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Definitions of the trigonometric functions

Sketches from the past

The first trigonometric calculations have appeared in the ancient times of Babylon and Egypt mostly for the needs of astronomy. In the Rhindov papyrus (Ahmes calculation) in the 18th Century B.C. there are some indications of the usage of the geometrical methods which were used by the Egyptians while building pyramids and measurement of fields. The special denomination SEGT is used, but it is still not clear whether this term is used to refer to today's cosine or cotangent.

The pioneers of trigonometry are considered to be the Greeks in the 3rd century B.C., especially the astronomer Aristarh and his pupil Hiparh from Nikea. He made the fist tables for the different axial angles. Menelai (1stcentury B.C.) for the first time shows trigonometry as the separate science in his book Sferika. In the 2nd century A.D. we can notice the big advance in trigonometry done by Ptolomei.
(~100. - ~178.),the creator of the geocentric system, in his work well known under the Arabic name of “Akmagest”. This work continued to be the basis for the science of the trigonometry for ages. Ptolomei himself made the tables of the chord values (with five decimals) in the relation with the axial angle, and with the distance of 30?. He used the method quite similar to the modern additional theorem in order to calculate the tables, which is itself closely connected to the theorem on square, which is today named after Ptolomei.
The first tables which were alike the tables of the sinus functions were made by the Indus in the 5th century A.D. Those tables include the lengths of the half-chords of the circle for the given axial angle. Aeyabhata (475. or 476:550.) already knows the basic equality sin2t + cos2t=1, as well as the formula for the halfway angle. Bhaskara (1114 : 1178.) knows the additional formulas for sine and cosine.
The knowledge of trigonometry is taken over by the Arabs in the 8th century. They introduce tangent and cotangent. It is of course still the spherical trigonometry. The theorem on sine was for the first time proven by Abu al-Wafa (940. - 998?.).
Europe acquires this knowledge by the Arabs, but is systematically worked on only by Regiomontan (1436.-1476.). He remade and supplemented the Arab tables of the value of the trigonometric functions, passing from hexadecimal to the decimal system. He was the first to learn the theorem on the spherical triangle, although it was first mentioned already in the time of Indus in the 5th century. Regiomontan works also with some formulas for the transformation of the sum into product.
The trigonometrical functions of the angle ( as the proportion of the arm of the rectangular triangle) are introduced by Rhaticus in the 16th century.
Francois Viete (1549 : 1503.) has brought up the theorem on cosine. He for the first time used all the six trigonometrical functions. He connected trigonometry with algebra.
The mathematicians who have completed the knowledge on the spherical trigonometry were John Napier (1550.- 1617.), Leonhard Euler (1770.-1783.), Karl Friedrich Gauss (1777. - 1855.). The important contribution was also given by Ru?er Boškovi? (1711.- 1787.).
They are first introduced as real functions of real variables by Euler who has first systematically connected them with the rectangular triangle with the hypotenuse. He is the first to observe the trigonometric functions according to the chosen angle.

About the names of the trigonometric functions

The name sine has arrived into the European languages through the technique of the broken telephone. The first name for the sine and cosine - jiva and kotjiva was given by the old Indus people. Jiva in their language Sanskrit means “chord” (so the first name was give ordhajiva which means “half chord”) and that name is really in the accordance with the meaning of sine. The Arabs take that name as jiba, which is the word that does not have any meaning in Arab language, so it is change into džaib (written also as džiba) and has the meaning of bay or armpit. The European medieval translator (Robert from Chester) translates that word directly into Latin sine (bay).
The name tangent (because of the connection to tangent) is introduced in 1583. by Fincke. The name cosine came into use in the 17th century (E.Gunter 1620.) as the short form of the complementi sinus. Cosine in the direct translation means: sine of the complementary angle. Cotangent and cosecans were given those names for the same reason.
The name of the trigonometric function was created by Klugel 1770. The signs for the trigonometric functions were introduced in the 17th century by J. Bernoulli. Since then different symbols are used but mainly: s,sc,t,tc. The modern symbols come from Euler (sin, cos, tang, cot). The symbols for minutes, degrees and seconds are introduced by Ptiscus at the end of the 16th century.

The four trigonometrical functions
Sine and cosine according to the chosen triangle

If t is according to the chosen triangle a real number, T = E (t) is the matching point on the numeral circle. Then T = (cos t, sin t). So the value of the function cosine (cos t) is x-axis, and the value of the function sine (sin t) is y-axis of the point T= E (t).
The sine of the angle in the rectangular triangle is the ratio of the corresponding catheti and hypothenuse. The cosine of the angle in the rectangular triangle is the ratio of the adjacent catheti and hypothenuse.

 The graph of the function sine.

