Public lesson in algebra and the beginnings of analysis on the topic: "The relationship between the sine and cosine of the same angle" (Grade 10)
Target: perception by students and primary awareness of the new educational material comprehension of connections and relationships in the 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, cosine for a given value of one of them.
Educational : to learn to analyze, compare, build analogies, generalize and systematize, prove and refute, define and explain concepts, develop and improve the ability to apply students' knowledge in different situations; to develop competent mathematical speech of students, the ability to give concise formulations
Educational: education of a conscientious attitude to work and a positive attitude to knowledge, to educate students in accuracy, the ability to listen, to express their opinion; 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 in quarters; sets of values of trigonometric functions; basic formulas of trigonometry.Atthe ability to choose the right formula for solving a specific task; work with simple fractions; perform the transformation of trigonometric expressions.
During the classes
Organizing time:
Check student readiness for the lesson. Opening a teacher's website on computers (Appendix 1).
Oral work on the 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 the sine, cosine, tangent and cotangent.
Independent work on the topic: "Signs of sine, cosine and tangent"
Students open the section "Assignments for the lesson in trigonometry" on the site. Self test
(Students complete task number 1, check their work and evaluate themselves)
Explanation of new material
On the desk:
х = … α , … ≤ cos α≤ … 2)* tg α = , α≠ …
y= … α, … ≤ sin α≤ … ctg α = , α≠ …
Exercise: add formulas
Teacher : “We have studied each concept separately. What do you think is the best topic to explore next?
( Suggested answer: "Dependence between these concepts")
The topic of the lesson is formulated: "Relationship between 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 consider both and choose the most rational"
On the desk:
Students draw the equationcos 2 α + sin 2 α = 1
Teacher : “We got equality 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 »
Consolidation of the studied material
A) teacher “Open the textbook p. 147, No. 457 (2; 4)” (called students decide at the blackboard)
B) Teacher: “Go to task number 2. We work according to options” (Discussion of the results obtained)
On the desk:
1 option 2 option
Teacher: “In these formulas, the signs are in front of the root”±» . What determines which sign to put in the formula?
(Suggested answer: “On the quarter in which the angle of rotation of the point P (1; 0) is located”)
B) teacher: "Go to task number 3." (Students solve tasks, check on the board)
Summing up the lesson
Teacher: “Well done! We will sum up 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: "Make a conclusion about your work in the lesson by completing the test."
8) Homework
§25, #456, 457(1;3),460(1;3).
Report
Let's try to find the relationship between the main trigonometric functions the same corner.
Relationship between cosine and sine of the same angle
The following figure shows the coordinate system Oxy with the part of the unit semicircle ACB depicted in it, centered at the point O. This part is the arc of the unit circle. The unit circle is described by the equation
- x2+y2=1.
As already known, the ordinate y and the abscissa x can be represented as the sine and cosine of the angle using the following formulas:
- sin(a) = y,
- cos(a) = x.
Substituting these values into the equations of the unit circle, we have the following equality
- (sin(a)) 2 + (cos(a)) 2 =1,
This equality holds for any values of the angle a. It is called the basic trigonometric identity.
From the main trigonometric identity, one function can be expressed in terms of another.
- sin(a) = ±√(1-(cos(a)) 2),
- cos(a) = ±√(1-(sin(a)) 2).
The sign on the right side of this formula is determined by the sign of the expression on the left side of this formula.
For instance.
Calculate sin(a) if cos(a)=-3/5 and pi Let's use the formula above: Since pi Now, let's try to find the relationship between the tangent and cotangents. By definition, tg(a) = sin(a)/cos(a), ctg(a) = cos(a)/sin(a). Multiplying these equalities, we get tg(a)*ctg(a) =1. From this equality, one function can be expressed in terms of another. We get: It should be understood that these equalities are valid only when tg and ctg exist, that is, for any a, except for a = k * pi / 2, for any integer k. Now let's try using the basic trigonometric identity to find the relationship between tangent and cosine. Divide the basic trigonometric identity, by (cos(a)) 2 . (cos(a) is not equal to zero, otherwise the tangent would not exist. We get the following equality ((sin(a)) 2 + (cos(a)) 2)/ (cos(a)) 2 =1/(cos(a)) 2 . Dividing term by term we get: As noted above, this formula is true if cos(a) is not equal to zero, that is, for all angles a, except a=pi/2 + pi*k, for any integer k. In this article, we will take a comprehensive look at . Basic trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other. We immediately list the main trigonometric identities, which we will analyze in this article. We write them down in a table, and below we give the derivation of these formulas and give the necessary explanations. Page navigation. Sometimes they talk not about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity. Before proving the basic 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 in transformation of 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 reverse order: the unit is replaced by the sum of the squares of the sine and cosine of any angle. Identities connecting the tangent and cotangent with the sine and