[2] Eine Minderheit von Forschern, vorwiegend aus hermeneutischen Forschungstraditionen, lehnt Triangulation aus erkenntnistheoretischen Grnden vollstndig ab. [60] Eine Minderheit von Autoren sieht dagegen in der Verwendung von CAQDAS eine Tendenz zur Rigiditt qualitativer Sozialforschung, die mglicherweise zur Verschlechterung des Forschungspotentials dieser Methoden fhrt. Denzin empfiehlt dabei, nicht wie blich Studierende oder Doktoranden die Kodierungsarbeit durchfhren zu lassen, sondern auf erfahrene qualitative Forscher zurckzugreifen.[31]. If you want data specific to your purposes with control over how it is generated, collect primary data. He is a founding trainer for the International Program for Development Evaluation Training, sponsored by The World Bank and other international development agencies each summer in Ottawa, Ontario. But if you want to synthesize existing knowledge, analyze historical trends, or identify patterns on a large scale, secondary data might be a better choice. | Guide & Examples, What Is Concurrent Validity? [25], Diesen epistemologischen Legitimierungen der Triangulation stehen pragmatischere Begrndungen gegenber, bei denen Forscher auf neue Methoden zurckgreifen, weil die mit den zuvor benutzten Methoden erreichten Vorhersageergebnisse sich als zu schlecht herausgestellt haben. The Indian text the Yuktibh contains proof for the expansion of the sine and cosine functions and the derivation and proof of the power series for inverse tangent, discovered by Madhava. Although triangulation may be one strategy used in a study, by examining the historical roots of triangulation, one can make a more informed decision about whether or not this concept is both useful and applicable for any particular study. You have control over the sampling and measurement methods. In the 13th century, Nasr al-Dn al-Ts was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. WebThis representativeness is enhanced by the triangulation procedures, resulting in increased validity. Kelle, U. [9] For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively. In reading Denzins account (1978), he insists that by using multiple forms of triangulation in a study, sociologists might develop propositions that permit discovery and verification (pp. Hufig wird bei der Methodentriangulation mit Norman Denzin zwischen einer Triangulation innerhalb einer Methode und zwischen Methoden unterschieden. Triangulation Process In order to use seismographs to locate the epicenter of a quake, at least 3 different instruments in different locations must be used. [64] Eine gut begrndete Lsung dieses Problems gibt es jedoch bis heute nicht. Change), You are commenting using your Twitter account. Flexible you can often adjust your methods as you go to develop new knowledge. [citation needed] In France, the law of cosines is still referred to as the theorem of Al-Kashi. If you want to develop a more mechanistic understanding of a topic, or your research involves hypothesis testing, collect quantitative data. https://benjamins.com/catalog/z.220 [citation needed] Ulugh Beg also gives accurate tables of sines and tangents correct to 8 decimal places around the same time. WebA simulation is the imitation of the operation of a real-world process or system over time. Biruni himself introduced triangulation techniques to measure the size of the Earth and the distances between various places. Developing your research methods is an integral part of your research design. Given a circle and an arc on the circle, the chord is the line that subtends the arc. Trenor, Yu, Waight, Zerda, and Sha (2008) [7] investigated the experience of female engineering students at university. Pelto (2017, pp. She teaches qualitative research methods, and has written on qualitative interviewing. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulation represents the evolution of the model over time.Often, computers are used to execute the simulation. Psychological Bulletin, 56(2), 81-105. doi:10.1037/h0046016. Reviews aren't verified, but Google checks for and removes fake content when it's identified, Qualitative Research & Evaluation Methods, Top Ten Pieces of Advice to a Graduate Student. Diese Seite wurde zuletzt am 16. [16], Norman K. Denzin entwickelte 1970 die bis heute am weitesten verbreitete Legitimation und Typologisierung der Triangulation. As these historical calculations did not make use of a unit circle, the length of the radius was needed in the formula. WebTriangulation means using more than one research method to improve the validity of the study. [3], The modern words "sine" and "cosine" are derived from the Latin word sinus via mistranslation from Arabic (see Sine and cosine#Etymology). [32] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.