The graph of the function cosine

Tangens and cotangens according to the chosen angle

If ,, T= E (t) it’s complementary point on the numeral circle and P intersection of the straight line OT with the tangent p then P= (1, tg,t). The value of the function tangens (tg t) is the y-axis in which straight line OT intersects tangent p.

If , , T=E(t) it’s complementary point on the numeral circle and Q intersection of the straight line OT with the tangent q then Q=(ctg t, 1). The value of the function cotangens (ctg t) is the x-axis in which straight line OT intersects tangent q.

 Graph of the function tangens Graph of the function cotangens

Basic identity

According to the Pitagora’s theorem we can say that for every real number t it is true that sin˛t+ cos ˛t=1.

The first tables connected to the trigonometry were made by the Greek astronomer Hiparh from Nikea. Those tables included the length of the chord for the different axial angles. Ptolomei made more detailed tables of the chord lengths (with five decimals) depending on the axial angle, and with the distance of 30?.

The first tables which were alike the tables of the sine functions were made by the Indus in the 5th century A.D. Those tables include the length of the half chord of the circle for the given axial angle.
Regiomontan (1436.-1476.) remade and supplemented the Arab tables for the value of the trigonometric functions. His tables of the sine have 7 decimals, with angles in distance of 1?.
Rhaticus (1514.-1576.), with the real name Georg Joachim von Lauchen, the young associate of N. Kopernik, has made the tables of the trigonometric functions value with 10 decimals and angles distanced 10?. Those tables were later remade by Bartholomaus Pitiscus (1561.-1613.) by calculating the value of 15 decimals.
After logarithms were introduced, except for the tables of the natural values, the tables of the logarithms of the trigonometric functions are also being made. The first tables of this kind were made by Henry Briggs (1561.-1630.).
Jurij Vega (1754.-1802.) has published logarithm tables with 7 decimals in 1783., and his famous tables of trigonometric functions with 10 decimals in 1794. Those tables named “Thesaurus logaritmorum completes”, had a mistake in the method, so many mistakes occurred in them.

The characteristics of the trigonometric functions
Even and odd parity

Points T?=E(t) and T?= E(-t) are symmetric in relation to axis Ox. Because of that their x-axis are cohere, and y-axis are different in their sign.
Cos(-t) = cos (t) =>even function
Sin (-t) = - sin(t) = >odd function
Tan (-t) = - tan (t) = >odd function
Ctg (-t) = -ctg (t) =>odd function

Periodicity of sine and cosine functions

For the function f we say that it is periodic if there is a real number P>0 that every t?D(f) it is true that f(t)=f(t+P). Number P is called the period of function f. The smallest number P (if it exists) is called the basic period of function f.
The basic period of functions sine and cosine is 2?, while the basic period of functions tangens and cotangens is ?.
The numbers t and t+2? has the corresponding point T on the numeral circle.
The points T?=E(t) and T2=E (t +?) are symmetric in correlation to the starting point 0. Because of that T1, O, T2 lie on the same straight line. In other words the straight lines OT1, OT2 cohere, so the value of tangens and cotangens coheres too, so it is true : tg (t + ?) = tg t I cg (t + ?) = cg t for every t defined by the functions.

Inverse function

 Arkus sine: for every number y from the interval [-1,1] there is just one angle a for which it is true sin a = y and . That angle is marked with a=arc sin y. Arkus cosin: for every number x from the interval [-1,1] there is just one angle a for which it is true sin a= x and . That angle is marked with a= arc sin x. Arkus tangens: for every real number y there is just one angle a for which it is true that tan a= y and . That angle is marked with a=arc tan y.

Trigonometric identities

Four trigonometric functions are interconnected: if we know the value of one of them we can easily determine the value of any other. There are also relations between the same trigonometric function calculated for different values of arguments. As the result of that fact, we can learn that trigonometric functions satisfy a large number of interesting and unusual identities, equalities which are true for every and each value of argument. Those identities crucial in the procedure of putting straight the trigonometric formulas, solving the trigonometric equations and in all other different applications of trigonometry.

Cosine of the sum and subtraction

Cos (t+s) = cos (t)cos(s)-sin(t)sin(s)
Cos(t-s)=cos(t)cos(s)+sin(t)sin (s)

Sine of the sum and subtraction
Sin(t+s)=sin(t)cos (s)+cos(t)sin(s)
Sin(t-s)=sin(t)cos(s)-cos(t)sin(s)

Tangens of the sum and subtraction

If the t and s are real numbers in order to ,. If plus that , than it is true that . And for the it is true that

The formulas for the reduction of the sine and cosine functions.

Cos(n -t) = - cos t
Cos (n +t) = - cos t

Sin ( + t) = cos t

Sin ( - t) cos t

Universal substitution

All the trigonometric functions can be expressed as rational functions of the variable tg.

for all the real numbers for which both sides of identity are defined.

