1 solving skills of trigonometric function in senior high school mathematics
Follow the analytic principle of trigonometric function
In the study of trigonometric function, students face different problems, implement different learning and realize different development. Get the necessary mathematical knowledge, gradually develop a scientific mathematical thinking, and provide everyone with equal learning opportunities. In the teaching process of trigonometric function in senior high school, we should follow the principle of from shallow to deep, and help students master the relevant knowledge of trigonometric function step by step. Because the content of trigonometric function is too abstract, it is difficult for most high school students to fully grasp it. This requires math teachers to start with the basic knowledge in the teaching process, never aim too high, carefully and patiently help students lay the basic knowledge, gradually guide students to think more deeply, gradually master the complex trigonometric function knowledge system, and master the trigonometric function knowledge more comprehensively, thus cultivating students' mathematical thinking.
As a two-way activity, mathematics teaching must attach importance to students' feedback and make constant adjustments according to the feedback. As participants in classroom teaching activities, teachers and students exchange information imperceptibly, and teachers constantly impart knowledge to students. In the process of learning, students also constantly give feedback to their teachers about the problems they don't understand. In the teaching process of trigonometric function in senior high school, we must pay attention to this feedback principle, summarize and analyze in time according to students' classroom reactions and test scores, grasp the main part of students' confusion, and deepen this part of teaching in a targeted manner.
Application of multiple choice questions in trigonometric functions
Multiple choice questions are common questions in high school mathematics, and the application of function knowledge is very common. These questions are similar in type, but in the process of solving problems, the methods used are diversified. When students are faced with the problem of using trigonometric function in multiple-choice questions, they must first master the basic knowledge of trigonometric function and practice various questions at different levels before they can effectively apply trigonometric function to the problem-solving process of multiple-choice questions. Through continuous practice, students have basically mastered certain problem-solving ideas, and can effectively summarize and summarize the relationship between trigonometric functions and multiple-choice questions within their own cognitive level.
Through the mastery and application of trigonometric functions, students constantly expand our own logical thinking and cultivate problem-solving ability and learning ability. Secondly, we should master the concept of trigonometric function in order to make full use of trigonometric function in the process of solving problems. By using the concept of trigonometric function, we can find out the hidden formulas of trigonometric functions in the topic, and increase the ideas and methods to solve multiple-choice questions. To use this method, we must first know how many problem-solving ideas we have mastered, so as to analyze and integrate these useful problem-solving methods in detail and find out the best problem-solving skills.
2 senior high school mathematics trigonometric function analysis skills
Make full use of the combination of numbers and shapes to solve problems
The figure of trigonometric function is related to the definition of coordinates, and then the algebraic problem in mathematics is transformed into the geometric problem on the coordinate axis, and then the number and shape are combined to solve it in the coordinate system. Generally speaking, the distance model and the slope model are commonly used in the combination of numbers and shapes of trigonometric functions. The following questions:
Solve the maximum value of the three-part function y=sinx/(2+cosx). When solving problems, we can use the method of graphic combination to establish a coordinate system, so that P(cosx, sinx) can be used, so that we can clearly know that P is a point on the unit circle. Then draw on the coordinate axis, we can know that the geometric meaning expressed by function Y is the slope of the line connecting the fixed point Q (-2,0) and P, and we can also know that the slope of the line PQ is the largest when it is tangent to the unit circle, and there are two.
Opportunistic, master some special trigonometric functions
In trigonometric function, although many knowledge points are difficult, there are still many skills to be used in solving problems, especially in multiple-choice questions, some "opportunistic" ways can be used to solve problems, thus reducing the time for solving problems. In teaching, teachers need to list some special trigonometric function values and some graphs, which require students to master. For some students with strong understanding ability, they can understand memory, and for those with good memory, they can choose to learn by rote.
After mastering some special numerical values, you can answer questions, especially some complex multiple-choice questions. You can choose to bring in some special values or directly bring in the option "Try Answers". Although it is necessary to write out the steps of solving the problem in detail, by mastering the values of some special functions, we can find the best way to solve the problem faster, and the final answer is generally not wrong. For trigonometric function in senior high school, the special value method is a problem-solving skill with tight examination time and high correct rate, which is worth mastering by students in trigonometric function learning.
3 senior high school mathematics trigonometric function teaching strategy
Effectively create situations and cultivate students' inquiry ability
In fact, the knowledge content of trigonometric function is closely and widely related to our life. Therefore, when teaching trigonometric functions, high school math teachers can make full use of the life characteristics of trigonometric functions, create situations closely related to real life on the basis of their knowledge content, and guide students to actively participate in classroom teaching, feel happy and have a strong desire to explore and apply for jobs. For example, in order to better teach students the image nature of trigonometric functions, guide students to actively participate in the learning process, and enhance their initiative in exploration, teachers can combine the knowledge content of this lesson with the problems in real life before teaching new knowledge, create certain teaching situations, and set the following questions:
Suppose a windmill with a radius of 2m rotates once every 12s. Its lowest point O is 0.5m from the ground, and a point A on the circumference of the windmill starts from O. After it moves t(s), its distance from the ground is set to h(m). What about the relationship between (1) function h=f(t)? (2) Can you draw an image with the function h=f(t)? Under the creation of this problem-based teaching situation, coupled with teachers' encouraging language and feelings of life situation, it will be easy to stimulate students' interest in learning, give full play to their inner feelings of learning, and their desire for exploration has been obviously strengthened. Under the condition of fully mobilizing students' enthusiasm, initiative and inquiry, their inherent initiative will encourage students to actively participate in teachers' overall teaching activities, which is conducive to improving students' ability to analyze and solve problems.
