TABLES OF VALUES OF THE TRIGONOMETRIC FUNCTIONS
The usual practice of obtaining the trigonometric functions of a given angle is to compute the trigonometric functions for many angles between 0° and 90° by more accurate methods and to compile these results in the form of a table to which the student may refer.
This table consists of angles from 0° to 45° listed by tenths of a degree in the left-hand column, reading downwards, and angles from 45° to 90° listed by tenths of a degree in the right-hand column, reading upwards. The angles horizontally opposite each other are complementary. For example, the angle 3° on the left has opposite it the angle 87°. In the four columns between are the sinus, the cosines, the tangents, and the cotangents of these two angles. The same numbers have been used for functions of each of the two complementary angles, for as was shown the functions of an angle are equal to the co-functions of its complement.; For the angles on the left, the trigonometric headings at the top of the table indicate the column in which each function may be found.
In general, the tables of trigonometric functions are used for one of the purposes:
1) to find the functional value when the angle is known;
2) to find the angle when the functional value is known.
EXERCISES
I.Read the following words paying attention to the pronunciation:
tangent, practice, angle, cotangent, accurate, inaccurate, opposite, table, compile, sine, cosine.
II.Add the suffixes and translate the words:
-ward(s): up, after, down, to;
-ly: usual, horizontal, accurate, equal, general.
III.Make up sentences of your own using the words and expressions given below:
method for obtaining, of a given angle, opposite, opposite each other, to compute the function.
IV.Answer the following questions:
1. How do we obtain a trigonometric function of a given angle? 2. Of what angles does the list of trigonometric functions consist? 3. What indicates the column in which each function may be found?
V.Translate into Ukrainian:
We have already discussed several methods for obtaining the trigonometric functions of a given angle. However, all the methods taken up were either inaccurate or restricted to special angles. Consequently, the usual practice is to compute the trigonometric functions for many angles between 0° and 90° by more accurate methods and to compile these results in a form of a table to which the student may refer.
VI.Translate into English:
Для отримання тригонометричної функції даного кута, потрібно вирахувати тригонометричні функції для багатьох кутів від 0° до 90° і скласти таблицю.
Таблиці тригонометричних функцій використовуються для знахождення величини функції, коли відомий кут, і для знахождення кута, коли відома величина функції.
TEXT 7
REVIEW
The trigonometric functions of an angle considered as certain ratios which were defined after placing the angle in standard positions on a rectangular coordinate system.
There are only a few angles for which the trigonometric functions can be found by geometrical considerations, and, therefore, tables must be used from which the values of the trigonometric functions of angles between 0° and 45°, but a few theorems about trigonometric functions of complementary and related angles permit us to find from these tables the functions of any angle.
One of the most important applications of the trigonometric functions is the solution of right triangles.
Two-other topics discussed are particularly worth mentioning.1 The first is the" possibility of measuring an angle in different units; and the examination of the, two most frequently used units; the degree and the radian. The second topic is the interpolation between two values given in a table, which is not only used for trigonometric tables but in all cases .where numerical tables are used and values have to be found which are between two values in the table.
Note:
1 are particularly worth mentioning — слід окремо згадати про
EXERCISES
I.Read the following words paying attention to the pronunciation:
trigonometric, system, prism, cylinder, polygon, coordinate, cofunctions, cosine, cotangent, cosecant.
II. Add suffixes and translate the newly formed words into Russian:
-ly: particular, possible, frequent;
-cal: geometry, trigonometry;
-tion: define, coordinate, apply;
-ing: place, relate, permit.
III.Make up sentences of your own using the words and expressions given below:
standard position, rectangular, consideration, frequently, interpolation, coordinate system, permit us to' find, have to be found, worth mentioning, worth studying, worth explaining, worth relating.
IV.Answer the following questions:
1. What is a trigonometric function of an angle? 2. What is one of the most important applications of the trigonometric functions? 3. How do we find the functions of any angle?
