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.
About trigonometric
tables
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
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.
Additional theorem
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.
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.
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 tan y.
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 
.