Matrix Theory and Linear Algebra (Несколько текстов для зачёта), страница 5

The variation of the values of the trigonometric functions for different angles may be represented by graphs, as in Fig. 5. It is readily ascertained from these curves that each of the trigonometric functions is periodic, that is, the value of each is repeated at regular intervals called periods. The period of all the functions, except the tangent and the cotangent, is 360°, or 2  radians. Tangent and cotangent have a period of 180°, or  radians.

Many other trigonometric identities can be derived from the fundamental identities. All are needed for the applications and further study of trigonometry.

The statement y is the sine of θ, or y = sin θ is equivalent to the statement θ is an angle, the sine of which is equal to y, written symbolically as θ = arc sin y = sin^-1y. The arc form is preferred. The inverse functions, arc cos y, arc tan y, arc cot y, arc sec y, arc csc y, are similarly defined. In the statement y = sin θ, or θ = arc sin y, a given value of θ will determine infinitely many values of y. Thus, sin 30° = sin 150° = sin (30° + 360°) = sin (150° + 360°). . .= 1/2; therefore, if θ = arc sin 1/2, then θ = 30° + n360° or θ = 150° + n360°, in which n is any integer, positive, negative, or zero. The value 30° is designated the basic or principal value of arc sin 1/2. When used in this sense, the term arc generally is written with a capital A. Although custom is not uniform, the principal value of Arc sin y, Arc cos y, Arc tan y, Arc cot y, Arc sec y, or Arc csc y commonly is defined to be the angle between 0° and 90° if y is positive; and, if y is negative, by the inequalities

Practical applications of trigonometry often involve determining distances that cannot be measured directly. Such a problem may be solved by making the required distance one side of a triangle, measuring othersides or angles of the triangle, and then applying the formulas below.

If A, B, C are the three angles of a triangle, and a, b, c the respective opposite sides, it may be proved that

The cosine and tangent laws can each be given two other forms by rotating the letters a, b, c and A, B, C.

These three relationships can be used to solve any triangle, that is, the unknown sides or angles can be found when one side and two angles, two sides and the included angle, two sides and an angle opposite one of them (usually there are two triangles in this case), or when three sides are given.

Spherical trigonometry, which is used principally in navigation and astronomy, is concerned with spherical triangles, that is, figures that are arcs of great circles (see Navigation) on the surface of a sphere. The spherical triangle, like the plane triangle, has six elements, the three sides a, b, c and the angles A, B, C. But the three sides of the spherical triangle are angular as well as linear magnitudes, being arcs of great circles on the surface of a sphere and measured by the angle subtended at the center. The triangle is completely determined when any three of its six elements are given, since relations exist between the various parts by means of which unknown elements may be found.

In the right-angled or quadrantal triangle, however, as in the case of the right-angled plane triangle, only two elements are needed to determine all of the remaining parts. Thus, given c, A in the right-angled triangle, ABC, with C = 90°, the remaining parts are given by the formula as sin a = sin c sin A; tan b = tan c cos A; cot B = cos c tan A. When any other two parts are given the corresponding formulas may be obtained by Napier's rules concerning the relations of the five circular parts, a, b, complement of c, complement of A, complement of B. With respect to any particular part, the remaining parts are classified as adjacent and opposite; the sine of any part is equal to the product of the tangents of the adjacent parts and also to the product of the cosines of the opposite parts.

In the case of oblique triangles no simple rules have been found, but each case depends on the appropriate formula. Thus in the oblique triangle ABC, given a, b, and A, the formulas for the remaining parts are

In spherical trigonometry, as well as in plane, three elements taken at random may not satisfy the conditions for a triangle, or they may satisfy the conditions for more than one. The treatment of certain cases in spherical trigonometry is quite formidable, because every line intersects every other line in two points and multiplies the cases to be considered. The measurement of spherical polygons may be made to depend upon that of the triangle. If, by drawing diagonals, one can divide the polygons into triangles, each of which contains three known or obtainable elements, then all the parts of the polygon can be determined.

Spherical trigonometry is of great importance in the theory of stereographic projection and in geodesy. It is also the basis of the chief calculations of astronomy; for example, the solution of the so-called astronomical triangle is involved in finding the latitude and longitude of a place, the time of day, the position of a star, and various other data.

The history of trigonometry goes back to the earliest recorded mathematics in Egypt and Babylon. The Babylonians established the measurement of angles in degrees, minutes, and seconds. Not until the time of the Greeks, however, did any considerable amount of trigonometry exist. In the 2nd century bc the astronomer Hipparchus compiled a trigonometric table for solving triangles. Starting with 7° and going up to 180° by steps of 7°, the table gave for each angle the length of the chord subtending that angle in a circle of a fixed radius r. Such a table is equivalent to a sine table. The value that Hipparchus used for r is not certain, but 300 years later the astronomer Ptolemy used r = 60 because the Hellenistic Greeks had adopted the Babylonian base-60 (sexagesimal) numeration system (see Mathematics).

