Teaching design of trigonometric function in the first unit of senior high school mathematics
Section 24 Teaching Time
Textbook: Double Angle Formula, Derivation? Sum-difference product And then what? Product and difference? formula
Objective: Continue to review and consolidate the double-angle formula and strengthen the training of flexible application of the formula; At the same time, let the students deduce the sum-difference product and sum-difference formula and understand it.
Process:
Firstly, the derivation process of double-angle formula, half-angle formula and universal formula is reviewed:
Example 1. It is understood that Tan? =, Tan? =, change 2? + ?
(Teaching and Testing P 1 15 Case 3)
Solution:?
Tan Er again? & lt0, Tan? & lt0 ? ,
2? + ? =
Example 2, known sin cos? =，，sum tan？ The value of
Solution: sin cos? = ?
Simplify:
namely
Second, the derivation of the product sum and difference formula
Sin (? + ? )+ sin(？ )= 2sin？ Cos crime? Because? = [sin (? + ? )+ sin(？ )]
Sin (? + ? ) ? Sin (? )= 2cos？ sin cos？ Sin? = [sin (? + ? ) ? Sin (? )]
cos(？ + ? )+ cos(？ )= 2cos？ cos cos？ Because? = [cos(？ + ? )+ cos(？ )]
cos(？ + ? ) ? cos(？ ) = ? 2sin？ Sin. Sin? Sin? = ? 【cos(？ + ? ) ? cos(？ )]
This set of formulas is called trigonometric function multiplication and difference formula, which is familiar with the structure and does not need memory. Its advantage lies in the integration? Product formula? Become? What is the difference? , which helps to simplify the calculation. (On the premise of informing the formula)
Example 3: verification: sin3? sin3？ + cos3？ cos3？ = cos32？
Certificate: left = (sin3? Sin? )sin2？ + (cos3？ Because? )cos2？
= ? (cos4 cos2？ )sin2？ + (cos4？ + cos2？ )cos2？
= ? cos4？ sin2？ + cos2？ sin2？ + cos4？ cos2？ + cos2？ cos2？
= cos4？ cos2？ + cos2？ = cos2？ (cos4？ + 1)
= cos2？ 2cos22？ = cos32？ = Right
? Proof of primitive form
Thirdly, the derivation of sum-difference product formula.
Ruo Ling? + ? = ? , ? = ? , and then replace with:
?
This set of formulas is called sum-difference product formula, which is characterized by using sine (cosine) strings with the same name. It is the complement and cooperation of sum-difference formula.
Example 4. Known cos cos? =, sin sin? =, looking for sin (? + ? The value of)
Solution: cos cos? = ,? ①
Sin. Sin? = ,? ②
∵
?
Fourth, summary: multiply with the difference, multiply with the difference
5. Homework: P36 in class exercises? 37 cases recommended 1? three
P38？ 39 cases recommended 1? three
P40 sample recommendation 1? three
Teaching design of induction formula of trigonometric function in senior high school mathematics
1 teaching material analysis
1. 1 the position and function of teaching materials
What is the teaching content of this course? Inductive formulas (2) and (3)? Is it the second chapter of the first volume of high school algebra published by People's Education Press? Section 2.6. It is not only the continuation and expansion of the knowledge of trigonometric function definition and inductive formula (1) that students have learned, but also the theoretical basis for deducing inductive formulas (4) and (5). Is this the chapter? Trigonometric function at any angle? This section and the whole chapter play an important role in connecting the preceding with the following. Finding the value of trigonometric function is an important content in trigonometric function. Inductive formula is the basic method to find the value of trigonometric function. The important function of inductive formula is to turn the problem of finding trigonometric function value at any angle into finding 0? ~90? The trigonometric function of angle and the derivation process of inductive formula reflect the combination of numbers and shapes in mathematics and the inductive transformation, and the form of mathematical inductive thinking from special to general, which is of great significance to cultivate students' innovative consciousness, develop their thinking ability and master their thinking methods in mathematics.
1.2 teaching emphases and difficulties
Teaching emphasis 1.2. 1
Derivation and application of inductive formula
1.2.2 Teaching difficulties
Geometric symmetry of the terminal edge of correlation angle and understanding of the structural characteristics of inductive formula.
