“Theorem of sines and cosines” - 1) Write down the theorem of sines for a given triangle: Find angle B. Write down the formula for calculation: Theorem of sines: Find the length of side BC. Theorems of sines and cosines. The sides of a triangle are proportional to the sines of the opposite angles. 2) Write down the cosine theorem to calculate the side of the MC: Independent work:
“Solving trigonometric inequalities” - All y values on the interval MN. 1. Constructing graphs of functions: The remaining intervals. The straight line y=-1/2 intersects the sinusoid at an infinite number of points, and the trigonometric circle intersects at point A. an infinite number of intervals. And on a sinusoid, the range of x values closest to the origin for which sinx>-1/2,
“Trigonometric formulas” - Formulas for converting the sum of trigonometric functions into a product. Formulas for converting the product of trigonometric functions into a sum. Addition formulas. By trigonometric functions of an angle?. Double angle formulas. Adding equalities (3) and (4) term by term, we obtain: Let us derive auxiliary formulas that allow us to find.
“Solving simple trigonometric inequalities” - cos x. Methods for solving trigonometric inequalities. sinx. Trigonometric inequalities are called inequalities containing a variable in the argument of a trigonometric function. Solving simple trigonometric inequalities.
“Sin and cos” - Is it true that the cosine of 6.5 is greater than zero? Sine of 60° is equal to?? Is it true that cos? x - sip? x = 1? A branch of mathematics that studies the properties of sine, cosine... Lesson on algebra and basic analysis in 10th grade. Solution trigonometric equations and inequalities. Abscissa of a point on the unit circle. The ratio of cosine to sine...
“The cosine theorem for a triangle” - Oral work. Unknown elements. Triangle. Square side of a triangle. State the cosine theorem. Theorem. Cosine theorem. Solving problems on square paper. Corners and sides. State the cosine theorem. Tasks based on finished drawings. Data shown in the figure.
There are 21 presentations in total
Open lesson on algebra and principles of analysis on the topic: “The relationship between the sine and cosine of the same angle” (grade 10)
Target: students’ perception and initial awareness of new educational material, understanding connections and relationships in objects of study
Educational : derivation of formulas for the relationship between the sine and cosine of the same angle (number); learning to use these formulas to calculate the values of sine and cosine from a given value of one of them.
Developmental : teach to analyze, compare, build analogies, generalize and systematize, prove and disprove, define and explain concepts, develop and improve the ability to apply students’ existing knowledge in different situations; develop students’ competent mathematical speech, the ability to give laconic formulations
Educational: fostering a conscientious attitude towards work and a positive attitude towards knowledge, instilling accuracy in students, the ability to listen, and express their opinions; culture of behavior.
Health-saving : creating a comfortable psychological climate in the classroom, an atmosphere of cooperation: student - teacher.
Knowledge and skills: definitions of basic trigonometric functions (sine, cosine); signs of trigonometric functions by quarters; sets of values of trigonometric functions; basic formulas of trigonometry.Uthe ability to choose the right formula to solve a specific problem; work with simple fractions; Convert trigonometric expressions.
During the classes
Organizing time:
Check students' readiness for the lesson. Opening the teacher’s website on computers (Appendix 1).
Oral work on the covered topic : “Signs of sine, cosine and tangent”
On the desk:
Exercise:
Arrange the numbers of the quarters of the coordinate plane and determine the signs of sine, cosine, tangent and cotangent.
Independent work on the topic: "Signs of sine, cosine and tangent"
Students open the section “Assignments for the trigonometry lesson” on the website. Self-test
(Students complete task No. 1, check their work and evaluate themselves)
Explanation of new material
On the desk:
x = … α , … ≤ cos α≤ … 2)* tan α = , α≠ …
y= … α, … ≤ sin α≤ … ctg α = , α≠ …
Exercise: add formulas
Teacher : “You and I studied each concept separately. What topic do you think is logical to study next?”
( Suggested answer: "Dependence between these concepts")
The topic of the lesson is formulated: “The relationship between the sine and cosine of the same angle”
Teacher : “There are several ways to solve this problem”
Using the unit circle equationUsing the Pythagorean theorem
Teacher : “Let’s look at both and choose the most rational one”
On the desk:
Students derive equalitycos 2 α + sin 2 α = 1
Teacher : “We have obtained an equality that is fair for any values of the letters included in it. What are such equalities called?
( Suggested answer : identities)
Teacher : “Remember what the identity is calledcos 2 α + sin 2 α = 1 »
Reinforcing the material learned
A) Teacher: “Open the textbook p. 147, No. 457 (2;4)” (the called students solve at the board)
B) Teacher: “Proceed with task No. 2. We are working on options" (Discussion of the results obtained)
On the desk:
Option 1 Option 2
Teacher: “In these formulas, the roots are preceded by the signs “±» . What determines which sign to put in the formula?”
(Suggested answer: “From which quarter the angle of rotation of the point P(1;0)” is located)
B) Teacher: “Proceed with task number 3.” (Students solve problems, check on the board)
Summing up the lesson
Teacher: “Well done! We will summarize the lesson with the help of a crossword puzzle” (Task 4) (Students work in pairs at the computer)
7) Reflection in the form of a questionnaire (Appendix 2)
Teacher: “Summarize your performance in class by completing the test.”
