Sharp trigonometric function 1. The concept of acute trigonometric function: in Rt△ABC, the ratio of the opposite side to the hypotenuse of (1) acute angle ∠A is the sine of ∠A, which is recorded as the opposite hypotenuse of Sina = ∠ A; (2) The ratio of the adjacent side to the hypotenuse of acute angle ∠A is the cosine of ∠A, which is recorded as the adjacent hypotenuse of COSA = ∠ A; (3) The ratio of the opposite side to the adjacent side of acute angle ∠A is the tangent of ∠A, which is recorded as the adjacent side of the opposite side of Tana = ∠A; (4) The ratio of the adjacent side to the opposite side of acute angle ∠A is the cotangent of ∠A, which is recorded as the opposite side of Cota = ∠A; (5) The included angle (α) between the slope and the horizontal plane is called the slope angle, and the ratio of the vertical height of the slope to the width of the horizontal plane is called the slope I (or the slope ratio), which means that the slope is equal to the tangent of the slope angle, and is recorded as; (6) The sine, cosine, tangent and cotangent of acute angle A are all called acute trigonometric functions of ∠A; Note: sinA, COSA, Tana and Cota are defined in the right triangle (pay attention to the combination of numbers and shapes to construct the right triangle). Her essence is a ratio, and its size is only related to the size of ∠ A.2. The trigonometric function relationship between two complementary angles: (1) The sine of the acute angle is equal to the cosine of the other angles, that is, sinA=cosB or sinB = cosA；; (2) The cosine of the acute angle is equal to the sine of the other angles, that is, cosA=sinB or cosB = sinA；; (3) The tangent of the acute angle is equal to the cotangent of the other angles, that is, tanA=cotB or tanB = cotB；; (4) The cotangent of acute angle is equal to the tangent of other angles, that is, cotA=tanB or cotB = tanA；; 3. trigonometric function relation between the same angles: (1) sum of squares relation:; (2) Reciprocal relationship: (3) Business relationship: 4. Trigonometric function value of special angle: α sin cos tancot 3045 1160 Solve right triangle1,and clarify the basis and thinking of solving right triangle. In a right triangle, we use the ratio of three sides to represent the definition of acute trigonometric function. Therefore, the definition of acute trigonometric function reveals the relationship between the angles in right triangle, which is the basis of solving right triangle. As shown in figure 1, in Rt△ABC, ∠ c = 90, and the sides of three internal angles A, B and C are A, B and C respectively (the following letters are the same), then the main basis for solving a right-angled triangle is the relationship between (1) angles: sinA=cosB= =. (2) The relationship between two acute angles: A+B = 90; (3) Trilateral relationship: Each of the above angular relationships can be regarded as an equation. The idea of solving the right triangle is to correctly select the relationship between the angles of the right triangle according to the known conditions and solve it by solving the unary equation. 2. Basic types and methods of solving right-angled triangles We know that the process of finding unknown elements from known elements in right-angled triangles is called solving right-angled triangles. In a right triangle, there are three sides and two acute angles * * *, so what kind of right triangle can be solved? If you know two acute angles, can you solve a right triangle? In fact, the solution of right triangle is essentially related to the judgment and drawing of right triangle, because it is known that two elements (at least one of which is an edge) can judge the congruence of right triangle or make right triangle, that is, right triangle is certain at this time, so such right triangle is solvable. Because it is known that two right-angled triangles with acute angles are uncertain and are countless similar right-angled triangles, it is impossible to find out the length of each side. So to solve a right triangle, at least one of the two elements except the right angle must be an edge. In this way, right-angled triangles can be divided into two categories, namely, a known side and an acute angle or two known sides. The four basic types and solutions are listed as follows: one side and one acute angle of the known conditional solution A and acute angle AB = 90-A, B = A? TanA, c= hypotenuse c and acute angle ab = 90-a, a = c? Sina, b=c? Two right-angle sides a and b on both sides of cosA, b = 90-a, right-angle side a and hypotenuse c sinA=, b = 90-a,

The general review of junior high school mathematics is the key link to systematically improve and deepen the learned content after completing the three-year mathematics teaching task in junior high school. Paying attention to and earnestly completing the teaching task at this stage is not only conducive to consolidating, digesting and summing up the basic knowledge of mathematics, improving the ability of analyzing and solving problems, but also conducive to the practical application of employed students. At the same time, for students with poor learning foundation, it is a kind of re-learning to find out the gaps and master the contents of the textbook. Therefore, it is one of the basic skills of junior high school mathematics teachers to arrange and implement the general review teaching in a planned and step-by-step manner.