In his great astronomical handbook, The Almagest, Ptolemy provided a table of chords for steps of °, from 0° to 180°, that is accurate to 1/3600 of a unit. He also explained his method for constructing his table of chords, and in the course of the book he gave many examples of how to use the table to find unknown parts of triangles from known parts. Ptolemy provided what is now known as Menelaus's theorem for solving spherical triangles, as well, and for several centuries his trigonometry was the primary introduction to the subject for any astronomer. At perhaps the same time as Ptolemy, however, Indian astronomers had developed a trigonometric system based on the sine function rather than the chord function of the Greeks. This sine function, unlike the modern one, was not a ratio but simply the length of the side opposite the angle in a right triangle of fixed hypotenuse. The Indians used various values for the hypotenuse.

Late in the 8th century, Muslim astronomers inherited both the Greek and the Indian traditions, but they seem to have preferred the sine function. By the end of the 10th century they had completed the sine and the five other functions and had discovered and proved several basic theorems of trigonometry for both plane and spherical triangles. Several mathematicians suggested using r = 1 instead of r = 60; this exactly produces the modern values of the trigonometric functions. The Muslims also introduced the polar triangle for spherical triangles. All of these discoveries were applied both for astronomical purposes and as an aid in astronomical time-keeping and in finding the direction of Mecca for the five daily prayers required by Muslim law. Muslim scientists also produced tables of great precision. For example, their tables of the sine and tangent, constructed for steps of 1/60 of a degree, were accurate for better than one part in 700 million. Finally, the great astronomer Nasir ad-Din at- Tusi wrote the Book of the Transversal Figure, which was the first treatment of plane and spherical trigonometry as independent mathematical sciences.

The Latin West became acquainted with Muslim trigonometry through translations of Arabic astronomy handbooks, beginning in the 12th century. The first major Western work on the subject was written by the German astronomer and mathematician Johann Müller, known as Regiomontanus. In the next century the German astronomer Georges Joachim, known as Rheticus introduced the modern conception of trigonometric functions as ratios instead of as the lengths of certain lines. The French mathematician François Viète introduced the polar triangle into spherical trigonometry, and stated the multiple-angle formulas for sin(nq) and cos(nq) in terms of the powers of sin(q) and cos(q).

Trigonometric calculations were greatly aided by the Scottish mathematician John Napier, who invented logarithms early in the 17th century. He also invented some memory aids for ten laws for solving spherical triangles, and some proportions (called Napier's analogies) for solving oblique spherical triangles.

Almost exactly one half century after Napier's publication of his logarithms, Isaac Newton invented the differential and integral calculus. One of the foundations of this work was Newton's representation of many functions as infinite series in the powers of x (see Sequence and Series). Thus Newton found the series sin(x) and similar series for cos(x) and tan(x). With the invention of calculus, the trigonometric functions were taken over into analysis, where they still play important roles in both pure and applied mathematics.

Finally, in the 18th century the Swiss mathematician Leonhard Euler defined the trigonometric functions in terms of complex numbers (see Number). This made the whole subject of trigonometry just one of the many applications of complex numbers, and showed that the basic laws of trigonometry were simply consequences of the arithmetic of these numbers.

Equation

Equation, statement of an equality between two expressions, used in almost all branches of pure and applied mathematics and in the physical, biological, and social sciences. An equation usually involves one or more unknown quantities, called variables or indeterminates. These are commonly denoted by letters or other symbols, as in the equations x² + x - 4 = 8, y = sin x + x, and 3y = log x. An equation is named for the number of variables it contains, called an equation in one, two, three, or more variables.

An equation is said to be satisfied or to be true for certain values of the variables if, when the variables are replaced by these values, the expression on the left side of the equals sign is equal to that on the right side. For example, the equation 2x + 5 = 13 is satisfied when x = 4. If one or more values of the variable fail to satisfy the equation, the equation is called conditional. The equation in two variables 3x + 4y = 8 is a conditional equation because it is not satisfied when x = 1 and y = 3. An equation is called an identity if it is satisfied by all possible values of the variables. For example, the equations (x + y)² = x² + 2xy + y² and sin²x + cos²x = 1 are identities because they are both true for all possible values of the unknowns. A solution of a conditional equation is a value of the variable, or a set of values of the variables, that satisfies the equation; thus, 3 is a solution of the equation x² - 2x = 3; and x = 2, y = 4 is a solution of the equation 3x² + 4y = 28. A solution of an equation in one variable is commonly called a root of the equation.

A polynomial equation has the form

in which the coefficients a₀, a₁, ..., a_n are constants, the leading coefficient a₀ is not equal to zero, and n is a positive integer. The greatest exponent n is the degree of the equation. Equations of the first, second, third, fourth, and fifth degrees are often called, respectively, linear, quadratic, cubic, biquadratic or quartic, and quintic equations. Other important types of equations are algebraic, as in = = 7; trigonometric, as in sin x + cos 2x = ; logarithmic, as in log x + 2 log (x + 1) = 8; and exponential, as in 3^x + 2x - 5 = 0. Diophantine equations are equations in one or more unknowns, usually with integral coefficients, for which integral solutions are sought. Differential and integral equations, which involve derivatives or differentials, and integrals, occur in calculus and its applications.

A system of simultaneous equations is a set of two or more equations in two or more unknowns. A solution of such a system is a set of values of the unknowns that satisfies every equation of the set simultaneously.