Senior high school mathematics comprehensive test III

(A) the statistical analysis of college entrance examination questions

1, trigonometric function test statistics in college entrance examination papers

Test content of test paper question score

Monotonicity of Five-Point Tangent Function in National Volume (1) and (5)

(6) Geometric series and cosine theorem with 5 points for multiple-choice questions.

(16) Fill-in-the-blank four-point derivative, trigonometric function parity, trigonometric transformation

(17) solving problems 12: simplification of trigonometric function, periodicity and maximum value of trigonometric function.

National Volume (2) (2) The five-angle formula of multiple-choice questions and the periodicity of trigonometric functions

(10) Five-point inductive formula and trigonometric function expression for multiple-choice questions

(14) Fill in the blanks 4 points arithmetic progression and Cosine Theorem.

(17) Solving 12 Fractional Vector and Triangulation Problem

Table1:Horizontal Statistics of National Volume, Beijing Volume and Shanghai Volume in 2006

Test content of test paper question score

Fill in the blanks with the sine theorem and cosine theorem of Beijing Volume (12)5.

(15) solution 12 definition domain of trigonometric function, simplification and evaluation of trigonometric function.

Shanghai Volume (6) Evaluation of Trigonometric Function of Multiple Choice Questions,

(17) solving problems 12: trigonometric transformation, range of trigonometric functions, minimum positive period.

(18) solving problems 12 using sine theorem and cosine theorem to solve practical problems related to measurement.

Table 2: Vertical statistics of Guangdong's turnover in recent three years

Examination content of annual question score

In 2004 (5), multiple-choice questions included trigonometric transformation, periodicity and parity of trigonometric functions.

(9) The relationship between the same angle of trigonometric function and the maximum value of quadratic trigonometric function in multiple-choice questions.

The Imagery and Monotonicity of Five-Point Tangent Function of Multiple Choice Questions (1 1)

(17) solving problems 12: the mean term of fractional arithmetic, the mean term of equal proportion, the double-angle formula, and the unary quadratic equation about trigonometric functions.

In 2005 (13), fill in the blanks with 5-point binomial theorem and trigonometric function values.

(15) Solve the problem 12 Simplify the trigonometric function and find the range and minimum positive period of the function.

In 2006 (3) the parity and monotonicity of five-point function.

(15) Solving the problem 14: the maximum value, period and trigonometric function value of trigonometric function.

2. Statistical analysis of trigonometric functions in college entrance examination papers.

Looking at the Guangdong examination papers in recent three years, the national college entrance examination papers in 2006, and the independent proposition papers of relevant provinces and cities, the proposition about trigonometric function has the following remarkable characteristics:

(1) Question type and score: Trigonometric function questions are generally two small questions and one solution question, which belongs to the conventional question type. The solution of trigonometric function is mostly in the position of 1 problem solving, and the average score of triangular part is about 22 points, accounting for about15%;

(2) Examination difficulty: Trigonometric function solution is generally a basic question and an intermediate question, which is not difficult and easy to deform, and is combined with exercises and examples in textbooks;

(3) Hot topics: First, the images and properties of trigonometric functions, especially the period, maximum, monotonicity and image transformation of trigonometric functions; The second is to simplify and evaluate by trigonometric identity transformation; The third is the synthesis of vector, sequence and quadratic function. Fourthly, using sine theorem and cosine theorem to solve practical problems related to measurement and geometry.

(B) Trigonometric function part of the college entrance examination proposition trend

1, trigonometric function proposition tends to be stable. The original examination style will still be maintained. Although the background of the proposition has changed, it is still a basic, intermediate and routine problem.

2. After the implementation of the new curriculum standard, the number of questions and scores of the triangle will decrease slightly. This is not to say that trigonometric function has lost its original position and importance, but that the new round of basic education reform has added many new contents suitable for modern life and scientific and technological development, which will attract the attention of proposers. For example, the content of derivative, limit, vector and linear programming was introduced in the last round of reform, which was fully reflected in 2004, because the test scores containing these knowledge points added up to 40 points. In fact, the triangle test questions in Guangdong in recent two years have been reduced to a small question and a problem solving. The third topic in 2006 is not a triangle in a strict sense, and it is expected to remain unchanged in 2007.