2 Target analysis
According to the requirements of the syllabus and the structural characteristics of the teaching content, according to the psychological laws of students' learning and the requirements of quality education, combined with the actual level of students, the teaching objectives of this course are as follows
2. 1 knowledge objectives
1) memorize inductive formulas.
2) Understand and master the connotation and structural characteristics of the formula, and initially find the value of trigonometric function by inductive formula, simplifying and proving the simple formulas of trigonometric functions.
2.2 Ability objectives
1) cultivate students' observation ability, analysis and induction ability, and master the inductive transformation thinking method of mathematics through the derivation of inductive formulas.
2) By deducing the inductive formula and analyzing the structural characteristics of the formula, students can experience and understand the thinking mode of mathematical inductive reasoning from special to general.
3) Improve students' practical ability to analyze and solve problems by practicing basic training questions and ability training questions.
2.3 emotional goals
1) Through the derivation of inductive formula, cultivate students' scientific spirit of active exploration and courage to discover, and cultivate students' innovative consciousness and spirit.
2) Through the training of inductive thinking, cultivate students' practical, meticulous, rigorous and scientific study habits, infiltrate from special to general, and transform the unknown into known dialectical materialism.
3 process analysis
3. 1 Create problem situations to guide students to observe, associate and introduce topics.
1) problem: the definition of trigonometric function, inductive formula (1) and its structural characteristics.
2) blackboard writing: inductive formula (1).
sin(k？ 360? +? ) = sin? ，cos(k？ 360? +? )=cos？ .
Tan (k? 360? +? ) = Tan? ，cot(k？ 360? +? )=cot？ (k？ z)
Structural features: ① The same trigonometric functions of the angles with the same terminal edge are equal.
② The problem of finding trigonometric function value at any angle is transformed into finding 0? ~360? The trigonometric function value problem of angle.
Teaching hypothesis makes students review and attach importance to existing knowledge by asking questions, paving the way for students to learn new knowledge.
3) Student Exercise: Try to find the following trigonometric function values.
sin 1 1 10？ ，sin 1290？ .
Teaching ideas lead to new questions from existing knowledge and create problem situations for learning new knowledge, thus stimulating students' learning needs and interests, stimulating students' thirst for knowledge and stimulating students' thinking sparks.
4) After introducing the concept of unit circle, guide students to observe and demonstrate (1) and think about the following questions:
①2 10? Can I use (180? +? ) in the form of expression (0? & lt？ & lt90? )? (2 10? = 180? +30? )
②2 10? With 30? What is the positional relationship between the terminal edges of angles? (mutually opposite extension lines or symmetrical about the origin)
③ Set 2 10? ,30? What is the positional relationship between point P and point P' when the terminal edge of an angle intersects the unit circle at point P and point P' respectively? (symmetrical about the origin)
(4) If the point P(x, y) is set, how to express the coordinates of the point p'? [P'(-x，-y)]
⑤sin2 10？ With sin30? What is the relationship between values?
The teaching idea is to guide students to discover 2 10 through dynamic demonstration by microcomputer. With 30? The terminal edge of an angle and its intersection with the unit circle are symmetrical about the origin. With the help of trigonometric function definition, find sin2 10? With sin30? The relationship between numerical values, from 0 to 0? ~90? The purpose of trigonometric function.
Students can experience and understand the mathematical thinking method of combination, induction and transformation of numbers and shapes by actively exploring and finding solutions to problems.
5) Import the theme
Any angle? , sin? Same crime (180? +? How's the relationship? Try to tell your guess.
3.2 Use the migration law to guide students to associate, analogize, induce and deduce formulas.
1) Guide students to observe the demonstration (2) and think about the following questions:
①? And (180? +? What is the terminal relationship of angle? (mutually opposite extension lines or symmetrical about the origin)
Two sets? And (180? +? The terminal edge of the angle intersects the unit circle at points p and P', respectively. What is the positional relationship between points P and P'? (symmetrical about the origin)
(3) If point P(x, y) is set, how do you express the coordinates of point P'? [P'(-x，-y)]
4 sin? Same crime (180? +? )，cos？ With cos( 180? +? What does it matter?
5 tan? With Tan (180? +? )，cot？ With cot( 180? +? What does it matter?
After exploration, can the above conclusions be summarized into a formula? What are the characteristics of its formula?
