks in Pavljan and Zahac The Perpetrators Ljubenic 11. -
Lesson 6: Trigonometric Identities
1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as a2 − b2 =(a − b)(a + b)or a3 + b3 =(a + b)(a2 − ab + b2): Identities such as these are used to simplifly algebriac expressions and to solve alge- a3 + b3 briac equations. For example, using the third identity above, the expression a + b simpliflies to a2 − ab + b2: The first identiy verifies that the equation (a2 − b2)=0is true precisely when a = b: The formulas or trigonometric identities introduced in this lesson constitute an integral part of the study and applications of trigonometry. Such identities can be used to simplifly complicated trigonometric expressions. This lesson contains several examples and exercises to demonstrate this type of procedure. Trigonometric identities can also used solve trigonometric equations. Equations of this type are introduced in this lesson and examined in more detail in Lesson 7. For student’s convenience, the identities presented in this lesson are sumarized in Appendix A 2. The Elementary Identities Let (x; y) be the point on the unit circle centered at (0; 0) that determines the angle t rad : Recall that the definitions of the trigonometric functions for this angle are sin t = y tan t = y sec t = 1 x y : cos t = x cot t = x csc t = 1 y x These definitions readily establish the first of the elementary or fundamental identities given in the table below. For obvious reasons these are often referred to as the reciprocal and quotient identities. -
World Bank Document
Document of The World Bank FOR OFFICIAL USE ONLY Public Disclosure Authorized Report 1.a: 47748-A .J PROJECT APPRAISAL DOCUMENT ON A PROPOSED INTERNATIONAL BANK FOR RECONSTRUCTIONAND DEVELOPMENT Public Disclosure Authorized PARTIAL RISK GUARANTEE IN THE AMOUNT OF EURO 60 MILLION (USD 78 MILLION EQUIVALENT) FOR THE PRIVATIZATION OF THE POWER DISTRIBUTION SYSTEM OPERATOR OPERATOR1 I SISTEMIT TE SHPERNDARJES SHA (OSSH) Public Disclosure Authorized IN ALBANIA April 14,2009 Sustainable Development Department South East Europe Country Management Unit Europe and Central Asia Region Public Disclosure Authorized This document has a restricted distribution and may be used by recipients only in the performance of their official duties. Its contents may not otherwise be disclosed without World Bank authorization. CURRENCY EQUIVALENTS (Exchange Rate Effective February 2009) CurrencyUnit = Lek Lek 102 = US$1 US$1.3 = €1 CZK1 = US$0.04 FISCAL YEAR January 1 - December31 ABBREVIATIONS AND ACRONYMS AMM Albanian Market Model APL Adaptable Program Loan CAS Country Assistance Strategy COOP1 Cooperazione Internazionale - Italian Bilateral Aid Agency DSO Distribution System Operator EBRD European Bank for Reconstruction and Development EC European Commission ECSEE Energy Community of South East Europe EIB European Investment Bank EIRR Economic Internal Rate of Return EMP Environmental Management Plan ERE Electricity Regulatory Entity ETSO European Transmission System Operators EU European Union FDI Foreign Direct Investment GOA Government of Albania IAS International -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Derivation of Sum and Difference Identities for Sine and Cosine
Derivation of sum and difference identities for sine and cosine John Kerl January 2, 2012 The authors of your trigonometry textbook give a geometric derivation of the sum and difference identities for sine and cosine. I find this argument unwieldy | I don't expect you to remember it; in fact, I don't remember it. There's a standard algebraic derivation which is far simpler. The only catch is that you need to use complex arithmetic, which we don't cover in Math 111. Nonetheless, I will present the derivation so that you will have seen how simple the truth can be, and so that you may come to understand it after you've had a few more math courses. And in fact, all you need are the following facts: • Complex numbers are of the form a+bi, where a and