3. The images and properties of trigonometric functions are the focus of examination. Because the images and properties of trigonometric functions are the basis for students to learn advanced mathematics and applied technology in the future, and they are also tools to solve practical production problems, and in recent years, the college entrance examination has lowered the requirements for trigonometric transformation, which is bound to increase the examination of trigonometric functions, making the images and properties of trigonometric functions a hot spot in the college entrance examination and a main type of trigonometric solution, with certain flexibility and comprehensiveness.

4. Simplified evaluation of trigonometric functions is a common question type. Often appear in minor problems, or as a minor problem in solving problems, it is bound to permeate the properties of simple trigonometric identity transformation and trigonometric function. Focus on the basic knowledge, skills and methods of trigonometric functions.

5. Test the application and build a triangle model.

The simple application of trigonometric function model is added in the new textbook, and the trigonometric problem of "tide and port depth" is taken as a reference case in the curriculum standard (in the original textbook, it is only a reading material). There are several places in the textbook that involve the application of trigonometry in physics, such as describing simple harmonic vibration and alternating current with the physical meaning of functions, which shows that trigonometric function is an important function model to describe periodic changes. It shows the intention of attaching importance to the application of triangle.

The practical problems of blending into the triangle often appear. This kind of question can not only examine the knowledge and methods of solving triangles, but also examine the skills of using trigonometric formulas to transform identities, so it has been favored by proposers in recent years, such as the typhoon attack in the national volume in 2003 and the rescue of fishing boats in the Shanghai volume in 2006. The main solution is to make full use of internal angle sum theorem, sine (cosine) chord theorem, area formula and so on. Triangular, and combined with the triangle formula for triangle transformation, so as to get the solution.

6. Comprehensive examination reflects the instrumentality of triangle.

In recent years, college entrance examination questions are often designed at the intersection of knowledge because they focus on ability and strengthen the comprehensive and applied examination of knowledge. The investigation of triangle knowledge is often combined with plane vector, sequence, solid geometry and analytic geometry to highlight the instrumentality of triangle. Especially the synthesis problem of plane vector and triangle, the probability is very high, because the content of the new textbook is very concerned about how to use vectors to deal with triangle problems. This clue can also be clearly seen from the college entrance examination questions in various provinces and cities in the past two years, which should be highly valued by teachers.

Third, strengthen basic training based on teaching materials.

There is a saying in our hometown: "If the textbook is not in place, review should be damned;" If you can't write the outline well, see you at the exam. "

Because the growing point of trigonometric function test questions in college entrance examination mostly appears in textbooks, the review of trigonometric function should adhere to textbooks and be higher than textbooks. So how do we do this?

First of all, our teachers should pay attention to returning to textbooks. The importance of textbooks in the first round of review is self-evident, but it is not easy to review textbooks frequently, because teachers have supporting review materials at hand and often throw them aside, and some may even have no textbooks. We might as well imagine this; What would I do if I were the proposer? Of course, I will hold a "chicken" (exam outline) in my left hand and a "duck" (textbook) in my right hand. Especially now, great changes have taken place in the new textbooks, so it is even more necessary for us to study them.

The second is to educate students to attach importance to teaching materials. I think: No matter how much we emphasize the importance of textbooks in front of students. Although it is the first round of review, it is impossible for us to repeat textbooks, and students are too tired to do review materials and have no time to look after textbooks, which will cause imbalance between textbooks and materials. In addition, there are many students who are "superior in eyes but inferior in skills" and have no patience to read textbooks carefully. So what should we do? We have to take some measures, for example, we can take out an exam from the textbook intact and let the students take a test to kill their spirit; You can also consciously infiltrate typical examples and exercises in textbooks into your study plan, and so on.