V.Translate into Ukrainian:
On the basis of the definitions of the trigonometric functions it is easy to determine whether a given function of an angle is positive or negative.
For example, if θ is any angle in the second quadrant,
cos θ = x/r is a negative quantity because r is always positive and x, in this case the abscissa of a point in the second quadrant, is negative.
VI.Translate into English:
Таблиці, в яких даються значення тригонометричних функцій, називаються натуральними (natural) тригонометричними таблицями. Для вирахування значень синуса і косинуса могуть слугувати одні й ті ж самі таблиці. Так, наприклад, sin 26° и cos 64° мають одне й те ж саме значення. В таблицях В. М. Брадіса значення аргумента синуса розміщені у порядку зростання зверху вниз, а значення аргумента косинуса – знизу вверх.
SUPPLEMENTARY READING
TEXT 1
EUCLID
Euclid is known to us almost exclusively from those of his works which have survived.
Euclid lived in Egypt approximately 300 B.C. He taught in Alexandria and was the founder of its illustrious mathematical school. His chief extent work is the Elements in 13 books. Books treat of plane geometry, of proportion in general, of the properties of numbers, of incommensurable magnitudes, of solid geometry. Besides the Elements there are the Data — a collection of geometrical theorems.
Euclid's Elements has been translated into many languages, and is probably better known than any other mathematical book, with many of its blemishes removed and its deficiencies supplied, it is still widely used in Britain as a text-book of geometry, though attempts have been made for the last 150 years to supersede it.
The first printed edition of Euclid was a translation from Arabic into Latin, which appeared at Venice in 1482. The first printed Greek text was published at Rasel in 1533. The most recent edition is that of Heiberg in 5 volumes (1883-88).
TEXT 2
PYTHAGORAS
Pythagoras is for us at once the glorified and the actual founder of the philosophical school (which exercised a great influence on the course of ancient science). He was also a great mathematician. Pythagoras investigated harmonies and properties of numbers. His attention was turned to the odd and even, to prime numbers, square numbers and so.
The great mathematical discovery of Pythagoras is of course a hypotenuse theorem, where the square is equal to the sum of two squares. "Pythagorian numbers" are such numbers as are related in the way the theorem indicates. Various other theorems are closely connected with this cardinal one; these concern chiefly the squares of the various perpendiculars which may be let fall from different angles of the right-angled triangle upon the hypotenuse and sides.
TEXT 3
NEWTON
The greatest of natural philosophers, was born on 25th December 1642 — year remarkable in the history of science by the birth of Newton and the death of Galileo. He was born at the hamlet of Woolsthorpe in Lincolnshire in the family of a farmer.
He received his early education at the grammar-school at the same hamlet. On the 5th June 1661 he left home for Cambridge, where he was admitted the same year. He applied himself there to the mathematical studies. After a few years he began to make some progress in the methods for extending the science.
In 1666, the fall of an apple, as he walked in the garden at Woolsthorpe, suggested the most magnificent of his subsequent discoveries — the law of universal gravitation.
He accordingly abandoned the hypotheses for other studies. He investigated the nature of light and was led to the conclusion that rays of light which differ in colour differ also in refractivity.
Newton became a professor of mathematics in 1669. In 1671 he resumed his calculations about gravitation.
In 1696 he was appointed warden of the Mint and was afterwards promoted to the office of Master of the Mint in 1699, an office which he held till the end of his life. He took a seat in parliament in the year 1701 as the representative of his university.
A mathematical feat is recorded of him so late as 1716 in solving a problem prpposed by Leibnitz for the purpose, as he expressed it, of feeling that pulse of the English analysts.
In 1699 Newton was elected a foreign associate of the Academy of Sciences. He died at Kensington on 20th March 1727 and was buried at Westminster Abbey, where a monument was erected to his memory in 1731.