[9]. In the 18th century, Leonhard Euler's Introduction in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their infinite series and presenting "Euler's formula"eix=cosx+isinx. Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. In an effort to unite the interrelated enterprises of sociology theory, research, and substantive interest (p. 3), Denzin outlined the ways in which qualitative researchers can use triangulation of method, investigator, theory and data in naturalistic inquiry (p. 294). [64] In der Praxis vertrauen Forscher dabei eher durch qualitative Methoden erzielten Ergebnissen, was aber wohl eher der emotionalen Involviertheit der Forscher mit qualitativen Daten denn einer formal-logischen Legitimationsbegrndung zu verdanken ist. Die Methodentriangulation ist die am weitesten verbreitete Methode der Triangulation,[35] wenngleich die Verwendung unterschiedlicher Methoden selbstverstndlich hufig auch die Verwendung unterschiedlicher Daten und somit Datentriangulation impliziert. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Contrast this with the modern use of the, Jacques Sesiano, "Islamic mathematics", p. 157, in, On the Sizes and Distances of the Sun and Moon, The Compendious Book on Calculation by Completion and Balancing, Muhammad ibn Jbir al-Harrn al-Battn, "A reconstruction of Gunter's Canon triangulorum (1620)", "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", "Abu Arrayhan Muhammad ibn Ahmad al-Biruni", "USI, NAIR-AL-DIN i. These have to be viewed as two independent philosophies that are different from one another. Whats the difference between method and methodology? [25] They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. WebIt was taken over by sociology in the 1930s when the Chicago school studied "deviant subcultures" in urban America in the great depression. [71], Dieser Artikel wurde am 28. Spezielle Formen dieser Art der Triangulation sind die Zeit- und Ortstriangulation,[30] bei der Daten zu verschiedenen Zeiten beziehungsweise an verschiedenen Orten erhoben werden, um so Idiosynkrasien des historischen Kontexts zu eliminieren. New to this edition: a) Twelve primary strategic themes of qualitative inquiry to clarify readers understanding of the different strands of qualitative research; b) Five distinct criteria-based frameworks for presenting and judging qualitative findings; c) Sixteen different theoretical and philosophical approaches to qualitative inquiry identified, compared and contrasted; d) Variations in observational methods, including historical perspectives, ethical issues, and case studies; e) Alternative interviewing strategies and approaches, including focus group interviews, group interviews, and cross-cultural interviews; f) Additional new coverage on: new issues in and approaches to fieldwork; in-depth treatment of emergent designs and purposeful sampling; detailed analytical guidelines, including software and computer-assisted options; strategies for enhancing quality and credibility of qualitative findings, mixed methods, and triangulation; and, a review and listing of the latest internet resources. WebTHREE TYPES OF RESEARCH AND RESEARCH PARADIGMS RESEARCH METHODS AND METHODOLOGY Kwesi Atta Sakyi TYPES OF RESEARCH AND PARADIGMS GSB 5011 Summer 2017 Prof. Richard Zigler ASSIGNMENT 4 Due Date: 18th June 2017 1 THREE TYPES OF RESEARCH AND RESEARCH PARADIGMS Research is a process Also in the 18th century, Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. For qualitative data, you can use methods such as thematic analysis to interpret patterns and meanings in the data. Dependent Variables | Definition & Examples, Inductive Reasoning | Types, Examples, Explanation, Inductive vs. Deductive Research Approach (with Examples), Internal Validity in Research | Definition, Threats & Examples, Internal vs. b Stephen Gorard, Katie Rushforth, Chris Taylor: Eugene J. Webb, Donald T. Campbell, Richard D. Schwartz, Lee Sechrest: Berth Danermark, Mats Ekstrm, Liselotte Jakobsen, Jan Ch. Positivism and post-positivism have to be viewed as philosophies used in science for scientific inquiry. It is described by Ramon Llull of Majorca in 1295, and laid out in the 1436 atlas of Venetian captain Andrea Bianco. These included: Denzin concludes the book with a discussion of the problems with multiple-triangulated investigations (pp. | Definition & Examples, What Is Data Cleansing? 