Third, give full play to the role of typical examples and exercises in teaching materials. In collective lesson preparation, if the teacher in charge of preparing lessons for each chapter can select typical examples and exercises from the textbook and let students do them again in the form of extracurricular homework, it will certainly achieve great results. Of course, our textbook this year is the first published experimental textbook, and there will inevitably be some imperfections. I made a comparison between this textbook and the next textbook, and found that there were some fine-tuning, and some slightly complicated, difficult and biased topics were deleted from the exercises, such as compulsory course 4, chapter 3, triangular equivalent transformation P161(group A). Compulsory 5 Chapter 1 Solving Triangle P 1 1 (Group B)1,P23 (Group A) 9, P29 (Group B)1,etc.

Relatively speaking, the triangle part of the college entrance examination is more prone to variations and combinations of exercises and examples in textbooks. This enlightens us that we should pay attention to two aspects when reviewing: one is "based on textbooks, focusing on improvement", and the other is to strengthen the induction and mastery of conventional questions. Only in this way can we ensure that these questions become the main scoring questions in the college entrance examination.

Fourth, pay attention to the outline and the key points of the exam to improve the review efficiency.

(A) closely follow the outline, grasp the lifeblood of the college entrance examination

"Examination Outline" is the main proposition basis of mathematics college entrance examination questions and the programmatic and guiding document of mathematics teaching in senior high schools. Therefore, we should carefully study the examination syllabus and accurately grasp the review direction.

Due to the shortage of class hours (especially in science), you should follow the contents and requirements stipulated in the outline in the review, and don't add deleted knowledge points at will. For example, trigonometric functions only talk about sine, cosine and tangent; Trigonometric functions with the same angle have only two basic relationships.

In the trigonometric function part, it is not required to introduce topics that are too difficult, too complicated and too skillful. Emphasis should be placed on the accuracy, proficiency and flexibility of knowledge understanding, and the review should focus on middle and low-grade topics.

(2) Master the concept, images and properties of trigonometric functions.

In the teaching of trigonometric function, we should play the role of unit circle and trigonometric function line. The unit circle can help students intuitively understand trigonometric functions at any angle, understand the periodicity of trigonometric functions, inductive formulas, the relationship between trigonometric functions and angles, and the images and basic properties of trigonometric functions. When reviewing, students are required to use the trigonometric function line in the unit circle to deduce the inductive formula, draw the image, and understand the influence of parameters on the function image transformation. The properties of trigonometric functions are range, periodicity, parity, monotonicity and maximum, among which monotonicity, maximum and minimum are the most prominent.

In recent years, the college entrance examination has lowered the requirements for trigonometric transformation, but strengthened the examination of images and properties of trigonometric functions. Therefore, the images and properties of trigonometric functions are a key point in this chapter. The review of trigonometric functions should make full use of the thinking method of combining numbers and shapes, that is, the properties of trigonometric functions can be obtained intuitively with the help of images (or trigonometric function lines), and the images of functions can be described by using the properties of trigonometric functions to reveal the algebraic properties of graphics.

(3) Master the basic transformation ideas of trigonometric functions.

The constant deformation of trigonometric functions is not only necessary in the simplification and evaluation of trigonometric functions, but also inevitable in the study of trigonometric functions' images and properties and the solution of triangles. The key to solve the problem of trigonometric constant deformation lies in mastering the basic transformation idea, using the main methods of trigonometric constant deformation-changing angle, changing function and changing structure, and paying attention to the flexible use of formulas.

The basic transformation idea is as follows: 1, transformed into "three ones": that is, transformed into the form of the first power of the trigonometric function of an angle; 2. Turn it into "two ones": that is, turn it into the quadratic structure of trigonometric function of an angle, and then solve it by matching method; 3. "Combination into one": For formulas with shapes, an auxiliary angle is introduced to form a combination (note that the difficulty is not increased here, only special values and special angles can be used); 4. Transform edges and angles with sine theorem, cosine theorem and area formula.

Trigonometric formula is the basic basis of trigonometric transformation. In the review of trigonometric identity transformation, students can be guided to derive the cosine formula of the difference between two angles by using the quantitative product of vectors, and the sine, cosine and tangent formulas of the sum and difference between two angles and the sine, cosine and tangent formulas of two angles from this formula, thus guiding students to derive the formulas of the sum and difference of products, the product of sum and difference and the half angle as the basic training of trigonometric identity transformation. Through the exploration of these formulas and the use of these formulas for trigonometric transformation, students can learn to predict the goal of transformation, choose the formula of transformation and design the way of transformation, and help students further improve their reasoning ability and operational ability.

