Notable Great Thinkers In Trigonometry

NOTABLE GREAT THINKERS IN TRIGONOMETRY • Aristotle • Isaac Newton • Ptolemy • Archimedes • Democritus Trigonometry (from Greek trigōnon "triangle" + metron "measure")[1] is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships. Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course. Trigonometry is informally called "trig". A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation. Development of Trigonometry is not the work of any one man or nation. Its history spans thousands of years and has touched every major civilization. It first originated in India and the basic concepts of angle and measurements have been noted in Vedic texts such as Srimad Bhagavatam.[2] However, trigonometry in its present form was established in Surya-siddhanta and later by Aryabhata 5th century CE. It should be noted that from the time of Hipparchus until modern times there was no such thing as a trigonometric ratio. Instead, the Indian civilization and after them the Greeks and the Muslims used trigonometric lines. These lines first took the form of chords and later half chords, or sines. These chord and sine lines would then be associated with numerical values, possibly approximations, and listed in trigonometric tables. If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure: •

The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

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The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

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The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides

respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA. The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and an angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles. Mnemonics A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, as in SOH-CAH-TOA. Sine = Opposite ÷ Hypotenuse Cosine = Adjacent ÷ Hypotenuse Tangent = Opposite ÷ Adjacent The memorization of this mnemonic can be aided by expanding it into a phrase, such as "Silly Old Hitler Couldn't Advance His Troops Over Africa", and "Some Officers Have Curly Auburn Hair Till Old Age"[3]. Any memorable phrase constructed of words beginning with the letters S-O-H-C-A-H-T-O-A will serve. [edit] Calculating trigonometric functions Main article: Generating trigonometric tables Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, Grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions. Pythagorean theorem – formula used in trigonometry and other trigonometric functions.

Economics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek οἰκονομία (oikonomia, "management of a household, administration") from οἶκος (oikos, "house") + νόμος (nomos, "custom" or "law"), hence "rules of the house(hold)".[1] Current economic models developed out of the broader field of political economy in the late 19th century, owing to a desire to use an empirical approach more akin to the physical sciences.[2] A definition that captures much of modern economics is that of Lionel Robbins in a 1932 essay: "the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses."[3] Scarcity means that available resources are insufficient to satisfy all wants and needs. Absent scarcity and alternative uses of available resources, there is no economic problem. The subject thus defined involves the study of choices as they are affected by incentives and resources. Economics aims to explain how economies work and how economic agents interact. Economic analysis is applied throughout society, in business, finance and government, but also in crime,[4] education,[5] the family, health, law, politics, religion,[6] social institutions, war,[7] and science.[8] The expanding domain of economics in the social sciences has been described as economic imperialism.[9][10] Common distinctions are drawn between various dimensions of economics: between positive economics (describing "what is") and normative economics (advocating "what ought to be") or between economic theory and applied

economics or between mainstream economics (more "orthodox" dealing with the "rationality-individualism-equilibrium nexus") and heterodox economics (more "radical" dealing with the "institutions-history-social structure nexus"[11]). However the primary textbook distinction is between microeconomics ("small" economics), which examines the economic behavior of agents (including individuals and firms) and macroeconomics ("big" economics), addressing issues of unemployment, inflation, monetary and fiscal policy for an entire economy.