TEXT 4
LEIBNITZ
Distinguished for almost universal scholarship especially in philosophy and mathematics Leibnitz was born on 1st July 1646 at Leipzig, where his father was professor of Moral Philosophy. He attended the school in Leipzig, but learned much more from independent study. He spent some time also in Jena working at mathematics. He graduated at Altdorf, the university town of Nurnberg.
Some years later he invented a calculating machine and devised what was in many respects a noble method of calculations. This gave rise to a controversy with Newton as to which of them first invented this valuable mathematical method.
In 1676 Leibnitz quitted the service of Mainz and was appointed a custodian of the library of Hanover. In 1687 he visited various cities in Germany, Austria and Italy.
Leibnitz was also a pioneer in the science of comparative philology. He died on 14th November 1716 at Hanover.
TEXT 5
GAUSS
Karl Friedrich Gauss, German mathematician was born at Brunswick, on 30th April 1777.
In 1801 he published an important work on the theory of numbers and other analytical subjects: Disquisitions Arithmetical. He was appointed as professor of Mathematics and director of the observatory at Gottingen. He also worked with equally brilliant success in the science of geodesy and astronomy.
Later in life (in 1843-46) he published a collection of valuable memoirs on surface geometry. He also studied the problems arising out of the earth's magnetic properties. In 1833 he wrote his first work on the theory of magnetism.
In applied mathematics he investigated the problems connected with the passage of light through a system of lenses in 1846. Besides the researches already mentioned he wrote papers or works on probability, the method of least squares, the theory of biquadratic residues, constructed tables for the conversion of fractions into decimals and of the number of classes of binary quadratic forms, and discussed hyper- geometric series, interpolation, curved surfaces, all of which he printed in the seven volumes of his collected works.
Gauss died at Gottingen, on 23rd February 1855.
TEXT 6
SOPHIA KOVALEVSKAYA
The outstanding Russian mathematician Sophia Kovalevskaya was born in Moscow on February 15,1850, in a well- off family of an artillery general, Korvin-Krukovsky.
Sophia's childhood was spent in Polibino, where the family used to live the greatest part of the year. When Sophia was eight an experienced teacher was invited to Polibino to instruct her in arithmetic, grammar, literature, geography and history. Though she liked literature so much that it seemed that literature would become her ultimate object in life, the girl showed an unusual gift in mathematics and at the age of twelve puzzled her teacher by suggesting a new solution for the determination of the ratio of the diameter of the circle to its circumference.
In 1867 Sophia and her elder sister were taken to St. Petersburg. There Sophia was allowed to go on with her studies privately. To attend lectures at the University a woman had to obtain a special permission, and even then by no means would she be allowed to take examinations, to say nothing of taking a degree.
This state of things remained unaltered despite the efforts of many scientists who voiced an urgent demand that women should be granted the right to education. The only way out for her was to go abroad, as some other Russian women did. But in this case there was a condition that the woman should be married. This made her marry Vladimir Kovalevsky, with whom she soon left for Vienna. There the Kovalevskys were given permission to attend lectures on physics at the Vienna University, but this did not satisfy Sophia. She made up her mind to go to the Heidelberg University to study under such scholars as Helmholz and Bunzen, as her intention was to take examinations for a Doctor's Degree in mathematics and mechanics. While in Heidelberg, she would attend eleven lectures a week, including eight lectures on mathematics and do a lot of practical work as well. In 1871 Sophia went to Berlin, where she read privately with professor Weierstrass, as the public lectures were not then open to women. During the four years spent in Berlin, Sophia succeeded not only in covering the university course of mathematics but also in writing three dissertations. In 1874 the University of Gottingen granted her a degree of Doctor of Philosophy in absentias excusing her from the oral examinations in consideration of the three dissertations sent in, one of which, on the theory of partial differential equations, was one of her most remarkable works.
