trigonometry

[Edit this paragraph] Name definition

A mathematical discipline that studies the relationship between the angles of a plane triangle and a spherical triangle. Trigonometry is a subject based on the study of the relationship between the sides and angles of a triangle, which is applied to measurement and also studies the properties and applications of trigonometric functions.

[Edit this paragraph] The origin of trigonometry

Trigonometry originated in ancient Greece. In order to predict the running route of celestial bodies, calculate the calendar, navigate and other needs, the ancient Greeks studied the relationship between the angles of a spherical triangle, and mastered the theorems that the sum of two sides of a spherical triangle is greater than the third side, the sum of the internal angles of a spherical triangle is greater than two right angles, and the angles are equal. Indians and Arabs also studied and popularized trigonometry, but it was mainly used in astronomy. In the 15 and 16 centuries, the study of trigonometry shifted to the plane triangle to achieve the purpose of measurement application. /kloc-The Vedas, a French mathematician in the 6th century, systematically studied plane triangles. He published a book about the mathematical laws applied to triangles. From then on, the plane triangle was separated from astronomy and became an independent branch. The contents of plane trigonometry mainly include trigonometric functions, solving triangles and trigonometric equations.

Triangulation also appeared very early in China, and it was explained in detail in the classic Weekly Parallel Computation as early as 100 BC. For example, in its first chapter, it is recorded that "Duke Zhou said," You are a good talker, so please use a moment's method. Shang Gao said that the flat moment is to use straight cable, the moment is to suppress the height, the complex moment is to measure the depth, and the lying moment is to know the distance. "(The moment mentioned by Shang Gao refers to two squares with vertical sides used by workers today, and the general idea of Shang Gao is that the height, depth and width of the target can be measured by putting the squares in different positions. ) 1 century "Nine Chapters of Arithmetic" has a chapter devoted to measurement.

[Edit this paragraph] The history of trigonometry

The early trigonometry was not an independent discipline, but attached to astronomy, which was a method to calculate astronomical observation results, so it was first developed by spherology. There is trigonometry in mathematics in Greece, India and Arabia, but most of them are by-products of astronomical observation. For example, Menelaus of Alexandria (AD 100) wrote "The Study of the Ball" and put forward the basis of trigonometry. Fifty years later, another ancient Greek scholar Ptolemy wrote Astronomy, which initially developed trigonometry. In 499 AD, Indian mathematician Ayabata also expressed the trigonometric thought of ancient India. Later, Varahamihira (about 505 ~ 587) first put forward the concept of sine and gave the earliest sine table. Some Arab scholars in 10 century further discussed trigonometry. Of course, all these works are part of astronomical research. It was not until Nasir-ud-Deen (1201~1274) that trigonometry began to break away from astronomy and become a pure mathematics. The first mathematician to separate trigonometry from astronomy was Reggio Montanus (1436 ~ 1476).

? The main work of Reggio Montanus is to study various triangles, which was completed in 1464. This is the first trigonometry work in Europe that is independent of astronomy. ***5 volumes, the first 2 volumes discuss plane trigonometry, and the last 3 volumes discuss sphericity, which is the source of trigonometry spread in Europe. Reggio Montanus also made some trigonometric function tables earlier.

? The work of Reggio Montanus laid a solid foundation for the application of trigonometry in plane and spherical geometry. After his death, his manuscripts were widely circulated among scholars and finally published, which had a considerable impact on mathematicians in the16th century, and also had a direct or indirect impact on a group of astronomers such as Copernicus.

? The English word trigonometric is trigonometry, which comes from the Latin tuigonometuia ... and was first used by the German mathematician Pittis Chius (B. Pitiscus,1561~1613) in the Renaissance. He coined this word in Trigonometry: A Concise Method for Solving Triangles published by 1595. Its composition method is composed of the words "triangle" and "measurement". The calculation of measurement is inseparable from trigonometric function table and trigonometric formula, which are developed as the main content of trigonometry.

? Rhaticus (G.J. Rhetucus, 15 14 ~ 1574) was the first Austrian mathematician to make trigonometric function tables in16th century. 1536 Graduated from the University of Tenberg and stayed there to teach arithmetic and geometry. 1539 went to Poland to study astronomy with the famous astronomer Copernicus, 1542 was hired as a professor of mathematics at Leipzig University. Rhaticus compiled tables of all six trigonometric functions for the first time, including the first detailed tangent table and the first printed secant table.

/kloc-After the invention of logarithm in the 7th century, the calculation of trigonometric function was greatly simplified. It is no longer difficult to make trigonometric function tables, and people's attention has turned to the theoretical research of trigonometry. However, the application of trigonometric function table has always occupied an important position and played an irreplaceable role in scientific research and production and life.

? The triangle formula is the relationship between sides and angles, or the relationship between sides and angles of a triangle. The definition of trigonometric function has embodied certain relations, and some simple relations have been studied by the ancient Greeks and later Arabs.