2) blackboard writing inductive formula
sin( 180？ +? ) =-sin? ，cos( 180？ +? )=-cos？ ,
Tan (180? +? ) = Tan? ，cot( 180？ +? )=cot？ .
Structural features: ① The function name is unchanged, and the symbol looks at the quadrant (put? At an acute angle).
② Find (180? +? ) into a solution? The value of the trigonometric function.
After the teaching hypothesis inspires students to guess, it inspires students to ask special questions (seeking sin2 10? Value) and general problems, realize method migration, guide students to observe and demonstrate, and find angles? And (180? +? ) and its symmetrical relationship with the intersection point of the unit circle about the origin, and the angle (180? +? ) into a solution? Train students' inductive thinking and cultivate their inductive thinking ability.
The dynamic demonstration of microcomputer enables students to watch from any angle. Have an accurate understanding, initially experience the inductive reasoning form from special to general, and understand the inductive transformation ideas and methods of mathematics.
3) Basic Training Problem Group 1
Find the following trigonometric function values (refer to table):
② Try to find fault [180? +(-2 10? )] value
Analysis:
For question 2, students may encounter the following situations:
sin[ 180？ +(-2 10? )]=-sin(-2 10？ ),
Or sin[ 180? +(-2 10? )]=sin(-30？ ).
(At this point, most students can't calculate.)
Teaching hypothesis leads to new unknowns on the basis of new knowledge, creates problem situations again, pushes students' interest in learning to a climax, encourages students to dare to meet challenges, overcome difficulties, pursue constantly, edify their sentiments and exercise their will.
4) Guide students to observe the demonstration (3) and think about the following questions:
①30? And (-30? What is the positional relationship between the terminal edges of angles? (About the X-axis symmetry)
② Set 30? And (-30? The terminal edge of the angle intersects the unit circle at points p and P' respectively. What is the positional relationship between points P and P'? (About the X-axis symmetry)
(3) If a point P(x, y) is set, how to express the coordinates of the point p'? [P'(x，-y)]
④sin(-30？ ) and sin30? What is the value relationship?
Teaching hypothesis guides students to find sin2 10? Ask and sin (-30? ) analogy, realize method migration. Through microcomputer dynamic demonstration, it is found that -30? With 30? The relationship between the terminal edge of an angle and the intersection point of the unit circle symmetrical about X axis is solved by the definition of trigonometric function. ) and sin30? The relationship between numerical values, from 0 to 0? ~90? The purpose of the value of trigonometric function.
5) Introduce a new question: For any angle? , sin? For sin (-? How's the relationship? Try to tell your guess?
6) Guide students to observe the demonstration (4) and think about the following questions: (Setting? At any angle)
①? Use (-? What is the positional relationship between the terminal edges of angles? (About the X-axis symmetry)
Two sets? Use (-? The terminal edge of the angle intersects the unit circle at points p and P', respectively. What is the positional relationship between points P and P'? (About the X-axis symmetry)
(3) If a point P(x, y) is set, how to express the coordinates of the point p'? [P'(x，-y)]
4 sin? For sin (-? )，cos？ With cos(-? What does it matter?
5 tan? With tan (-? )，cot？ With cot(-? How's the relationship?
7) Students discuss in groups, try to deduce formulas, and teachers patrol, giving feedback, correcting and commenting in time.
8) blackboard writing inductive formula
Sin (-? ) =-sin? ，cos(-？ )=cos？ .
Tan (- ) =-Tan? ，cot(-？ )=-cot？ .
Structural characteristics: the function name is unchanged, and the symbol looks at the quadrant (put? As an acute angle)
Ask (-? ) into a solution? The value of the trigonometric function.
9) Basic Training Problem Group (2): Find the following trigonometric function values (look-up table)
③cos(-240？ 12'); ④cot(-400？ ).
3.3 Building a knowledge system, mastering methods and strengthening capabilities.
Class summary: (Ask questions and fill in the blanks, and let the students finish it by themselves)
1) inductive formula:
sin(k？ 360? +? ) = sin? .
cos(k？ 360? +? )=cos？ .
Tan (k? 360? +? ) = Tan? .
cot(k？ 360? +? )=cot？ . (k？ z)
sin( 180？ +? ) =-sin? .
cos( 180？ +? )=-cos？ .