(D) Strengthen the application awareness of trigonometric functions.

Trigonometric function is a basic and important function, which is widely used in mathematics, other sciences and production practice. The new textbook arranges practical examples and exercises to understand triangles, involving practical problems such as measurement and navigation, and also adds simple application of trigonometric function model, with a clear intention: to highlight the application of trigonometric function. In recent years, the application test questions with trigonometric function as the background have formed a bright spot in the college entrance examination.

When reviewing trigonometric functions, we should pay attention to the relationship between disciplines. We can link the periodic phenomena in physics, biology and nature (such as simple pendulum motion, wave propagation and alternating current), and let students realize that trigonometric function is an important model to describe periodic phenomena through concrete examples.

In the teaching of triangle solution, we should pay attention to the role of sine theorem and cosine theorem in exploring the relationship between angles of triangle, and guide students to understand that they are a method to solve practical problems related to measurement and geometric calculation, without too complicated constant deformation training.

(5) Effectively improve the comprehensive ability of trigonometric functions.

Trigonometric function has a strong penetration, which can be integrated with other mathematical knowledge, especially with vector and geometry. Pay attention to the comprehensive test questions of trigonometry and geometry, and introduce the angle as an independent variable into geometry to establish a functional model or solve several models, which can make it difficult to solve problems easily (see the compulsory textbook P 156 Case 4); Pay attention to the comprehensive test questions of triangle and vector. Plane vector has an extremely rich practical background, and it is a tool to communicate algebra, geometry and trigonometric functions. Therefore, we should review the knowledge through the integration of trigonometric function, plane vector and downward triangle, and conduct comprehensive training through the integration of trigonometric function, plane vector and downward triangle.

Five, the test center case analysis, to provide students with exemplary problem-solving guidance.

1 test center trigonometric function image

Trigonometric function image is a framework to support the knowledge system of trigonometric function, and it is also a powerful lever for students to learn trigonometric function well.

Part of the image of Zhenti 1(05 Tianjin) function is shown in the figure, and the function expression is

(A) (B)

Analytical solution 1: From the function image, we can see the function passing point, amplitude,

Period, frequency, shift the function to the right by 6 units, and get

choose one

Solution 2: You can substitute the coordinates of points for filtering. Choose one.

Comment 1. This topic examines the image transformation of sinusoidal curves and the equivalent transformation ability of graphics and shapes.

2. Generally speaking, if the analytical expression of sine curve is obtained from an image, the parameters are determined, that is, the amplitude is obtained from the highest or lowest point of the image and determined by the period or half period (the distance between the abscissas of adjacent maximum points). Considering the uniqueness of, the coordinates of the maximum point are substituted into the analytical expression of sine function on the basis of determination, and the value is taken in a given interval.

Properties of Trigonometric Function in Test Site 2

If the image of trigonometric function is the skeleton of trigonometric function, the essence of trigonometric function is the flesh and blood of trigonometric function. Therefore, the examination of trigonometric function in college entrance examination has been enduring for a long time.

Monotonicity and periodicity of trigonometric functions

Known functions of Zhenti No.2 (Fujian 2006)

(i) Find the minimum positive period and monotonically increasing interval of the function;

(II) How to convert an image of a function from an image of a function?

Analysis (1)

minimal positive period

Get it from the meaning of the question

The monotone increasing interval of is

(2) Method 1: First, move all the points on the image to the left by one unit length to obtain an image, and then move all the points on the obtained image upward by one unit length to obtain an image.

Method 2: Translate all points on the image according to the vector to get the image.

This topic mainly examines the basic formula of trigonometric function, trigonometric identity transformation, the properties and image transformation of trigonometric function, as well as reasoning and operation ability.

Maximum value of trigonometric function

The minimum positive period, maximum value and minimum value of Zhenti 3(04 national).

Analysis, so

Comment 1, flexibly apply the boundedness of y=sinx and y=cosx to study the maximum (or range) of some types of trigonometric functions.