153-166). My books include: How Hard Should Interviewers Push for Sensitive Information? [8] It is due to the Babylonian sexagesimal numeral system that each degree is divided into sixty minutes and each minute is divided into sixty seconds. [12] Disziplinbergreifend ist Triangulation allerdings eher ein Desiderat in Lehrbchern denn ein in der Praxis verbreitetes Forschungsprogramm,[13] obgleich beispielsweise 2004 in einer Umfrage unter britischen Bildungsforschern 71% aller Befragten angaben, sie wrden auf Triangulation zurckgreifen. Denzin argued strongly for interactionist research that used multiple approaches, commenting that. One half of the bisected chord is the sine of one half the bisected angle, that is,[12], and consequently the sine function is also known as the half-chord. For questions about ideas, experiences and meanings, or to study something that cant be described numerically, collect qualitative data. Eine weitere Mglichkeit ist es, dass verschiedene Forscher die Daten analysieren. In den 1990er Jahren schlug Valerie Janesick eine fnfte Art der Triangulation, die interdisziplinre Triangulation, vor. He has conducted applied research and evaluation on a broad range of issues, including antipoverty initiatives, leadership development, education at all levels, human services, the environment, public health, medical education, employment training, agricultural extension, arts, criminal justice, mental health, transportation, diversity initiatives, international development, community development, systems change, policy effectiveness, managing for results, performance indicators, and effective governance. | Examples & Definition, What Is Probability Sampling? Original theoretical works, research reports, literature reviews, software reviews, book reviews and announcements are published. [47] Dies hat zur Folge, dass die Kombination verschiedener Methoden immer auch zunchst die Entscheidung fr ein bestimmtes erkenntnistheoretisches Forschungsparadigma voraussetzt. In mixed methods research, you use both qualitative and quantitative data collection and analysis methods to answer your research question. Methods are the specific tools and procedures you use to collect and analyze data (for example, experiments, surveys, and statistical tests). With this in mind, the identity upon which trigonometry is based turns out to be the Pythagorean Theorem. | Definitions & Examples, What Is a Focus Group? [66] Manche Autoren, darunter Norman W. H. Blaikie und Yvonna S. Lincoln, glauben daher sogar, dass Triangulation in der Praxis nur innerhalb (post-)positivistischer Anstze erfolge. Allows you to describe your research subject without influencing it. Priority should be equal but can be given to either approach. -- Description from http://www.amazon.co.uk (Feb. 11, 2015). [29] In einer Untersuchung zu Sprachkursen knnen so beispielsweise Schler, Lehrer und das administrative Personal befragt werden,[28] da jede dieser Quellen vermutlich eine andere Herangehens- und Sichtweise zu den Sprachkursen aufweist. Ein besonders bekanntes Beispiel dieser Art innerhalb der Ethnomethodologie ist Cicourels unbegrenzte Triangulation, bei der sowohl Forscher als auch Forschungssubjekte sowie Forschungsassistenten bei der Generierung verschiedenartigen Datenmaterials und dessen Auswertung mitwirken. [42] He also developed the following trigonometric formula:[43], In his original text, Ab al-Waf' states: "If we want that, we multiply the given sine by the cosine minutes, and the result is half the sine of the double". Web triangulation; any other reasonable response. To understand how something occurs in its natural setting. [45], The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying[46] and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century. Data integration can also occur during analysis. {\displaystyle \sin \left(a-b\right)} [69], Von Vertretern der Postmoderne wird dagegen angemahnt, dass Triangulation verkenne, dass jede Methode eine andere Sicht auf ein Phnomen wirft; sie schlagen daher statt Triangulation Kristallisation in der Methodenkombination vor, bei der eben gerade nicht quivalente Ergebnisse erzielt werden sollen. To understand general themes in the data and how they are communicated. In statistics, sampling allows you to test a hypothesis about the characteristics of