First, stick to the outline and carefully prepare the review plan.

The content of junior high school mathematics is complex, and the basic knowledge and skills are scattered in the three-year textbooks. Students often learn new things and forget old things. Therefore, the review plan must be carefully formulated according to the contents stipulated in the outline and the systematic knowledge points. The preparation of the plan must conform to the students' reality. We can use the problem-solving method of basic knowledge, according to the practical application knowledge of students in normal teaching, and compile a test that permeates the main knowledge points, so that students can complete it independently within the specified time. Then determine the focus of the plan according to the contents that appear in the exam that are difficult for students to understand, have a high forgetting rate, are easy to be confused and make mistakes. After the review plan is made, we should make a good choice of examples in the review class and screen out the supporting homework for the exercises. The review plan made by the teacher should be handed over to the students, who are required to make specific review plans according to their own learning reality and determine their own goals.

Second, trace the source and systematically master the basic knowledge.

The first stage of review, first of all, emphasizes that students systematically master the basic knowledge and skills in textbooks and make them better. Put forward clear requirements for students: ① Basic concepts, laws, formulas and theorems should not only be described correctly, but also be used flexibly; (2) The exercises after the textbook must be passed one by one; The review questions after each chapter are more comprehensive, requiring most students to complete them independently, and a few students with difficulties can complete them under the guidance of the teacher.

Thirdly, systematic arrangement to improve the review efficiency.

In the second stage of general review, the leading role of teachers should be particularly reflected. The junior high school mathematics knowledge will be systematically sorted, classified, classified and reorganized according to the mutual connection and transformation of basic knowledge, and become a systematic and organized knowledge point. For example, algebra in grade three can be divided into function definition, positive and negative proportional function and linear function; Quadratic equation, quadratic function, quadratic inequality; Three parts of statistics. Geometry is divided into four 13 lines: the first line is 1 line with right triangle as the main body. The second similar shape is divided into three lines: (1) proportional line segment; (2) similar triangles's judgment and nature. (3) Determination and properties of similar polygons; The third circle contains seven lines: (4) the nature of the circle; (5) straight lines and circles; (6) circle and circle; (7) Angle and circle; (8) Triangle and circle; (9) Quadrilateral and circular; (10) polygons and circles. The fourth block is a drawing problem, which has two lines: (1 1) making a circle, making the internal and external common tangent of the circle, etc. (12) point trajectory. This kind of induction and summary can be done by both teachers and students under the guidance of teachers, that is, students make the finishing touch and teachers make the finishing touch. Classes in middle schools and below are classified by teachers and compared. Block exercises and comprehensive exercises are carried out alternately, so that students can really master the contents of junior high school mathematics textbooks.

Fourth, concentrate on practicing and strive for the best results.

After finishing in blocks and mastering the content of the textbook, the third stage of comprehensive review will begin. In this stage, in addition to paying attention to the key chapters in the textbook, it is mainly based on repeated exercises to give full play to the students' main role. Usually, comprehensive exercises based on chapter comprehensive exercises and systematic knowledge are given priority to, and the weight of simulation questions is appropriately increased. For teachers, the main task at this time is to carefully select exercises, carefully correct the exercises completed by students, make comments in time, find out the missing contents, consolidate the review effect and achieve the goal of self-improvement. Two problems should be paid attention to when choosing comprehensive exercises: First, the selected exercises should be purposeful, typical and regular. Second, exercises should be enlightening, flexible and comprehensive. For example, the proof and application of the angle bisector theorem, the application of the angle of a circle, the central angle of a circle, the central angle of a chord, the power theorem of a circle and the projective theorem in the proof of a circle are all comprehensive and important topics that should be mastered, and we should grasp and produce results.