? In the late Renaissance, French mathematician F Vieta became a master of trigonometric formulas. His Mathematical Laws Applied to Triangles (1579) is one of the earliest monographs that systematically discussed planes and spheres. The first part lists six trigonometric function tables, some of which are separated by fractions and degrees. The trigonometric function values accurate to 5 digits and 10 digits are given, and the multiplication table and quotient table related to trigonometric values are attached. In the second part, the method of making the table is given, and the calculation formula of the relationship between the number of river streamlines in the triangle is explained. In addition to summarizing predecessors' achievements, I also added my own new formulas, such as tangent law and sum-difference product formula. He listed these formulas in a general table, so that after giving some known quantities at will, the values of unknown quantities can be obtained from the table. This book is based on a right triangle. For the oblique triangle, David imitated the method of the ancients and turned it into a right triangle. For spherical right triangle, a complete calculation formula and memory rules, such as cosine theorem, are given. 159 1 year, David got the relationship of multiple angles, 1593, and deduced the cosine theorem by triangle method.

1722, the British mathematician de Moivre got the trigonometry theorem named after him.

? (cosθ isinθ)n=cosnθ+isinnθ，

? It is proved that this formula holds when n is a positive rational number. In 1748, Euler proved that the formula also holds when n is an arbitrary real number, and he also gave another famous formula.

? eiθ= cosθ+isθ，

? It has played an important role in promoting the development of trigonometry.

Modern trigonometry began with Euler's introduction to infinite analysis. He defined the unit circle and trigonometric function by the ratio of function line to radius. He also created lowercase Latin letters A, B and C to represent the three sides of a triangle, and uppercase Latin letters A, B and C to represent the three angles of a triangle, thus simplifying the triangle formula, and further transforming trigonometry from studying triangle solutions to studying trigonometric functions and their applications, becoming a relatively complete branch of mathematics.

[Edit this paragraph] The characteristics and application of trigonometry

The early trigonometry was not an independent discipline, but attached to astronomy, which was a method to calculate astronomical observation results. Therefore, it was first developed in sphericity. There are trigonometry in mathematics in Greece, India and Arabia, but most of them are by-products of astronomical observation. Until the13rd century, the Central Asian mathematician Nasuraddin wrote the book "Complete Quadrilateral" on the basis of summarizing the achievements of predecessors. It was not until the15th century that trigonometry was separated from astronomy. The publication of On Triangle by German J. Regio montan us (1436- 1476) marks that ancient trigonometry has officially become an independent discipline. This book not only contains very accurate sine tables and cosine tables, but also gives modern trigonometry.

/kloc-in the 6th century, the French mathematician F. Viet (1540-1603) further systematized trigonometry. In his first book on trigonometry, Mathematical Rules Applied to Triangle, there are detailed solutions of right triangle and oblique triangle (Swiss mathematician L Euler (25438+08 century). 1707- 1783), he first studied trigonometry, which liberated trigonometry from the original static solution of triangle research and became a subject with modern mathematical characteristics, reflecting some movements and changes in the real world. Euler not only defined trigonometry with rectangular coordinates, completely solved the symbolic problem of four-quadrant trigonometry, but also introduced rectangular coordinates. There is a bridge between algebra and geometry, and the combination of numbers and shapes provides an important way of thinking for the study and research of mathematics. The famous Euler formula connects trigonometric functions and exponential functions that people thought were unrelated to each other, adding new vitality to trigonometry.

Therefore, trigonometry originated from surveying practice, and after a long period of brewing, with the continuous efforts of many Chinese and foreign mathematicians, it gradually enriched and evolved into the present trigonometry.

[Edit this paragraph] Calculation method of trigonometric function

There are six trigonometric functions in trigonometry, which are defined by geometric methods. In the rectangular coordinate system, let the angle with ray Ox as the starting edge and OP as the ending edge be θ, and the coordinate of point P be (x, | op | = R. At this time, the six ratios are all functions of θ, which are called trigonometric functions of angle θ, and are recorded as sine, cosine, tangent, cotangent, secant and cotangent of angle θ respectively. Tg, ctg and csc are also denoted as tan, cot and cosec respectively.

There are three groups of operational relationships between trigonometric functions with the same angle, namely

Trigonometric functions are all periodic functions with a period of 2π.

The basic identity of trigonometric function is the sum angle formula;

sin(a+β)=sinαcosβ+cosαsinβ

cos(a+β)=cosαcosβ-sinαsinβ

From these two formulas, we can deduce the formula of difference angle, double angle, half angle, sum-difference product and sum-difference formula.

When some elements (edges and angles) of a triangle are known, solve the triangle and find out the remaining unknown elements. Let three angles of a triangle be A, B and C, and their opposite sides be A, B and C respectively, then there are

Sine theorem: a/sinA=b/sinB=c/sinC=2R(2R is a constant in the same triangle, twice the radius of the circumscribed circle of the triangle).

Cosine theorem: A2 = B2+C2-2bccosa is the main basis for solving triangles.

Trigonometric equations generally refer to equations containing some trigonometric functions, and the independent variables of trigonometric functions contain unknowns. Since every trigonometric function is a periodic function, any trigonometric equation has infinite solutions as long as there are solutions.

triangulation

Triangulation refers to the technology of accurately measuring distance and angle in navigation, surveying and civil engineering, which is mainly used for positioning ships or aircraft. Its principle is that if one side and two angles of a triangle are known, then the other two angles can be calculated by plane trigonometry. In the west, Pythagoras, a famous mathematician in ancient Greece, proved the Pythagorean theorem about right triangle for the first time, that is, China's Pythagorean theorem, which made great contributions to the research and application of geometry.