a population. [41] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. If you want to analyze a large amount of readily-available data, use secondary data. Sandra Mathison (1988) has since outlined an alternative way of conceptualizing triangulation, suggesting that there are three possible outcomes of using the approaches to triangulation discussed by Denzin (1978): When a research study yields not convergence on a single explanation, but data that supports inconsistent or contradictory explanations, Mathison (1988, p. 17) asserts that the researchers need to make sense of what we find. In Mathisons view, this moves the focus on triangulation away from a technological solution for ensuring validity and places the responsibility with the researcher for the construction of plausible explanation about the phenomena being studied (p. 17). (2014). [53] hnlich der theoretischen Triangulation werden hier Anstze verschiedener Fachdisziplinen verwendet, um die Erklrung eines Phnomens zu entwickeln. The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above. The MRCA of a set Traditionally clubs host over the board, face to face chess more than play on internet chess servers or computer chess. For example, if you are researching the opinions of students in your university, you could survey a sample of 100 students. The Pythagorean Theorem, p2 + b2 = h2 is a representation of the fundamental trigonometric identity sin2(x) + cos2(x) = 1. | Types & Examples, What Is Predictive Validity? Professional editors proofread and edit your paper by focusing on: Your data analysis methods will depend on the type of data you collect and how you prepare it for analysis. multiple methods should be used in every investigation, since no method is ever free of rival causal factors (and thus leads to completely sound causal propositions), can ever completely satisfy the demands of interaction theory, or can ever completely reveal all the relevant features of empirical reality necessary for testing or developing a theory (p. 28). [17] In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". | Tips, How to Do Thematic Analysis | Step-by-Step Guide & Examples, How to Write a Literature Review | Guide, Examples, & Templates, How to Write a Strong Hypothesis | Steps & Examples, Inclusion and Exclusion Criteria | Examples & Definition, Independent vs. | Definition & Examples, What Is Convergent Validity? To statistically analyze the results of a large collection of studies. In experimental research, you systematically intervene in a process and measure the outcome. WebUsing qualitative research can often help clarify quantitative results in triangulation. Rent and save from the world's largest eBookstore. [20] A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. Reliability Test-retest reliability Assessing the same person on two different occasions which shows the extent to which the test produces the same answers. Methoden-Triangulation in der qualitativen Forschung. How do I decide which research methods to use? To understand general characteristics of a population. [19] The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity. It provides background for those who will encounter this methodology in their reading WebThis paper provides a concise but comprehensive review of research on religion/spirituality (R/S) and both mental health and physical health. In this text, Denzin clearly situates his work in the field of sociology, employing principles of symbolic interactionist theory. Requires training in data collection methods. Interactional studies of qualitative interviews. Frequently asked questions about research methods, Between-Subjects Design | Examples, Pros & Cons, Cluster Sampling | A Simple Step-by-Step Guide with Examples, Confounding Variables | Definition, Examples & Controls, Construct Validity | Definition, Types, & Examples, Content Analysis | Guide, Methods & Examples, Control Groups and Treatment Groups | Uses & Examples. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748). VS Verlag fr Sozialwissenschaften. [20], Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.