Teachers should guide students to master trigonometric functions comprehensively.
Mathematical knowledge is interrelated. Therefore, in the teaching of trigonometric functions, teachers must hold the overall concept, put trigonometric functions in a broader knowledge framework, flexibly use diversified teaching methods, combine the requirements of new curriculum standards and students' learning characteristics, innovate teaching plans, and guide students to fully understand the relationship between trigonometric functions and non-trigonometric functions, so as to form a better understanding and mastery of trigonometric functions concepts and knowledge more comprehensively and concretely.
Senior high school math teachers should pay attention to strengthening students' reflective and abstract abilities through comprehensive exercises, and guide students to fully understand trigonometric functions, and understand that trigonometric functions such as sin are not just simple operation symbols, but should be mastered as a whole concept. Only in this way can we really understand the experts of trigonometric function and lay the foundation for the deformation and formula derivation after trigonometric function. Senior high school math teachers should make full use of the time and space of classroom teaching and strengthen students' ability of abstract generalization and comprehensive application of trigonometric functions. In addition, the comprehensive analysis method is also one of the effective methods to solve the trigonometric function problem. Because the combination of numbers and shapes is also a commonly used basic mathematical idea, teachers can guide students to comprehensively analyze and apply all available mathematical knowledge when solving mathematical problems, and organically combine them to effectively solve trigonometric function problems.
4 senior high school mathematics trigonometric function line concept teaching
Introduce the concept of trigonometric function line through the history of mathematics
The early solution of triangle was due to the need of astronomical observation, because people needed to cross boundless and uninhabited grasslands and virgin forests at that time, or make an adventurous long-distance voyage along the coastline by waterway. First, they must have a clear direction. /kloc-before the 0/8th century, sine, cosine, tangent, cotangent, secant and cotangent were considered as some line segments related to the same arc in a known circle, that is, trigonometry was expressed by geometric features. This is the classic face of trigonometry. 1748, Euler pointed out in his famous book Introduction to Infinitesimal Analysis: "Trigonometric function is the ratio of the radius of a function line to a circle." That is to say, the trigonometric function of any angle can be regarded as a line segment OP, which is obtained by taking the vertex of the angle as the center, making a circle with a certain length as the radius, and making a vertical line PM from the intersection point P between one side of the angle and the circumference to the other side. The ratios between MP (i.e. function lines) are sinα=MPOP, cosα=OMOP, tanα=MPOM, etc. If the radius is the unit length, all six trigonometric functions can be greatly simplified. Euler's definition is extremely scientific, which liberates trigonometry from the narrow world of static triangle solution, makes it possible to reflect the process of motion change, and thus integrates trigonometry into one.
Positive migration introduces the concept of trigonometric function line.
Do you remember how to define the sine, cosine and tangent of an acute triangle in junior high school? According to the function of positive transfer in educational psychology, we might as well draw trigonometric function lines with special acute angles such as π6, π4 and π3 in rectangular coordinate system by using the unit circle, and then transition from special to general, so as to get trigonometric function lines with any angle, which makes students feel that trigonometric function lines have a sense of deja vu. In the process of learning, experience how to combine the number and shape of trigonometric function naturally, achieve the perfect combination of number and shape, and form a feeling of mathematical beauty.
Grasp the essential attribute of trigonometric function line and guide it skillfully layer by layer.
The stage of introducing unit circle to construct trigonometric function line
For teachers, the two-step leap from the ratio yr to Y, xr to X, and then to the sine line and cosine line seems simple, but it is hard for students to think of it. Here, the construction process of knowledge is reproduced as clearly as possible, so that students can clearly understand the principle and grasp the formation of concepts. From the level of mathematical thinking, we can highlight the thinking method of "simplifying" trigonometric function into "one variable". Then the trigonometric function line, an intuitive graphic tool, is successfully used to express the main line of trigonometric function, and the requirements of "simplification" and "unification" are repeatedly emphasized in the teaching process. Such views or thoughts can not only embody the characteristics of this mathematical method, but also occupy an important position in the process of mathematics teaching and have universality.
The sine line and cosine line point to the tangent line.
Sine lines and cosine lines are easier for students to understand and master because of their intuitive feelings, while tangent lines are difficult to understand and master. The key to breaking through this difficulty is to help students fully understand the number of directed line segments and related concepts. Then when talking about some difficult-to-express mathematical concepts such as the number of directed line segments and the number of directed line segments, students' attention can mainly focus on the corresponding relationship between "shape" and "quantity", which naturally highlights the formation process of "sine and cosine function lines" and explores the basic methods to determine them. Friedenthal pointed out that students are not passively accepting knowledge, but re-creating. At this stage, if we can provide students with more open space, let them learn "tangent function line" consciously or consciously.
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