b are real numbers and i is defined to be a square root of −1. That is, i2 = −1. (Of course, (−i)2 = −1 as well, so −i is the other square root of −1.) • The number a is called the real part of a + bi; the number b is called the imaginary part of a + bi. All the real numbers you're used to working with are already complex numbers | they simply have zero imaginary part. • To add or subtract complex numbers, add the corresponding real and imaginary parts. For example, 2 + 3i plus 4 + 5i is 6 + 8i. • To multiply two complex numbers a + bi and c + di, just FOIL out the product (a + bi)(c + di) and use the fact that i2 = −1. -
Complex Numbers and Functions
Complex Numbers and Functions Richard Crew January 20, 2018 This is a brief review of the basic facts of complex numbers, intended for students in my section of MAP 4305/5304. I will discuss basic facts of com- plex arithmetic, limits and derivatives of complex functions, power series and functions like the complex exponential, sine and cosine which can be defined by convergent power series. This is a preliminary version and will be added to later. 1 Complex Numbers 1.1 Arithmetic. A complex number is an expression a + bi where i2 = −1. Here the real number a is the real part of the complex number and bi is the imaginary part. If z is a complex number we write <(z) and =(z) for the real and imaginary parts respectively. Two complex numbers are equal if and only if their real and imaginary parts are equal. In particular a + bi = 0 only when a = b = 0. The set of complex numbers is denoted by C. Complex numbers are added, subtracted and multiplied according to the usual rules of algebra: (a + bi) + (c + di) = (a + c) + (b + di) (1.1) (a + bi) − (c + di) = (a − c) + (b − di) (1.2) (a + bi)(c + di) = (ac − bd) + (ad + bc)i (1.3) (note how i2 = −1 has been used in the last equation). Division performed by rationalizing the denominator: a + bi (a + bi)(c − di) (ac − bd) + (bc − ad)i = = (1.4) c + di (c + di)(c − di) c2 + d2 Note that denominator only vanishes if c + di = 0, so that a complex number can be divided by any nonzero complex number. -
Customary Laws, Folk Culture, and Social Lifeworlds: Albanian Studies in Critical Perspective Albert Doja
Customary Laws, Folk Culture, and Social Lifeworlds: Albanian Studies in Critical Perspective Albert Doja To cite this version: Albert Doja. Customary Laws, Folk Culture, and Social Lifeworlds: Albanian Studies in Critical Perspective. Luka Breneselovic. Spomenica Valtazara Bogišića o stogodišnjici njegove smrti [Gedächt- nisschrift für Valtazar Bogišić zur 100. Wiederkehr seines Todestages], Beograd: Sluzbeni & Institute of Comparative Law, vol. 2, pp. 183-199., 2011. halshs-00692739 HAL Id: halshs-00692739 https://halshs.archives-ouvertes.fr/halshs-00692739 Submitted on 1 May 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Customary Laws, Folk Culture, and Social Lifeworlds: Albanian Studies in Critical Perspective A D The Institute of Comparative Law in Belgrade launched this year a Festschrift in honour of Valtazar Bogišić (1834–1908), a jurist, law historian and folklorist. His most notable works are researches on family structure and he is primarily known as a codifier of civil law in Montenegro, which he prepared on the basis of a voluminous questionnaire.1 One of his main informants was the leader of the Kuci lineage, at the border of Montenegro and Albania and reputed of Albanian stock. -
The Call for Albanian Civic Unity in Gjergj Fishta's The
