The sine theorem and cosine theorem are a set of very important theorems we encountered in high school mathematics, which reveal the angular relationships of triangles, and the combination of the two can solve many (oblique) triangle problems.
You may have wondered why you wanted to study these two theorems, but were they just for the sake of understanding? No! In fact, these two theorems have very wonderful applications, and it is a pity that we brushed up on many problems and ignored the mathematical beauty behind them.
Today, Xiaobian will talk to you about why we have to learn the sine theorem and the cosine theorem.
1 Sine theorem and cosine theorem
First, let's review what is the sine theorem and the cosine theorem:
The sine theorem is in the middle , and if the edge lengths of the opposite sides of the angle are respective , then there is
The cosine theorem is in the middle , if the edge lengths of the opposite sides of the angle are respective , then there is
Using the regular and cosine theorems, we can solve a large number of practical problems.
2 Positive and cosine theorems and mysterious meteors
Meteors are an astronomical phenomenon, a fact that almost every modern person is familiar with, but when we walk through the fog of history, we will find that human cognition of meteors is not clear from the beginning.
It was once speculated that the meteors streaked in the sky were an evaporation of the Earth, or a phenomenon of phosphorus fire burning on the Earth after it was lifted into the air. It was not until the turn of the 18th and 19th centuries that german astronomers Bensenberg and Branders used trigonometry to brilliantly argue that meteors were actually "alien visitors".
As shown in the figure, there are two observers at two observation points on The Earth, and they observe the same meteor, which is obtained by the Radius of the Earth
therefore
The elevation angles of the two observers are known to be respective
rule
Derived from the sine theorem
It can be calculated
This can be obtained from the cosine theorem
That is, the meteor is about the height of the earth's surface
However, science has found that the height of the clouds does not exceed, so we can conclude that the meteor cannot be some kind of evaporation on Earth, it must be an alien visitor! It can be seen that it is the sine theorem and the cosine theorem that help human beings take the first step in correctly understanding this mysterious astronomical phenomenon.
3 Positive and cosine theorems and measurement problems
After the 17th century, with the development of trigonometry, people used trigonometry more to solve many measurement problems.Especially in the early 18th century, the French mathematician Mare (1630-1706) discussed several types of classical trigonometric application problems in his book "Practical Geometry".
Question I.: As shown in the figure, how to measure the height of a building on an island?
On the one hand, the difficulty of this problem is that it is impossible to measure the distance from the observation point to the bottom of the building, but on the other hand, with the help of the goniometer that was invented at that time, we can measure the various angles between the two observation points and the bottom of the building and the top of the building, and the distance between the two land observation points can also be known.
In this regard, we can abstract the following mathematical model:
Known as well, seek.
Solution: In, by the sine theorem:
So
The same principle, in the middle, is derived from the sine theorem:
After calculating the sum, the cosine theorem is used in the sum:
In this way, the problem of altimeter measurement is solved.
Correspondingly, if there is an altimeter measurement problem, there is a ranging problem.
Question II. : As shown in the figure, how to measure the distance between two island buildings?
In fact, with the foreshadowing of problem I, we can more easily understand and solve problem II, abstracting it into the following model:
Following the solution to the altimeter problem above, we only need to use the two sine theorems in the sum and then use the cosine theorem in it.
It can be seen that the altimeterization problem and the ranging problem run through the entire development process of trigonometry. In fact, the important influence of trigonometry in the field of surveying can be seen in its English name "Trigonometry": the word was first coined by the German mathematician B. Pitiscus (1561-1613) in 1595, composed of the Greek words "trigono" (trigono) and "metrein" (measurement), the original meaning of which is trigonometric measurement. Various measurement problems are the basic problems to be studied in trigonometry, and later the meaning of trigonometry has become more and more abundant, and gradually become a mathematical branch for the study of trigonometric functions and their applications.
4 Positive and cosine theorems and planar geometry
Some elementary geometric problems are often difficult to solve with pure geometry, but when we use the positive and cosine theorems, the problem can be simplified. For example, the ancient Greek mathematician Helen proposed the famous "Helen formula" in his book Surveying:
“
Known three sides, notation called half perimeter, the area of the triangle is
This beautiful formula has a beautiful geometric argument that will not be repeated here. In fact, we can also deduce them by using the regular and cosine theorems:
The two sides and their angles are known, and we have
Thanks to the cosine theorem, it is available
It is worth mentioning that this formula is called "three oblique products", discovered by the famous mathematician Qin Jiushao of the Southern Song Dynasty of the mainland, and further deformation can obtain Helen's formula, the two are equivalent.
Further processing of the above formula, obtained
Order, then there is
WANG Xiaoqin. HPM: History of Mathematics and Mathematics Education[M].Science Press, 2017. Wang Xiaoqin,Shen Zhongyu. History of Mathematics and The Teaching of Mathematics in High Schools: Theories, Practices and Cases[M].East China Normal University Press, 2020.
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Source: Big and Small Wu's Math Classroom
EDIT: just_iu