cosine of one angle of the form and Due to this obviousness of the identities and To conclude this section, it should be noted that the identities and An even more obvious trigonometric identity than the two previous ones is the identity connecting the tangent and cotangent of one angle of the form Proof of the formula So, the tangent and cotangent of one angle, at which they make sense, is. Trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, which allows you to find any of these functions, provided that any other is known. tg \alpha = \frac(\sin \alpha)(\cos \alpha), \enspace ctg \alpha = \frac(\cos \alpha)(\sin \alpha) tg \alpha \cdot ctg \alpha = 1 This identity says that the sum of the square of the sine of one angle and the square of the cosine of one angle is equal to one, which in practice makes it possible to calculate the sine of one angle when its cosine is known and vice versa. When converting trigonometric expressions, this identity is very often used, which allows you to replace the sum of the squares of the cosine and sine of one angle with one and also perform the replacement operation in reverse order. tg \alpha = \frac(\sin \alpha)(\cos \alpha),\enspace These identities are formed from the definitions of sine, cosine, tangent and cotangent. After all, if you look, then by definition, the ordinate of y is the sine, and the abscissa of x is the cosine. Then the tangent will be equal to the ratio \frac(y)(x)=\frac(\sin \alpha)(\cos \alpha), and the ratio \frac(x)(y)=\frac(\cos \alpha)(\sin \alpha)- will be a cotangent. We add that only for such angles \alpha for which the trigonometric functions included in them make sense, the identities will take place, ctg \alpha=\frac(\cos \alpha)(\sin \alpha). For instance: tg \alpha = \frac(\sin \alpha)(\cos \alpha) is valid for \alpha angles that are different from \frac(\pi)(2)+\pi z, a ctg \alpha=\frac(\cos \alpha)(\sin \alpha)- for an angle \alpha other than \pi z , z is an integer. tg \alpha \cdot ctg \alpha=1 This identity is valid only for angles \alpha that are different from \frac(\pi)(2) z. Otherwise, either cotangent or tangent will not be determined. Based on the points above, we get that tg \alpha = \frac(y)(x), a ctg\alpha=\frac(x)(y). Hence it follows that tg \alpha \cdot ctg \alpha = \frac(y)(x) \cdot \frac(x)(y)=1. Thus, the tangent and cotangent of one angle at which they make sense are mutually reciprocal numbers. tg^(2) \alpha + 1=\frac(1)(\cos^(2) \alpha)- the sum of the square of the tangent of the angle \alpha and 1 is equal to the inverse square of the cosine of this angle. This identity is valid for all \alpha other than \frac(\pi)(2)+ \pi z. 1+ctg^(2) \alpha=\frac(1)(\sin^(2)\alpha)- the sum of 1 and the square of the cotangent of the angle \alpha , equals the inverse square of the sine of the given angle. This identity is valid for any \alpha other than \pi z . Find \sin \alpha and tg \alpha if \cos \alpha=-\frac12 and \frac(\pi)(2)< \alpha < \pi
; Show Solution Solution The functions \sin \alpha and \cos \alpha are linked by the formula \sin^(2)\alpha + \cos^(2) \alpha = 1. Substituting into this formula \cos \alpha = -\frac12, we get: \sin^(2)\alpha + \left (-\frac12 \right)^2 = 1 This equation has 2 solutions: \sin \alpha = \pm \sqrt(1-\frac14) = \pm \frac(\sqrt 3)(2) By condition \frac(\pi)(2)< \alpha < \pi
. In the second quarter, the sine is positive, so \sin \alpha = \frac(\sqrt 3)(2). To find tg \alpha , we use the formula tg \alpha = \frac(\sin \alpha)(\cos \alpha) tg \alpha = \frac(\sqrt 3)(2) : \frac12 = \sqrt 3 Find \cos \alpha and ctg \alpha if and \frac(\pi)(2)< \alpha < \pi
. Show Solution Solution Substituting into the formula \sin^(2)\alpha + \cos^(2) \alpha = 1 conditional number \sin \alpha=\frac(\sqrt3)(2), we get \left (\frac(\sqrt3)(2)\right)^(2) + \cos^(2) \alpha = 1. This equation has two solutions \cos \alpha = \pm \sqrt(1-\frac34)=\pm\sqrt\frac14. By condition \frac(\pi)(2)< \alpha < \pi
. In the second quarter, the cosine is negative, so \cos \alpha = -\sqrt\frac14=-\frac12. In order to find ctg \alpha , we use the formula ctg \alpha = \frac(\cos \alpha)(\sin \alpha). We know the corresponding values. ctg \alpha = -\frac12: \frac(\sqrt3)(2) = -\frac(1)(\sqrt 3). A sine graph wave by wave From a student song. GOALS AND OBJECTIVES OF THE LESSON: 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 office. DIDACTIC SUPPORT OF THE LESSON: textbook, notebook, posters on the topic of the lesson, tables, computer, disks, screen, projector. ACTIVITY METHODS: group and individual work at the desk and at the blackboard. TYPE OF LESSON: a lesson in mastering new knowledge. DURING THE CLASSES 1. Organizational moment: greeting, checking the attendance of students, filling out the journal. 2. Checking the readiness of students for the lesson: setting the students to work, bringing the lesson plan to them. 3. Analysis of homework errors. On the screen - a picture with a correctly completed homework. Each student checks with a detailed frontal explanation and notes the correctness of the implementation in the work card of the lesson. WORKING LESSON CARD. C / o - self-esteem. O / t - assessment of a friend. 4. Actualization of knowledge, preparation for the perception of new material. The next stage of our lesson is dictation. We write down the answers briefly - the drawing is on our slide. Dictation (oral repetition of the necessary information): 1. Define: 2. Define: 3. Write down the signs of the sine, cosine, tangent, cotangent for the angles obtained by turning the point P (1; 0) by the angle 4. For all these angles, indicate the quarters of the coordinate plane. The guys check the dictation on the slide together with the teacher, explaining each statement and grading themselves on the lesson worksheet. 