 Sinusoid The function is called sinusoid. Here C >0 a positive constant called amplitude, is called circular frequency, and phase lag. Sinusoid is the periodic function. Her period is .

The usage of trigonometry

 The necessity of exact measurements were through centuries together with the astronomic measurements the most important reason for the development of trigonometry. The basic problem of measurement is to determine the distance of the two mostly unreachable points. The general scheme looks like this: on the reachable part of terrain we determine two prominent points (elevation) and measure their distance. By theodolite we can measure the angles between any three visible points, among which we can also have unreachable points. After that, it is the task of trigonometry to calculate the given distances.
 The way to solve this example was applied in 1752. by French astronomers La Lande and La Caille, trying to calculate the distance to the Moon. They have calculated the angles a' and b' that are closed by the Moon, through two points A and B that are situated on the same meridian, according to the zenithal directions (verticals in A and B). The position of the Moon must be such that it is situated in the ley of the meridian, so, points 0, A, B and M must be complanary. The latitude tp1 and tp2 of the places A and B are known so on the picture the values |AO| and |BO| =R,?=?1+?2 and angles ?= 180ş- ??, ?= 180ş- ??. In this way we got the data from the preceding example and can determine unknown values from the picture. We must also mention that angles x and y are called the parallaxes of the Moon for the points A and B.

Mathematicians

Leonhard Euler (Basel, April, 15th, 1707. - Petersburg, September, 18th, 1783.), was a great Swiss mathematician, physicist and astronomer. He put a great influence on the mathematics as a whole. He learned mathematics from Johann Bernoulli. After the Petersburg science academy is founded, in 1726. he went to live in Russia, where he stays for the rest of his life. He is, together with Cauchy, the mathematician with the biggest number of published scientific works. His text book are well known all over the world: “Introductio in analysin infinitorium”, “Institutiones calculi differentialis”, “Institutiones calculi integralis” and “Arithmetica universalis”. Although he was almost blind, most of his works were written by the end of his life. He was the first to observe the complex variables and connected trigonometric with the exponential functions. He introduced signs for the trigonometric functions mostly alike the ones used nowadays: sn, cos, tang, cot.
His formula V - B + S = 2, on relation of the number of peaks , edges and arms in the polyhedron is well known among mathematicians. He is also considered to be the founder of the graph theory.

Regiomontan (Regiomontanus) is the pseudonym of Johann Miiller (Konigsberg, June 6th 1436. - Rome, July 6th 1476.), was the German astronomer and mathematician. He took part in the programmes of improving the calendar. He introduced trigonometry into the European mathematics. He remade and supplemented the Arab tables of value of the trigonometric functions, and passed from the hexadecimal to decimal system. His sine tables have 7 decimals with angles distanced 1?. His main work is “De train gulis omnimodis libri quinque” ( Five books on triangles of all kinds). The theorem on tangenses: is called after him. He first recognized the theorem on cosine for the spherical triangle, as well as some formulas for the transformation of the sum of trigonometric functions into product of trigonometric functions.

John Napier (Merchiston Castle, Edinburg, 1550.- April 4th 1617.), a Scottish mathematician. He discovered logarithms. He defined logarithms in the way somewhat different from the method used nowadays. In the logarithm tables published in 1614, there is a definition of logarithms, description of their characteristics, logarithms of sine, cosine and tangens, and the usage of the logarithm calculation in the spherical trigonometry. Napier’s tables were not decimal, so it was the reason for the calculation to be so difficult. The decimal table were made by Henry Briggs in 1617. The logarithm calculation was a great improvement in relation to calculation through the natural values. The most of the trigonometric formulas came in the form of a product ( or the quotient) or can be transformed in such forms. The calculation through logarithms enables the operations of multiplication and division to be replaced by addition and subtraction which makes a calculation a lot easier.

Karl Friedrich Gauss (Braunschweig, April 30th, 1777.- Gottingen, February 23rd 1855.), according to many people the greatest mathematician of all times. He was only 19 years old when he solved the problem of constructing the regular heptagon. He was 22 years old when he was the first to prove the basic poster of algebra, according to which each polynom has at least one (complex) zero point. He has made important improvements in all fields of mathematics, so many postulates and theorems were named after him. He was the chief of the mathematics and astronomy department on the Gottingen University since 1807., and besides he managed the local observatory. He was a great practitioner too. He was only 24 years old when he calculated the orbit of the planetoid Ceresa that got lost soon after it was discovered; hiding behind the Sun. It is rediscovered a year later on the exact place foreseen by Gauss. He also tried to check the way that light spread through geodesic measurements by calculating the angles in a triangle which make three prominent points (elevation). Although their sum is not equal to exactly 180? the difference is so small that Gauss could not notice it. He also prepared the mathematical tables which were used for more than hundred and fifty years.

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