2. Generally look for the problems of trigonometric function properties, such as symmetry, monotonicity, periodicity, maximum value, value range, drawing, etc. You can use trigonometric formula to change the solved function into a new form, and then solve it according to the known conditions and their properties. This kind of questions are freely tested in the college entrance examination almost every year.

Test center 3 trigonometric function evaluation

Zhenti 4(05 Tianjin) is known.

Analytical solution 1: Based on the conditions set by the problem, it is obtained by applying the sine formula of the difference between two angles.

, namely (1)

The conditions based on topic setting and the application of double-angle cosine formula.

Therefore, ② is derived from ① and ②; So ② is derived from the tangent formula of the sum of two angles.

Solution 2: Set the conditions from the problem, and apply the double-angle cosine formula to get the solution, that is,

You can get it because, and, therefore? Therefore, in the second quadrant, the following solutions are the same.

Comment 1. This topic examines the ability of trigonometric transformation and operation through the evaluation of trigonometric function. The evaluation of trigonometric function can be obtained by transforming the internal relationship between the known angle and the angle to be solved (both included).

2. When calculating the value of trigonometric function, we must use formulas flexibly and pay attention to the use of implicit conditions to prevent multiple solutions or missing solutions.

Test site 4 solution triangle

Question 5(05 Hubei) In △ABC, the median line BD= on the side is known, and the value of Sina is found.

Analytical solution 1: Let E be the midpoint of BC, followed by DE, and then by DE//AB, with DE=

Using the cosine theorem in △BDE, we can get: BD2 = BE2+ED2-2BE? EDcosBED，

Comment 1. This small topic mainly examines the basic knowledge of sine theorem and cosine theorem, and at the same time examines the skills and operational ability of constant deformation using trigonometric formulas.

2. The definition of acute trigonometric function, Pythagorean theorem, sine theorem and cosine theorem are commonly used tools when solving problems about triangles. Pay attention to the use of triangle area formula and the limitation of triangle internal angle sum.

Test site trigonometric function synthesis

Trigonometric function is an important elementary function. Because of its special nature and close connection with other algebra and geometry knowledge, it has become an important tool for learning other parts of knowledge and one of the important contents of the college entrance examination.

Triangle sum vector

Question 6(06 Sichuan) It is known that a triangle has three internal angles, a vector, and.

(i) Find the angle; (2) If, ask

Analysis (a) ∵ ∴, that is ∵ ∴ ∴

(2) From the topic, I sorted it out.

∴∴∴ or, and make, give up∴∴∴

Comment on this topic, integrate the coordinate operation of vector product into trigonometric function, mainly investigate the concept of trigonometric function, the relationship between trigonometric function and the same angle, the formula of sum and difference between two angles of trigonometric function, summarize the formula, solve the equation and find the value of trigonometric function.

Triangle sum sequence

Zhenti 7(06 Shaanxi) "Equation sin(α+γ)=sin2β holds" is "α, β, γ becomes arithmetic progression" ().

A. Necessary and insufficient conditions B. Sufficient and unnecessary conditions C. Sufficient and necessary conditions D. Insufficient and unnecessary conditions

Analysis If the equation sin(α+γ)=sin2β holds, then α+γ = kπ+(- 1) k? 2β, at this time, α, β and γ are not necessarily arithmetic progression.

If α, β and γ become arithmetic progression, 2β=α+γ, and the equation sin(α+γ)=sin2β holds, so "Equation sin(α+γ)=sin2β holds" is a necessary but not sufficient condition for α, β and γ to become arithmetic progression. Choose a.

Comment on this topic is at the intersection of triangle and series, which plays a transitional role, with emphasis on triangle. Designing test questions at the intersection of knowledge networks is easy to test mathematical ability, which is a common form of proposition in college entrance examination and needs attention.

Triangle sum equation

Question 8: It is known that the equation sinx+cosx=k has two solutions at 0≤x≤π, so find the value range of K.

Analyze the original equation sinx+cosx=k sin(x+ )=k, and make the image of functions y 1= sin(x+) and y2=k in the same coordinate system. For the image with y= sin(x+), let x=0 and get y= 1. ∴ pawn ∴. king

Comment on this topic is to judge the number of real solutions of the equation by the number of intersections of function images, so we should pay attention to this method.