When the Kovalevskys returned to Russia they planned to live and work in St. Petersburg, but despite the efforts of Mendeleyev, Butlerov and Chebyshev, Sophia Kovalevskaya, a great scientist could not find a position there and was obliged to turn to journalism.
In 1878 Sophia gave birth to a daughter and as her husband was promised a lectureship at the Moscow University, she decided to take her Magister's Degree there. Great was her disappointment when she learned that her application had not been accepted, though her personal experience should have suggested her that there was no use in trying to get a degree in Russia. Again she went to Berlin to complete her work on the refraction of light in crystals, but the news of her husband's bankruptcy and suicide caused her to return home.
In 1883 she was given an opportunity to report on the results of her research at a session held in Odessa, but no post followed. Therefore, when she was offered lectureship at Stockholm University she willingly accepted the offer and went there with her little daughter.
In 1888 she achieved the greatest of her successes winning the highest prize offered by the Paris Academy. The problem set was: "to perfect in one important point the theory of a movement of a solid body about an immovable point." The solution obtained by her made a valuable addition to the results submitted by Euler and Lagrange.
In 1889 she was awarded another prize by the Swedish Academy of Science. Soon, in spite of her being the only woman-lecturer in Sweden, she was elected professor of mechanics and held the post until her death.
Unfortunately Sophia Kovalevskaya did not live to reap the full reward of her labour, for she died on February 10, 1891, at the age of 41, just as she had attained the height of her fame and had won recognition even in her own country by election to membership of the St. Petersburg Academy of Sciences.
TEXT 7
NIKOLAI LOBACHEVSKY
Lobachevsky (1792-1856) will always be regarded as one of the greatest thinkers. Like Archimedos, Galileo, Copernicus and Newton, he is one of those who laid the foundations of science.
Lobachevsky became seriously interested in mathematics while still a schoolboy. He remained true to this science all his life.
During the 2000 years before Lobachevsky, geometry was based on The Elements of Euclid. Euclid gave the axiom on parallel lines asserting that there can be only one parallel to the given line through the point outside that line. It was this theory that Lobachevsky attacked.
Lobachevsky proved that there could be several parallels to the given line through a point outside that line.
The revolution in geometry achieved by Lobachevsky is the most significant example of the radical transformation which our conceptions of space have undergone. He demonstrated the need to study the properties of real space experimentally. Lobachevsky's ideas exercised a profound influence in the development not only of geometry but also of other mathematical sciences, as well as on mechanics, physics and astronomy.
TEXT 8
MATHEMATICIAN No. 1
At an international mathematical symposium a French scientist came up to academician Andrei Kolmogorov and said: "I thought that I would never be able to see you, I could not believe that all the books written by Kolmogorov actually belong to one person. I thought that it was a pen-name for a whole group of authors. I'm very glad indeed to meet you."
Kolmogorov's colleagues, many of whom are scientists of worldwide fame, call him "Mathematician No. 1." Kolmogorov has written more than 220 works. He is the creator of the modern theory of probability, and has applied it in diverse fields of human activities, ranging from the theory of directed firing (during the war artillerymen used Kolmogorov's tables) to the theory of information.
Kolmogorov has lately begun to work on the application of mathematical statistics in the analysis of verse. His reports on this subject have stimulated great interest among linguists and poets.
One rarely meets Kolmogorov alone. He is always surrounded by his colleagues and pupils at the University, where he heads the department of the theory of probability, and in his summer cottage in the small Komarovka village near Moscow.
His cottage in Komarovka is like a hiking centre, a place for the start and finish of long hikes and skiing and boating excursions which he often organises with his pupils. New maths theories had often been outlined in talks by the camp fires. Kolmogorov's pupils return from these "maths hikes" not only with fresh ideas and high spirits. They also have rich impressions of landscapes and people in the Moscow countryside and of the architectural ancient relics there.
TEXT 9
ABOUT COMMON FRACTIONS
"Fractio" is a Latin word meaning "to break". When a bone is fractured, it is broken. Fractions, in arithmetic, are used to express parts.