8) Homework
§25, No. 456, 457(1;3),460(1;3).
Report
And the sine graph is wave by wave
The x-axis runs away.From a student song.
GOALS AND OBJECTIVES OF THE LESSON:
- EDUCATIONAL: derivation of formulas for the relationship between sine, cosine and tangent of the same angle (number); learning to use these formulas to calculate the values of sine, cosine, tangent of a number from a given value of one of them.
- DEVELOPMENTAL: teach to analyze, compare, build analogies, generalize and systematize, prove and disprove, define and explain concepts..
- EDUCATIONAL: fostering a conscientious attitude towards work and a positive attitude towards knowledge.
HEALTH SAVING: creating a comfortable psychological climate in the classroom, an atmosphere of cooperation: student - teacher.
METHODOLOGICAL EQUIPMENT OF THE LESSON:
MATERIAL AND TECHNICAL BASE: mathematics room.
DIDACTIC SUPPORT FOR THE LESSON: textbook, notebook, posters on the topic of the lesson, tables, computer, disks, screen, projector.
METHODS OF ACTIVITY: group and individual work at a desk and at the blackboard.
TYPE OF LESSON: lesson on learning new knowledge.
DURING THE CLASSES
1. Organizational moment: greeting, checking student attendance, filling out the register.
2. Checking students’ readiness for the lesson: getting students in the mood for work, bringing the lesson plan to them.
3. Analysis of homework errors. On the screen is a picture of correctly completed homework. Each student checks with a detailed frontal explanation and notes the correctness of execution in the lesson work card.
WORKING CARD OF THE LESSON.
S/o – self-esteem.
O/t – assessment of a comrade.
4. Updating knowledge, preparing to perceive new material.
The next stage of our lesson is dictation. We write down the answers briefly - we have the drawing on the slide.
Dictation (oral repetition of necessary information):
1. Define:
- sine of acute angle A of a right triangle;
- cosine of acute angle B of a right triangle;
- tangent of acute angle A of a right triangle;
- cotangent of acute angle B of a right triangle;
- what restrictions do we impose on sine and cosine when determining the tangent and cotangent of an acute angle right triangle.
2. Define:
- sine of the angle a a.
- cosine of the angle a through the coordinate (which) of the point obtained by rotating the point (1;0) around the origin by an angle a.
- tangent of the angle a.
- cotangent of the angle a.
3. Write down the signs of sine, cosine, tangent, cotangent for angles obtained by rotating the point P(1;0) by an angle
4. For all these angles, indicate the quarters of the coordinate plane.
The children check the dictation on the slide together with the teacher, explaining each statement and giving themselves a grade on the lesson card.
5. From the history of trigonometry. The modern form of trigonometry was given by the greatest mathematician of the 18th century Leonard Euler- Swiss by birth, long years worked in Russia and was a member of the St. Petersburg Academy of Sciences. He introduced well-known definitions of trigonometric functions, formulated and proved reduction formulas that you have yet to encounter, and identified classes of even and odd functions.
6. Introduction of new material:
The main thing is not just to inform students of the final conclusions, but to make students, as it were, participants in a scientific search: by posing the question so that, having awakened their curiosity, they become involved in the research, which helps to achieve a higher level of mental development of students.
Therefore, when introducing new material, I create a problematic situation - how can it be easier and more rational to establish the relationship between the sine and cosine of the same angle - through the equation of the unit circle or through the Pythagorean theorem.
The class is divided into options into the first and second options - on the screen there is a slide with the conditions and drawings, there is no solution yet.
Option 1 establishes the relationship between sine and cosine through the equation of a circle with a center at the origin and a radius equal to 1x 2 +y 2 =1; sin 2 + cos 2 =1.
Option 2 establishes the relationship between sine and cosine through the Pythagorean theorem - in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: OB 2 +AB 2 =OA 2 - and we get sin 2 +cos 2 =1.
They compare the results and draw conclusions: the main thing is that equality is true for any values of the letters included in it? Students must answer that this is an identity
(the slide shows the correct solution for both the first and second options).
We have obtained an equality that is valid for any values of the letters included in it. What are such equalities called? That's right - identities.
Let's remember what other identities we know in algebra - abbreviated multiplication formulas:
a 2 -b 2 =(a-b)(a+b),
(a-b) 2 =a 2 -2ab+b 2,
(a+b) 3 =a 3 +3a 2 b+3ab 2 +b 2 ,
(a-b) 3 =a 3 -3a 2 b+3ab 3 -b 3 ,
a 3 -b 3 =(a-b)(a 2 +ab+b 2),
a 3 +b 3 =(a+b)(a 2 -ab+b 2).
The next problem is why we derived the main trigonometric identity - sin 2 +cos 2 =1.
That's right - to find from one known value of sine, cosine or tangent - the values of all other functions.
Now you and I can always use the basic trigonometric identity, but the main thing is for the same argument.
