△ In △ABC, the following triangular relations are often used to solve related triangular problems, which should be accurately remembered, well remembered and flexibly used.
4. To solve the oblique triangle problem, it is usually necessary to abstract one or several triangles from the actual problem according to the meaning of the problem, and then get the required quantities by solving these triangles, so as to get the solution of the actual problem. Among them, the thinking method of establishing mathematical model is also the destination of our learning mathematics, and solving practical problems by mathematical means is the fundamental purpose of learning mathematics.
When solving problems, we should reasonably choose sine and cosine theorems according to the known and unknown, so as to make the problem-solving process concise, the algorithm concise, the formula neat and the calculation accurate.
(1). Steps to solve the application problem of oblique triangle
(1) Accurately understand the meaning of the question, distinguish the known from the unknown, and accurately understand the related nouns and terms in the application question, such as elevation angle, depression angle, visual angle, direction angle, azimuth angle and slope, latitude and longitude, etc. ;
(2) Draw a picture according to the meaning of the question;
③ The problem to be solved is reduced to one or several triangles, and the mathematical model is established by using the knowledge of sine theorem and cosine theorem reasonably, and then it is solved correctly. The calculation process should be concise and accurate, and the answer should be given at last.
(2) Relevant nouns and terms in practical application.
① Elevation angle and depression angle: the included angle between the horizontal line of sight and the target line of sight in the same vertical plane. When the target line of sight is higher than the horizontal line of sight, it is called elevation angle, and when the target line of sight is lower than the horizontal line of sight, it is called depression angle.
② Direction angle: the horizontal angle from the specified direction line to the target direction line.
③ Azimuth: the horizontal angle from the clockwise direction of the specified direction line to the target direction line.
④ Inclined plane: the degree of dihedral angle formed by inclined plane and horizontal plane.
(3) Be familiar with the related formulas in the triangle.
Sine theorem and cosine theorem are mainly used to solve oblique triangles, and sometimes perimeter formula and area formula are also used, such as:
(Is the perimeter of a triangle)
(Indicate the height on the side)
(can be deduced from sine theorem)
(is the radius of the inscribed circle)
You must also be familiar with the sine, cosine and tangent formulas of the sum and difference of two angles.
Fifth, pay attention to the key points
1. In our textbook, sine theorem is derived from right triangle, which shows that sine theorem is also true for right triangle. We should also know that the traditional derivation method of sine theorem is to derive first and then divide by various formulas to get sine theorem.
In teaching materials, the formula of sine theorem is derived by using vector knowledge. It is the concrete application of plane vector knowledge.
2. Pay attention to the deformation application of sine theorem.
It is not difficult for us to prove (where r is the radius of the circumscribed circle).
In this way, the sine theorem can have the following changes:
, , ;
, , ;
;
, , ;
, , 。
These relationships can be applied according to the conditions of questions and conclusions.
3. Discussion on knowing the diagonal of two sides and one of them and solving the triangle.
Given the diagonal of two sides and one of them, the shape of a triangle cannot be uniquely determined. There will be no solution, one solution and two solutions to solve this triangle problem, which should be discussed in different situations. Figures 1 and 2 show various situations when solving triangles in the middle, known and a.
When alpha is an acute angle,
When α is a right angle or an obtuse angle,
4. Each equation of cosine theorem contains four different quantities, which are three sides and an angle of a triangle. If we know three of them, we can get the fourth quantity. If we know three sides, we can get the angle. At this time, we can use cosine theorem to get another form.