Tan (180? +? ) = Tan? .
cot( 180？ +? )=cot？ .
Sin (-? ) =-sin? .
cos(-？ )=cos？ .
Tan (- ) =-Tan? .
cot(-？ )=-cot？ .
2) Structural features of the formula: the name of the function remains unchanged, and the symbol depends on the quadrant (put? As an acute angle)
3) Methods and steps:
By asking questions and filling in the blanks, teaching hypothesis leads students to generalize the existing knowledge, form a knowledge system, discover the law of knowledge and its structural characteristics, deepen their understanding of the connotation and essence of inductive formulas and strengthen their memory.
Mining knowledge system embodies the inductive transformation thinking method of mathematics, cultivates students' generalization and abstraction ability, and forms knowledge network and method network.
4) ability training problem group: (test students' comprehensive ability to use knowledge)
5) Think after class.
① Find the following trigonometric function values:
6) Homework and extracurricular thinking
Homework: P 162 exercise 13 (1)? (6)
The teaching idea is to test students' comprehensive ability to use knowledge, cultivate students' creative thinking ability and improve students' practical ability to analyze and solve problems through ability training questions and extracurricular thinking questions.
Stay for students after class? Aftersound? Cultivate students' good study habits of consciously learning and actively exploring, and prepare for learning inductive formulas (4) and (5) in the next class.
4 Analysis of teaching methods
According to the structural characteristics of teaching content and the psychological law of students learning mathematics, this course adopts? Question, analogy, discovery, induction? Inquiry thinking training teaching method.
4. 1 Use the existing knowledge to derive new problems, create problem situations, stimulate students' interest in learning, stimulate students' thirst for knowledge, and achieve the purpose of expanding the old with the new.
4.2 by ( 180？ +30? ) and 30? ,(-30? ) and 30? A special case of terminal symmetry, using multimedia dynamic demonstration, students at any angle? Understanding is more complete. Through association, guide students to carry out problem analogy and method transfer, and find any angle. And (180? +? ),-? The symmetry of terminal edge, from special to general inductive reasoning training, students' inductive thinking is more objective, rigorous and profound, and students' innovative ability is cultivated.
4.3 Inquiry thinking training teaching method, using questions, observations and demonstrations, leads to association, analogy, discovery and induction step by step, aiming at making students fully feel and understand the generation and development process of knowledge. Under the timely inspiration of teachers, students actively explore and discover mathematical laws (formulas) in the process of analogy and induction, thus cultivating students' innovative consciousness and spirit and cultivating students' thinking ability.
4.4 The application of inductive formulas (1), (2) and (3) is further expanded through ability training problem groups and extracurricular thinking problems, which prepares the theoretical basis for deductive inductive formulas (4) and (5), organically combines inductive reasoning with deductive reasoning, and develops students' thinking ability.
5 Evaluation and analysis
In the teaching process of this course, students are guided to gradually associate, analogize and summarize from the special to the general, and find mathematical formulas, which embodies the learning process of teacher-led, student-centered and positive thinking.
In the thinking training process of problem analogy, method transfer and inductive reasoning, teachers and students have smooth information exchange, timely feedback, timely evaluation and correction, active students' thinking, and teaching activities are always under the control of teachers' expectations.
5 teaching plan design instructions
5. 1 On the teaching guiding ideology of this lesson
Inductive reasoning is the basic thinking form of discovering and acquiring knowledge. Laplace once said: The main tools for discovering truth are also induction and analogy? Inductive thinking plays a special and important role in the formation of innovative consciousness, and inductive thinking often gets pioneering creation (re-creation). Trigonometric function evaluation is one of the important problems in trigonometric function, and inductive formula is the basic method to solve this kind of problem. In the teaching process, through questioning, multimedia dynamic demonstration and other teaching methods, problem situations are created to guide students to learn from special and individual attributes. Through association, analogy and induction, it has a universal and general integrity, which reflects students' full feelings and understanding of the process of knowledge generation and development and encourages students to actively think, explore, discover and innovate. Through inductive thinking training from special to general, students actively acquire new knowledge in the process of acquiring knowledge, form good thinking quality and develop students' thinking ability.
5.2 On the design of teaching process
1) Reproduce the existing related knowledge and pave the way for learning new knowledge.