V (2018) 1, 203–214 This work is licensed under a Creative Commons Attribution 4.0 International License. Ovaj rad dostupan je za upotrebu pod licencom Creative Commons Imenovanje 4.0 međunarodna. Ismet BUJUPAJ UDK 821.18.09 Fishta, G.-13 University of Prizren Faculty of Philology DOI: https://doi.org/10.29162/ANAFORA.v5i1.8 Rruga e Shkronjave, nr. 1 20 000 Prizren Izvorni znanstveni članak Republic of Kosovo Original Research Article [email protected] Primljeno 3. veljače 2018. Received: 3 February 2018 Prihvaćeno 10. svibnja 2018. Accepted: 10 May 2018 “ALWAYS BAYRAM, ALWAYS EASTER”: THE CALL FOR ALBANIAN CIVIC UNITY IN GJERGJ FISHTA’S THE HIGHLAND LUTE Abstract Gjergj Fishta’s verse narrative, Lahuta e Malcis (ca. 1907) is often regarded as the national epic of Albanians. The epic produces a fresh vision of the shared cultural identity of Albanians by telling the story of their struggle for independence blended with elements of regional and local culture which had never before entered into high literary language. The paper argues that Fishta creates a national epic not just by the events he narrates in the poem but also by the language and imagery he uses. The poem’s language puts Albanian modes of expression into high literature, depicts Muslim and Christian Albanian practices in a manner that emphasizes their common ground, and utilizes Albanian folklore such as the mythological creatures called zana and ora, which lie outside both the Christian and Muslim religions, but are closely associated with the land. In emphasizing the Albanian land, language, and culture, Fishta’s epic poem crafts a civic Albanian identity that embraces Albanian religious diversity and transcends sectarian identifications. -
Inverse Trig Functions Summary
Inverse trigonometric functions 1 1 z = sec ϑ The inverse sine function. The sine function restricted to [ π, π ] is one-to-one, and its inverse on this 3 − 2 2 ϑ = π ϑ = arcsec z interval is called the arcsine (arcsin) function. The domain of arcsin is [ 1, 1] and the range of arcsin is 2 1 1 1 −1 [ π, π ]. Below is a graph of y = sin ϑ, with the the part over [ π, π ] emphasized, and the graph − 2 2 − 2 2 of ϑ y. π 1 = arcsin ϑ = 2 π y = sin ϑ ϑ = arcsin y 1 π π 2 π − 1 ϑ 1 1 z − 1 π 1 π ϑ 1 1 y − 2 2 − 1 − 1 1 3 π ϑ = 2 π ϑ = 2 π − 2 By definition, By definition, 1 1 1 3 ϑ = arcsin y means that sin ϑ = y and π 6 ϑ 6 π. ϑ = arcsec z means that sec ϑ = t and 0 6 ϑ< 2 π or π 6 ϑ< 2 π. − 2 2 Differentiating the equation on the right implicitly with respect to y, gives Again, differentiating the equation on the right implicitly with respect to z, and using the restriction in ϑ, dϑ dϑ 1 1 1 one computes the derivative cos ϑ = 1, or = , provided π<ϑ< π. dy dy cos ϑ − 2 2 d 1 (arcsec z)= 2 , for z > 1. 1 1 2 2 dz z√z 1 | | Since cos ϑ> 0 on ( 2 π, 2 π ), it follows that cos ϑ = p1 sin ϑ = p1 y . -
Rregjimi Komunist Vrau Dhe Pushkatoi 226 Malësorë
E PAVARUR, INFORMATIVE, HISTORIKE DHE PATRIOTIKE Botues; Vasel GILAJ, Viti i botim XXI, Nr 57, Çmimi 2 USD, New York Mars 2016 [email protected] Janar 1945 forcat e Prek Calit vrasin 57 partizanë FAQE 20-21 Zenel Shabani, “Gjenerali me xhamadan” i atdhetarisë FAQE 22-25 Pretash Zeka Ulaj, bajraktari i Kojës dhe kufinjve të Shqipërisë FAQE 26-29 Lukë Kaçaj, Shaljapini sss Shqipetar FAQE 32-35 Malësorët e Plavë e Gucisë një shekull të masakruar Rregjimi komunist FAQE 36-41 Shasi vendbanimi i hershëm ilir, qyteti vrau dhe pushkatoi i 365 kishave FAQE 42-45 Adem Demaci, Profeti i 226 malësorë Kosoves FAQE 46-48 MARS 2016 2 Ju fton të kontriboni, për promovimin dhe mbrojtjen e vlerave të patjetërsueshme që trashëguam si amanet nga të parët! E PAVARUR, INFORMATIVE, HISTORIKE DHE PATRIOTIKE INFORMATIVE, E PAVARUR, MARS 2016 3 Urdhër lirie apo ikje për liri ite me pare ne Bashkin e qytetit te shkodres,keshilli bashkiak dha nje titull nderi. Gazetaret e pranishem e pyeten kryetarin e keshillit se c’ fare ka bere ky person qe nuk e njohim,nuk e dini i pyeti Kryetari gazetaret jo than serish ata,po ky ka Vbere kerkese sqaroi kryetari. Keshtu me duket se ka ndollur edhe me Dekoraten e dhen Shoqatave Malesia e Madhe ne Detroid dhe New York,tashme nga ishe Presidentja e Kosoves A.Jahjaga ne muajin Shkurt te ketij viti,me titullin Urdheri i Lirise. Nisur nga ky fakt dhe vecanarishte nga trumbetimi dhe tundja qe po i bejne keto shoqata kesaj dekorate,dhe posacerishte ata te Detroidit ketyre zoterinjeve deshiroj tu jape dhe pa dyshim miqesishte,disa argumente per mendimin time kundershtues.