5. From the history of trigonometry. The modern form of trigonometry was given by the largest mathematician of the 18th century Leonard Euler- Swiss by origin, who worked in Russia for many years 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 meet, and distinguished classes of even and odd functions. 6. Introduction of new material: The main thing is not just to inform the students of the final conclusions, but to make the students, as it were, participants in a scientific search: by posing a question, so that they, having awakened their curiosity, are included in the study, which contributes to the achievement of a higher level of mental development of students. Therefore, when introducing new material, I create a problem situation - how easier and more rational to establish the relationship between the sine and cosine of the same angle - through the unit circle equation or through the Pythagorean theorem. The class is divided into options for the first and second options - on the screen there is a slide with a condition 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 \u003d OA 2 - and we get sin 2 + cos 2 \u003d 1. They compare the results, draw conclusions: the main thing is equality is fulfilled for any values of the letters included in it? Students must answer that this is the same (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 these equalities called? That's right - identities. Recall - what other identities do we know in algebra - the formulas for abbreviated multiplication: a 2 -b 2 \u003d (a-b) (a + b), (a-b) 2 \u003d 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 \u003d (a-b) (a 2 + ab + b 2), a 3 + b 3 \u003d (a + b) (a 2 -ab + b 2). The next problem - why did we derive the main trigonometric identity - sin 2 + cos 2 =1. That's right - to find one known value of the sine, cosine or tangent - the values of all other functions. Now we 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 in terms of the cosine of the angle. Option 2 - Express the cosine in terms of the sine of the angle. The correct answer is on the slide. Question of the teacher - no one forgot to put down the + and - signs? What could be the angle? - anyone. In these formulas, the sign in front of the root depends on what? on which quarter the angle (argument) of the trigonometric function that we are defining is located. We perform at the blackboard 2 students No. 457. - 1st option - 1, 2nd option - 2. The slide shows the correct answer. Independent work on recognition of the basic trigonometric identity 1. find the value of the expression: 2. express the number 1 through the angle a, if There is a mutual check - according to 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 the technology of G.E. Khazankin - technology of supporting tasks). PROBLEM 1. Calculate ……….. if …………………………………………………………………. 1 student at the blackboard on their own - then slide with the correct solution. TASK 2. Calculate……………., if……………………………………………………………….. 2nd student at the blackboard, then 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 so active that the time for us seems to be lengthening - according to A. Einstein's theory of relativity, but let's do gymnastics for cerebral vessels: Let us now find out the relationship between the tangent and the cotangent………………………………………………………………………………………………………… There is a new study on the topic - what can be the angle in the second trigonometric identity? THE MAIN THING IS CLAIMING THE SET ON WHICH THESE EQUALITIES ARE CARRIED OUT. MARK ON THE PICTURE THE POINTS WHERE THE TANGENT AND COTENGENCE OF THE ANGLE DO NOT EXIST. 3rd student at the blackboard. The equalities are valid for………………………. TASK3. 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 should be fun ... To digest knowledge, you need to absorb it with appetite." To test your knowledge on this topic, you are offered a crossword puzzle. After checking the crossword puzzle, the guys give themselves grades in the work card of the lesson. The teacher grades those students who are especially active in the lesson. The result is the average score for the work in the lesson. 9. Instructing the teacher to do homework. 10. Summing up the lesson by the teacher. 11. Homework: paragraph 25 (before task 5), No. 459 (even), 460 (even), 463 * (4). Textbook Sh.A Alimov "Algebra and the beginning of analysis", 10-11, "Enlightenment", M., 2005.The ratio between the tangent and cotangent of the same angle
Relationship between sine and cosine of one angle
. The explanation for this fact is quite simple: the equalities are obtained from the basic 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 will discuss this in more detail in the following paragraphs.Tangent and cotangent through sine and cosine
immediately follow from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, the sine is the ordinate of y, the cosine is the abscissa of x, the 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, 
.often the definitions of tangent and cotangent are given not through the ratio of the abscissa and the ordinate, but through the ratio of the 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.hold for all such angles for which the trigonometric functions in them make sense. So the formula is valid for any other than (otherwise the denominator will be 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
. It is clear that it takes place for any angles other than , otherwise either the tangent or the cotangent is not defined.very simple. By definition and from where 
. The proof could have been carried out in a slightly different way. Since and , then 
.Finding tangent and cotangent through sine and cosine
Relationship between tangent and cotangent
Relationships between tangent and cosine, cotangent and sine
Examples with solutions to problems using trigonometric identities
Example 1
Example 2
The abscissa runs away.