Application of acquired knowledge:
OPTION 1 – express the sine through the cosine of the angle.
Option 2 – express the cosine through the sine of the angle. The correct answer is on the slide
Teacher's question: did anyone forget to put the + and - signs? What could the angle be? - anyone.
In these formulas, the sign in front of the root depends on what? on which quadrant the angle (argument) of the trigonometric function we are defining is located.
We perform at the board 2 students No. 457. – 1st option - 1, 2nd option - 2.
The slide shows the correct solution.
Independent work on recognizing the basic trigonometric identity
1. find the meaning of the expression:
2. express the number 1 through an angle a, If
There is a mutual check - on the finished slide and evaluation of the work - both by self-assessment and by the assessment of a friend.
6. Consolidation of new material (according to G.E. Khazankin’s technology - technology of support tasks).
TASK 1. Calculate ……….. if ……………………………………………………………….
1 student at the board independently - then a slide with the correct solution.
TASK 2. Calculate……………., if…………………………………………………………………………………..
2nd student at the board, then a slide with the correct solution.
7. Physical education minute. I know that you are already adults and think that you are not tired at all, especially now, when the lesson is going on so actively that time seems to be lengthening for us, according to A. Einstein’s theory of relativity, but let’s do gymnastics for cerebral vessels:
- turns and tilts the head right - left, up - down
- massage of the shoulder girdle and scalp - arms from the hand, face and back of the head - from top to bottom.
- raise your shoulders up and relax down. We perform each exercise 5-6 times!
Let us now find out the relationship between tangent and cotangent……………………………………………………………………………………………………………………………
There is a new study on the topic - what could be the angle in the second trigonometric identity?
THE MAIN THING IS TO DETERMINE THE SET ON WHICH THESE EQUALITIES ARE FULFILLED. MARK THE POINTS IN THE FIGURE AT WHICH THE TANGENS AND COTENGENTS OF THE ANGLE DO NOT EXIST.
3rd student at the blackboard. The equalities are valid for……………………….
TASK 3. Calculate………if………………………….
TASK 4. Calculate…………….. if………………………………………………………………
The rest of the students work in their notebooks.
1 SUPPORT…………………………………………………………………………………………
2 SUPPORT…………………………………………………………………………………………………………………
3 SUPPORT. Application of the basic trigonometric identity to problem solving.
8. Crossword. Anatole France once said: “Learning must be fun... To digest knowledge, you must absorb it with appetite.”
To test your knowledge on this topic, you are offered a crossword puzzle.
- A branch of mathematics that studies the properties of sine, cosine, tangent...
- Abscissa of a point on the unit circle.
- The ratio of cosine to sine.
- Sine is…..points on the unit circle.
- An equality that does not require proof and is true for any values of the letters included in it.
It's called...... After checking the crossword puzzle, the children give themselves grades on the lesson map. The teacher gives grades to those students who were especially active in the lesson. Bottom line - GPA
for work in class.
9. Instructing the teacher on completing homework.
10. The teacher sums up the lesson.
11. Homework: paragraph 25 (before problem 5), No. 459 (even), 460 (even), 463*(4). Textbook by Sh.A Alimov “Algebra and the beginnings of analysis”., 10-11, “Enlightenment”., M., 2005. In this article we will take a comprehensive look. Basic trigonometric identities
Let us immediately list the main trigonometric identities that we will analyze in this article. Let's write them down in a table, and below we'll give the output of these formulas and provide the necessary explanations.
Page navigation.
Relationship between sine and cosine of one angle
Sometimes they do not talk about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind 
. The explanation for this fact is quite simple: the equalities are obtained from the main trigonometric identity after dividing both of its parts by and, respectively, and the equalities 
And 
follow from the definitions of sine, cosine, tangent and cotangent. We'll talk about this in more detail in the following paragraphs.
That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.
Before proving the main trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.
The basic trigonometric identity is very often used when converting trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in the reverse order: unit is replaced by the sum of the squares of the sine and cosine of any angle.
Tangent and cotangent through sine and cosine
Identities connecting tangent and cotangent with sine and cosine of one angle of view and 
follow immediately from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, sine is the ordinate of y, cosine is the abscissa of x, tangent is the ratio of the ordinate to the abscissa, that is, 
, and the cotangent is the ratio of the abscissa to the ordinate, that is, 
.
Thanks to such obviousness of the identities and Tangent and cotangent are often defined not through the ratio of abscissa and ordinate, but through the ratio of sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.
In conclusion of this paragraph, it should be noted that the identities and take place for all angles at which the trigonometric functions included in them make sense. So the formula is valid for any , other than (otherwise the denominator will have zero, and we did not define division by zero), and the formula - for all , different from , where z is any .
Relationship between tangent and cotangent
An even more obvious trigonometric identity than the previous two is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it holds for any angles other than , otherwise either the tangent or the cotangent are not defined.
Proof of the formula very simple. By definition and from where 
. The proof could have been carried out a little differently. Since , That 
.
So, the tangent and cotangent of the same angle at which they make sense are .