[21]. The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae. The validity of your research will depend on your experimental design. [48], Schlielich ist bei der Methodenwahl immer zu beachten, dass die gewhlten Methoden kongruent mit den jeweils benutzten Theorien sind; Methodenmix sollte nicht zum Selbstzweck werden. Primary data is any original information that you collect for the purposes of answering your research question (e.g. Quantitative research deals with numbers and statistics, while qualitative research deals with words and meanings. He was on the faculty of the University of Minnesota for 18 years, including 5 years as director of the Minnesota Center for Social Research, where he was awarded the Morse-Amoco Award for innovative teaching. Quantitative analysis uses numbers and statistics to understand frequencies, averages and correlations (in descriptive studies) or cause-and-effect relationships (in experiments). [10], Parallel zur Bedeutung qualitativer Methoden in den verschiedenen sozialwissenschaftlichen Disziplinen ist die Triangulation vor allem weit verbreitet in deren Hochburgen wie der Ethnographie;[11] in mehr quantitativ ausgerichteten Disziplinen wie der Volkswirtschaftslehre ist sie dagegen seltener anzutreffen. Alternativ oder in Verbindung mit dieser Vorgehensweise, knnten zum Beispiel experimentelle Daten, Umfragedaten oder aus einer teilnehmenden Beobachtung ber die verschiedenen Personengruppen gesammelt werden. Can be collected to answer your specific research question. Michael Quinn Patton is an independent consultant with more than 40 years experience conducting applied research and program evaluations. [40][41] Muhammad ibn Jbir al-Harrn al-Battn (Albatenius) (853-929 AD) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1 to 90. WebTriangulation ist eine Forschungsstrategie in der empirischen Sozialforschung, bei der verschiedene Methoden oder Sichtweisen auf das gleiche Phnomen angewendet werden oder verschiedenartige Daten zur Erforschung eines Phnomens herangezogen werden, um mit den Strken der jeweils einen Vorgehensweise die Schwchen der jeweils anderen WebCovers the significant developments in the field of astronomy and astrophysics, including: the sun; solar system and extrasolar planets; stars; the interstellar medium; galaxy and galaxies; active galactic nuclei; cosmology; and instrumentation and techniques, and the history of the development of new areas of research. [39] Bei der Triangulation innerhalb einer Methode wird die gleiche Methode auf verschiedene Art und Weise angewandt. [23] Obwohl letztere Forscher mehr in der Tradition des Relativismus und erstere eher einen kritischen Realismus vertreten, finden sich beide Traditionen in dem in der Triangulationsliteratur hufig zitierten[24] Aphorismus von Richard Levins wieder: [O]ur truth is the intersection of independent lies. Writing from a symbolic interactionist perspective, Denzin argued in his 1978 text that triangulation, or the use of multiple methods, is a plan of action that will raise sociologists above the personalistic biases that stem from single methodologies (p. 294). Seale, C. (1999). One mark for each method correctly identified (up to a maximum of two). Correlation vs. Causation | Difference, Designs & Examples, Correlational Research | When & How to Use, Critical Discourse Analysis | Definition, Guide & Examples, Cross-Sectional Study | Definition, Uses & Examples, Data Collection | Definition, Methods & Examples, Descriptive Research | Definition, Types, Methods & Examples, Ethical Considerations in Research | Types & Examples, Explanatory and Response Variables | Definitions & Examples, Explanatory Research | Definition, Guide, & Examples, Exploratory Research | Definition, Guide, & Examples, External Validity | Definition, Types, Threats & Examples, Extraneous Variables | Examples, Types & Controls, Guide to Experimental Design | Overview, 5 steps & Examples, How Do You Incorporate an Interview into a Dissertation? Ein immer wiederkehrender Kritikpunkt ist die Wahl der Metapher (der Begriff Triangulation stammt ja aus der Geodsie), deren Zweidimensionalitt fr ein veraltetes epistemologisches Weltbild stnde. The completely revised and updated edition of this methodological classic continues to provide practical, comprehensive and strategic guidance on qualitative design, purposeful sampling, interviewing, fieldwork, observation methods, and qualitative analysis and interpretation while integrating the extensive qualitative literature of the last decade. You can use it to interpret data that was collected: Qualitative analysis tends to be quite flexible and relies on the researchers judgement, so you have to reflect carefully on your choices and assumptions. View all posts by qualpage. When planning your methods, there are two key decisions you will make. Biruni himself introduced triangulation