First, the relationship between solving triangle and judging triangle congruence
Solving a triangle discusses the relationship between various geometric quantities in a triangle, such as the relationship between sides, angles, areas, circumscribed circle radius and inscribed circle radius. Sine theorem and cosine theorem are the main tools to solve a triangle. Plane geometry mainly studies triangles from the qualitative point of view, solving triangles mainly studies the relationship between various geometric quantities in triangles from the quantitative point of view, and studies triangles by analytical methods. The two research angles are different and can complement each other.
Axioms for judging triangle congruence are: edge axiom (SAS), edge axiom (SSS), edge axiom (ASA) and edge axiom (AAS). At least one element is an edge, and only three angles (AAA) correspond to two equal triangles, which are similar but different. The geometric significance of judging the congruence condition of triangle is that other variables of triangle can be expressed by a given set of variables. For example, the geometric significance of SSS axiom in determining triangle congruence is that the lengths of three sides of △ABC can uniquely determine its three internal angles. For example, if the three sides of △ABC are known, a triangle can be obtained through the inference of cosine theorem. The geometric significance of judging triangle congruence by SAS axiom lies in that the lengths of two sides of △ABC and their included angles uniquely determine the length of the third side, and then uniquely determine the lengths of the other two sides. If both sides of △ABC and their included angle c are known, the third side can be found by cosine theorem. At this time, the three sides are known, and the remaining two angles can be calculated by the inference of cosine theorem. This is exactly two kinds of problems that cosine theorem can solve: knowing three sides and finding triangle (SSS); Knowing two sides and their included angles, find the third side and the other two angles (SAS).
With the help of triangle interior angle sum theorem, we can think that angle axiom (ASA) and angle edge axiom (AAS) are essentially the same. Its geometric meaning is that two angles and any edge of △ABC can uniquely determine the remaining angles and edges, such as two angles A and B and clamping edge C of △ABC. It can be found that this is a kind of problem that can be solved by sine theorem: knowing two angles and any edge, find the remaining edges and angles (Asa, AAB). Sine theorem can also solve a kind of problem: knowing the diagonal of two sides and one of them, finding the third side and the other two angles (SSA). Geometrically speaking, SSA can't determine the congruence of triangles, so it can't uniquely determine a triangle, which shows that there will be two solutions, one solution and no solution, when solving triangles with sine theorem.
From sine theorem and cosine theorem, edge axiom (SAS), edge axiom (SSS), edge axiom (ASA) and edge axiom (AAS) are equivalent.
As can be seen from the above, when learning textbooks, we should grasp the textbooks from the overall and overall height, and understand the structure, status, function and interrelation of the textbooks, so that they can interpret and complement each other and produce new opinions. In teaching, analyzing the relationship between triangle congruence judgment axiom and triangle solution can improve students' cognitive structure and sublimate junior high school knowledge.
Second, mathematical thinking methods
The teaching of mathematical thinking method is an important part of mathematics teaching, which is conducive to deepening students' understanding and mastery of mathematical knowledge and improving their ability to solve mathematical problems. The two main conclusions of this section are sine theorem and cosine theorem. In teaching, we should attach importance to the teaching of mathematical thinking methods closely related to the content, and give concrete demonstration and guidance to students in raising questions and thinking about strategies to solve problems.
In the part of sine theorem, considering that it is not easy to directly get the relationship between each side and each angle in a general triangle, students can be guided to discover this law by considering the knowledge of trigonometric functions related to each angle in a right triangle, and then guess the generality of this law, and then prove it in an acute triangle and an obtuse triangle, so as to get the sine theorem. This process embodies the mathematical thought of classification discussion from special to general. When proving the conclusion in acute triangle and obtuse triangle, it is also proved by transforming it into right triangle, which embodies the mathematical thought of reduction.
In the part of cosine theorem, after obtaining cosine theorem, the form of cosine theorem is analyzed and the problem of finding angles on three sides is put forward. Combined with the idea of equation, the inference of cosine theorem is obtained, and the conclusion of judging the "edge, edge, edge" of triangle congruence is described quantitatively. After proving cosine theorem and its inference, the textbook compares cosine theorem with Pythagorean theorem. Put forward a thinking problem: "Pythagorean theorem points out the relationship between three squares in a right triangle, and cosine theorem points out the relationship between three squares in a general triangle." How to see the relationship between these two theorems? "Combined with the analysis of the properties of cosine function, it is concluded that cosine theorem is a generalization of Pythagorean theorem, and Pythagorean theorem is incorporated into the knowledge system of cosine theorem, which embodies the idea from general to special.
This paper introduces the application of sine theorem and cosine theorem through two different types of examples. Sine theorem mainly introduces two types of "corner edge" and "edge edge", while cosine theorem mainly introduces two types of "edge edge" and "edge edge", which embodies the idea of classified discussion.
Third, the relationship between mathematical knowledge
The proof and application of sine theorem and cosine theorem involve many mathematical knowledge, such as vector, trigonometric function, analytic geometry and so on. This should be paid attention to in teaching.
Sine theorem and cosine theorem describe the quantitative relationship between the angles of a triangle, which is closely related to the basic relationship between the angles of a triangle and the knowledge of judging the congruence of a triangle. When introducing the content of sine theorem into the textbook, students are required to start from the existing geometric knowledge and ask an exploratory question: "There is a corner relationship between a big side and a big angle, and a small side and a small angle in any triangle. Can you get an accurate quantitative representation of the relationship between edges and angles? " When introducing the content of cosine theorem, starting from the congruence of triangle learned in junior high school, this paper qualitatively explains that the triangle is completely determined when both sides and included angles of the triangle are known, thus raising the question: Can the third side be calculated quantitatively when both sides and included angles of the triangle are known? Finally, sine theorem and cosine theorem are based on solving triangles, which makes the axiom of judging triangle congruence learned in junior high school reasonably explained. It is a sublimation from qualitative to quantitative. It can also be said that the two have found resonance here and merged into one. In this way, looking at past problems from a new perspective with a contact point of view will enable students to have a new understanding of past knowledge, and at the same time make new knowledge on the basis of existing knowledge and form a good knowledge structure.
Mathematics Curriculum Standard for Compulsory Education arranges Sine Theorem and Cosine Theorem after Compulsory 5. Before this content, students have learned trigonometric functions, plane vectors, analytic geometry and other contents closely related to this chapter, which makes this part of the content have more tools, such as the proof of sine theorem. The textbook uses the trigonometric function relationship between the angles of a right triangle. In fact, it can also be proved by the circumscribed circle and vector of a triangle. Cosine theorem can be proved not only by vector method used in textbooks, but also by coordinate method, with the help of distance formula between two points and triangle knowledge. In teaching, paying attention to the application of all kinds of proof methods can not only consolidate all parts of knowledge, understand the internal relationship between mathematical knowledge, and reflect the role and strength of mathematical knowledge such as vectors and trigonometric functions, but also broaden the thinking, extract the essence and refine the optimal problem-solving method by comparing various methods.
Therefore, when teaching sine theorem and cosine theorem, we should pay attention to the connection with the contents of the previous chapters, review and apply what we have learned to prepare for the contents of the following chapters. In this way, the whole set of teaching materials can become an organic whole, improve the teaching effect and help students learn and consolidate mathematics knowledge.