Several thousand years before the beginning of our era the Egyptians living in the valley of the Nile River, had a highly developed civilization. The fractions which they used were unit fractions. A unit fraction has 1 for its numerator. When the Egyptians wished to express the quantity which we call 3/4, they used 1/2+1/4.
The Babylonians, who lived in southwestern Asia, thought of the whole as being broken into sixty equal parts. Each of these sixty parts was thought of as broken into sixty equal parts. These fractions were called sexagesimal fractions, from "sexaginta" meaning 60. We still use the idea of these fractions in dividing our hour into 60 minutes and each minute into 60 seconds. The quantity which we call 3/4 would have been 45/60.
The Greeks used sexagesimal fractions too. They also used unit fractions which were represented by writing the denominator followed by an accent ('). 1/4 would have been written as 4 with an accent after it, thus 4'. When they used fractions which were not unit fractions they wrote the numerator once and the accented denominator twice. Using then symbols for 3 and 4 they would have written 3/4 as 34'4'. The Romans divided the whole into 12 parts. Englishmen still divide one foot into 12 parts or inches. In the seventh century of our era, a Hindu writer used the plan of writing the numerator over the denominator, 3/4 would then be 3/4.
The Arabs made one more change, inserting a bar between the numerator and denominator, giving us a present form for writing fractions 3/4.
TEXT 10
THE TRUTH IN GODEL'S PROOF
Ever since the time of Euclid, 2,200 years ago, mathematicians have tried to begin with certain statements called "axioms" and then deduce from them all sorts of useful conclusions.
In some ways it is almost like a game with two rules. First, the axioms must be as few as possible. If you can deduce one axiom from the others, that„deduced axiom must be dropped. Second, the axioms must be self-consistent. It must never be possible to deduce two conclusions from the axioms with one the negative of the other.
Any high school geometry book begins with a set of axioms: that through any two points only one straight line can be drawn; that the whole is equal to the sum of the parts, and so on. For a long time, it was assumed that Euclid's axioms were the only ones that could build up a self-consistent geometry to that they were "true".
In the 19th century, however, it was shown that Euclid's axioms could be changed in certain ways and that different "non-Euclidean geometries" could be build up as a result. Each geometry was different from the others, but each was self-consistent. After that it made no sense to ask which was "true". One asked instead which was useful.
In fact there are many sets of axioms out of which a self- consistent system of mathematics could be built; each one different, each one self-consistent.
In any such system of mathematics you must not be able to deduce from its axioms that something is both so and not- so, for then the mathematics would not be self-consistent and would have to be scrapped. But what if you make a statement that you can't prove to be either so or not-so?
Suppose, that I say: "The statement I am now making is false."
Is it false? If it is false, then it is false that I am saying something false and I must be saying something true. But if I am saying something true then it is true that I am saying something false, and I am indeed saying something false. I can go back and forth forever. It is impossible to show that what I have said is either so or not-so.
Suppose you adjust the axioms of logic to eliminate the possibility of my making statements like that. Can you find some other way of making such neither-so-nor-not-so statements?
In 1931, an Austrian mathematician, Kurt Godel presented a valid proof that showed that for any set of axioms you can make statements that cannot be shown to be so from those axioms and yet cannot be shown to be not-so either. In that sense, it is impossible to work out, ever, a set of axioms from which you can deduce a complete mathematical system.
Does that mean that we can never find "truth"? Not at all.
First: Just because a mathematical system isn't complete doesn't mean that what it does contain is "false". Such a system can still be extremely useful provided we do not try to use it beyond its limits.
Second: Godel's proof applies only to deductive systems of the types used in mathematics. But deduction is not the only way to discover "truth". No axioms can allow us to deduce the dimensions of the solar system. Those dimensions were obtained by observations and measurements — another route to "truth".
TEXT 11