Trigonometric function and quadratic function

When Question 9 (04 Guangdong) is true, the maximum value of the function is ().

A. The 4th day of the 2nd year BC

Analysis, select (d).

Comments are transformed into a quadratic function about tanx, and the maximum value is obtained by matching method.

Application of Trigonometric Function of Test Center 6

As shown in Figure 10 (Shanghai, June), when Ship A was located at A, it was learned that there was a fishing boat in distress at B, 20 nautical miles east of it. Ship A immediately went to the rescue, and at the same time informed Ship B, which is 30 southwest of Ship A and 30 nautical miles away from 10, how many degrees northeast ship B should go to the rescue (accurate angle).

Analyzing the connection BC from cosine theorem, bc2 = 202+102-2× 20×10cos120 = 700.

Therefore BC= 10. * ∴sin∠acb=，

∫∠ACB & lt； 90∴∠ACB = 4 1 ∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴8

Comments turn practical problems into mathematical models, and then use sine theorem and cosine theorem to solve measurement and triangulation problems.

Trigonometric function test paper (full mark 150, test time 120 minutes)

First, multiple-choice questions (this big question * * 10, 5 points for each small question, 50 points for * * *)

The values of 1 and tan600 are ().

(A) (B) (C) (D)

2. The minimum positive period of the function y = sin (2x+) is ().

(A) (B) (C) 2 (D) 4

3. "Equation holds" is "arithmetic progression" ().

(a) Sufficient and unnecessary conditions (b) Necessary and insufficient conditions (c) Sufficient and necessary conditions (d) Insufficient and unnecessary conditions.

4, when the range is ()

(A) (B) (C) (D)

5. If it is odd function, it can be ().

(A) (B) (C) (D)

6. Translate the image of the function according to the vector, and the translated image is as shown in the figure, then the analytical formula of the function corresponding to the translated image is ().

(A) (B) (C) (D)

7. If △ABC area S=, ∠C= ()

(A) (B) (C) (D)

8. Yes, if, it must be ()

(a) isosceles triangle (b) right triangle (c) isosceles right triangle (d) isosceles or right triangle

9. The maximum and minimum values of are () respectively.

(A)7、5 (B)7 、( C)5 、( D)7 、-5

10. The included angle between the known vector sum is ().

(A) (B) (C) (D)

2. Fill in the blanks (4 small questions in this big question, 5 points for each small question, 20 points for * * *)

1 1. If =, and it is the angle of the fourth quadrant, then =;

12. If it is known, then _ _ _ _ _ _ _;

13. The known three internal angles A, B and C form a arithmetic progression, and the length of the midline AD on the BC side is;

14. If it is a odd function with a period of 5, =4, cos, =.

Third, answer the question (this big question is ***6 small questions, ***70 points)

15, (12 minutes) is known, and, are the two roots of the equation, and find the value of COS ().

16( 14 minutes) △ Three internal angles A, B and C of △ABC, find the maximum value when A is, and find this maximum value.

17.( 14 points) Minimum positive period of known function (Ⅰ);

(ii) the maximum and minimum values of; (iii) the value, if any.

18.( 12 minutes) As shown in the figure, when a ship is located at A, it is known that there is a ship at B, which is located 20 nautical miles east of it.

The fishing boat is in distress, waiting for rescue. Ship A immediately went to the rescue and told The Paper that it was 30 southwest of Ship A, with a distance of 10.

For ship B at C in the sea, how many degrees east of north should ship B rescue in a straight line (accurate angle)?

19.( 14 points). As we all know, it is a triangle with three internal angles and vectors.

(i) Find the angle; (2) If, ask

20.( 14 points) It is known that b and c are real numbers, and the function f(x)= has is for any α and β R: and.

(1) Find the value of f (1); (2) proof: c; (3) Let the maximum value be 10 and find f(x).

Trigonometric function test paper reference answer

First, multiple choice questions

DBBDB CCDDA

Second, fill in the blanks

1 1、 12、 13、 14、—4

Third, answer questions.

15、