techniques to measure the size of the Earth and the distances between various places. Qualitative methods allow you to explore concepts and experiences in more detail. Can be used to systematically describe large collections of things. | Tips & Examples, What Is a Controlled Experiment? In the 7th century, Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. Quantitative methods allow you to systematically measure variables and test hypotheses. To conduct an experiment, you need to be able to vary your independent variable, precisely measure your dependent variable, and control for confounding variables. Systematic Error | Definition & Examples, Reliability vs. Validity in Research | Difference, Types and Examples, Reproducibility vs Replicability | Difference & Examples, Sampling Methods | Types, Techniques & Examples, Semi-Structured Interview | Definition, Guide & Examples, Simple Random Sampling | Definition, Steps & Examples, Single, Double & Triple Blind Study | Definition & Examples, Stratified Sampling | Definition, Guide & Examples, Structured Interview | Definition, Guide & Examples, Survey Research | Definition, Examples & Methods, Systematic Review | Definition, Example & Guide, Systematic Sampling | A Step-by-Step Guide with Examples, Textual Analysis | Guide, 3 Approaches & Examples, The 4 Types of Reliability in Research | Definitions & Examples, The 4 Types of Validity in Research | Definitions & Examples, Transcribing an Interview | 5 Steps & Transcription Software, Triangulation in Research | Guide, Types, Examples, Types of Interviews in Research | Guide & Examples, Types of Research Designs Compared | Guide & Examples, Types of Variables in Research & Statistics | Examples, Unstructured Interview | Definition, Guide & Examples, What Is a Case Study? 27-50). [13][14], Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas. [2] In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function. (LogOut/ [49][50][51], In the 15th century, Jamshd al-Ksh provided the first explicit statement of the law of cosines in a form suitable for triangulation. For quantitative data, you can use statistical analysis methods to test relationships between variables. Secondary data is information that has Some of the early and very significant developments of trigonometry were in India. And Book III contains the "theorem of Menelaus". Heute wird jedoch zumeist zwischen Methoden kombiniert, und hierbei ist dies zumeist die Kombination einer oder mehrerer quantitativer Methoden mit einer oder mehreren qualitativen Methoden, in der Tat ist diese Variante der Methodentriangulation mittlerweile so dominant, dass manche Autoren sie als einzige Form der Triangulation definieren. Where did the term triangulation come from, and how did it come to be used in qualitative inquiry? To gain an in-depth understanding of a specific group or context, or when you dont have the resources for a large study. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form. You can use quantitative research methods for descriptive, correlational or experimental research. Can only be applied to studies that collected data in a statistically valid manner. [15] It seems that the systematic use of the 360 circle is largely due to Hipparchus and his table of chords. | Definition & Examples, What Is Ethnography? Triangulation kommt heute zumeist in der Qualitativen Sozialforschung zum Einsatz, innerhalb deren Paradigmen ber das vergangene Halbjahrhundert die grte Entwicklung der Triangulation stattgefunden hat. [3] Ab Mitte jener Dekade fand die Triangulation Einzug in viele Standardlehrbcher zur empirischen Sozialforschung,[1] ohne dass jedoch systematische methodische Vorgehensweisen gelehrt wurden. [26][27] Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. Mixed methods in ethnographic research: Historical perspectives. He has authored six other SAGE books: Utilization-Focused Evaluation, Creative Evaluation, Practical Evaluation, How to Use Qualitative Methods for Evaluation, Essentials of Utilization-Focused Evaluation, and Family Sexual Abuse: Frontline Research and Evaluation. [29][33] Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that: Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice pointBy such methods he was able to obtain the du l (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha l (difference between chords of arcs differing by 1 degree). [9] To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the statement that, in modern notation, sin/sin