Trigonometric function calculation problem

Daily use is to look up the table. And accurate numerical calculation is a formula supported by theory.

Trigonometric function.

Although trigonometric knowledge originated in ancient times, it was first given by Euler (1707- 1783) in the book Introduction to Infinitesimal Analysis. Before Euler, the research of trigonometric function was mostly carried out in a circle with a certain radius. For example, Ptolemy in ancient Greece set the radius at 60; India's Ayabata (about 476-550) has a radius of 3438; German mathematician Giovannas (1436- 1476) once set a radius of 600,000 in order to accurately calculate the value of trigonometric function. Later, in order to make a more accurate sine table, the radius was set to 107. So the trigonometric function at that time was actually the length of some line segments in the circle.

Italian mathematician Le Tix (15 14- 1574) changed the previous practice, that is, in the past, AB was generally called sine, and sine and circle were firmly connected, but Le Tix called it ∠AOB sine, which made the sine value directly linked to the angle and the circle O became subordinate.

It was not until Euler defined the radius of the circle as 1, that is, put the angle in the unit circle, that the trigonometric function was defined as the ratio of the radius of the corresponding line segment to the circle.

Sine and cosine

Sine Theorem was first discovered and proved by the famous Iranian astronomer Abu Rewi (940-998). Alberta Rooney of Central Asia (973- 1048+05) gave a proof of triangle sine theorem.

It is also said that the proof of sine theorem is the first time that Hilding discussed trigonometry as an independent subject in On Complete Quadrilateral in the 3rd century/kloc-0, and clearly demonstrated sine theorem for the first time. He also pointed out that three sides can be obtained from three angles of a spherical triangle, or three angles can be obtained from three sides. This is an important symbol to distinguish spherical triangle from planar triangle. At this point, trigonometry began to break away from astronomy and embark on the road of independent development.

The first volume of Claudius Ptolemy's "Astronomical Masterpieces" includes not only some major astronomical data, but also the string table mentioned above. It gives the length of the chord subtended by all central angles from (1/2) to 180. The radius of a circle is divided into 60 equal parts and the chord length is equal.

For example, crd 36 = 37p4'55 ",which means that a chord with a central angle of 36 is equal to the radius (or 37 small parts), plus a small part, plus a small part. As can be seen from the figure below, the chord table is equivalent to the sine function table.

At the beginning of the 6th century, Indian mathematician aryabhata made a sine table with an interval of 3 45' in the first quadrant. According to the custom of Babylonians and Greeks, the circumference is divided into 360 degrees, each degree is 60 minutes, and the whole circumference is 265,438+0,600 copies. Then according to 2πR = 2 1 600, R = 3,438. Among them, using the same unit to measure radius and perimeter gave birth to the earliest concept of arc system. When calculating the sine value, he takes the half chord length of the arc opposite the central angle, which is closer to the modern concept of sine than the Greeks take the full chord length. Indians also used sine vectors and cosines, and gave some approximate fractions of trigonometric functions.

2. Tangent and cotangent

Al Albatani (850-929), a famous Syrian astronomer and mathematician, made a cotangent table with an interval of 1 from 0 to 90 around 920.

In 727 AD, Emperor Xuanzong of the Tang Dynasty instructed monks and their entourage to compile the Grand Nautical Calendar. In order to find out the length of solar terms in any part of the country in one year, a table corresponding to the zenith distance of the sun and the length of the shadow is compiled with an eight-foot pole, and the relationship between the zenith distance of the sun and the length of the shadow is a tangent function. Albatani compiled a cotangent function table, and the height of the sun is a tangent function.

/kloc-In the middle of the 4th century, Arub (1393- 1449), originally a descendant of Genghis Khan, organized large-scale astronomical observation and calculation of mathematical tables. His sine watch is accurate to nine decimal places. He also created a distance of 65440 between 30 and 45.

In Europe, the British mathematician and Archbishop of Canterbury Bravadine (1290? -1349- First, tangent and cotangent are introduced into his trigonometric calculation.

3. Secant and cotangent

The concepts of "﹝secant﹞" and "﹝cosecant﹞" were first put forward by Abreu-Weaver. The abbreviation of sec is 1626, and the Dutch number Kirader+0595- 1630.

During the Renaissance in Europe (14th century-16th century), the great astronomer Copernicus (1473- 1543) advocated the theory of the earth motion, and his student Litex Lovech Pcf saw that astronomical observations at that time became more and more accurate, every/kloc-0. There were no logarithms at that time, let alone calculators. The task is heavy. Littix and his assistants worked hard with perseverance 12 years. Unfortunately, he didn't finish the work until 1596. It was completed and published by his student Otto (1550- 1605), and published in Heidelberg that year, worldwide,161515665438+.

In modern times, Taylor series is generally used to expand. According to how many decimal places you need to be accurate, the more projects you expand, the more accurate you will be.

arcsin x=∑(n=~∞)[(n)！ x^(n+)/[^n*(n！ *(n+)]

arctan x=∑(n=~∞)[(-)^n]x^(n+)/(n+)