2) Thinking always starts with the problem, in sin 1290? In the process of evaluation, from the known to the unknown, it raises new questions, creates an atmosphere, arouses students' learning needs and interests, and stimulates students' thirst for knowledge.
3) Mathematical thinking method is the core of mathematical quality. What is the definition of sin2 10? In the process of evaluation, the unknown is transformed into the known, and students are guided to find ways and means to deduce inductive formulas and understand the inductive transformation thinking method of mathematics.
4) Through the multimedia intuitive and dynamic demonstration, complete the classification of all situations from special to general, guide students to associate, carry out problem analogy, method transfer, inductive reasoning to draw general conclusions, form formulas, and conduct inductive thinking training.
5) By analyzing the structural characteristics of inductive formula, we should strengthen our understanding and memory of inductive formula, deeply understand the connotation and essence of inductive formula, build a knowledge system, and cultivate students' ability of generalization and abstraction.
6) Through the practice of basic training questions and extracurricular thinking questions, master problem-solving methods, form skills and improve students' ability to analyze and solve problems.
Teaching plan design of double-angle trigonometric function in senior high school mathematics
I. Knowledge and skills
1. We can deduce the half-angle formula from the sine, cosine and tangent formulas of double angles and understand their internal relations; Reveal the knowledge background, stimulate students' interest in learning, stimulate students' learning attitude of analysis and inquiry, strengthen students' awareness of participation and cultivate students' comprehensive analysis ability.
2. Master the formula and its derivation process, and use the formula to simplify, evaluate and prove.
3. Through formula derivation, master the relationship between half-angle and double-angle, half-angle formula and double-angle formula, and cultivate the ability of logical reasoning.
Second, the process and methods
1. Let students deduce the half-angle formula from the double-angle formula, understand the special mathematical ideas from generalization, experience the harmonious beauty contained in the formula, and stimulate students' interest in learning mathematics;
2. By giving examples, summarize the methods and consolidate the learned knowledge by doing the questions.
Third, feelings, attitudes and values.
1. Through the derivation of the formula, we can understand the internal relationship between the half-angle formula and the double-angle formula, thus cultivating the ability of logical reasoning and dialectical materialism.
2. Cultivate the viewpoint of looking at the problem from the perspective of connection.
Teaching emphases and difficulties:
Emphasis: derivation and application of half-angle formula (evaluation, simplification and proof)
Difficulties: the internal relationship between the half-angle formula and the double-angle formula, and the choice of symbols when using the formula.
Learning methods and teaching tools;
1. Learning method:
(1) Autonomous+inquiry learning: Let students derive the double-angle formula from the sum-angle formula, understand the special mathematical ideas from generalization, appreciate the harmonious beauty contained in the formula, and stimulate students' interest in learning mathematics.
(2) Feedback exercise method: practice the application of testing knowledge and find out the places and gaps that have not been mastered.
2. Teaching method: a teaching method combining observation, induction, inspiration and inquiry.
Guide students to review the double-angle formula, set questions according to the textbook knowledge structure, and guide students to deduce the half-angle formula. In class, under the guidance of teachers and with students as the main body, the structural characteristics of the formula are analyzed, the application of the formula is obtained according to its characteristics, and the formula is simplified, proved and evaluated. Teachers create problem scenarios for students and encourage them to explore actively.
3. Teaching tools: multimedia, physical projector.
Teaching type: new teaching
Class schedule: 1 class hour
Teaching philosophy:
First, create a scene to reveal the theme.
Second, explore new knowledge.
Fourth, consolidate and deepen, feedback and correct mistakes.
Five, induction, overall understanding
1. Consolidate the double-angle formula, and you will derive the half-angle formula, sum-difference product, product and difference formula.
2. Familiar with the relationship between "double angle" and "quadratic" (ascending angle-descending order, descending angle-ascending order).
3. Pay special attention to the triangular expression of the formula and be good at deformation:
4. The left side of the half-angle formula is a square. As long as you know the quadrant where the terminal edge of the angle is located, you can square it; What is the "essence" of the formula? The cosine of an angle represents the sine, cosine and tangent of the angle.
5. Pay attention to the structure of the formula, especially the symbols.
Sixth, connecting the preceding with the following, leaving suspense.
Seven, blackboard design (omitted)
Eight, after-class notes: omitted
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