Basic trigonometric identities. Relationship between trigonometric functions of the same angle
LESSON MAP “DEPENDENCE BETWEEN SINE, COSINE AND TANGENT OF THE SAME ANGLE”
Student ________________________________________________________________________________
| 1. I know the material from previous lessons | Points |
| I answered all the questions correctly without notes. | |
| I answered without a note with one mistake. | |
| I answered without taking notes and made more than one mistake. | |
| I answered all the questions correctly using the notes. | |
| I answered using my notes, with one mistake. | |
| I answered using my notes and made more than one mistake |
| 2. I have completed recording the examples. | Points |
| I completed all tasks without errors | |
| I completed with one error | |
| I completed the tasks and made more than two mistakes |
| 3. I deduced the formula for finding sine and cosine | Points |
| I got the formulas right | |
| I derived the formulas and made one mistake | |
| I derived the formulas with the help of my teacher |
| 4. I applied my knowledge on the topic: “The relationship between the sine, cosine and tangent of the same angle” when solving independent work | Points |
| I solved the examples of option 1 without errors. | |
| I solved the examples of option 1 and made a mistake. | |
| I solved the examples of option 2 without errors. | |
| I solved the examples of option 2 and made a mistake. | |
| I solved examples 3 options without errors | |
| I solved the examples of option 3 and made a mistake. | |
| I solved examples 4 options without errors. | |
| I solved the examples of option 4 and made a mistake. |
| 5. Rate yourself: | |
| I understood the derivation of the formulas and can solve examples on this topic with a notebook and the help of a teacher. | |
| I understood the derivation of the formulas and can solve examples on my own without a notebook, just by looking at the formulas. | |
| I understood the derivation of formulas and can solve examples on my own without a notebook; if I forget a formula, I can deduce it myself. |
My points: __________
Maximum number of points – 22
18 – 22 points - score “5”
15 – 17 points - score “4”
11–14 points - score “3”
Less than 11 points - you need to come for a consultation in the coming days, the material has not yet been mastered.
"Brief Plan"
Vera Anatolyevna Golovatova, mathematics teacher
GB POU "Okhta College"
Summary of two lessons for studentsI course (10th grade) on the topic:
“The relationship between sine, cosine and tangent of the same angle”
Target: study the relationship between sine, cosine and tangent of the same angle.
To achieve this goal it is necessary:
Know:
formulations of definitions of basic trigonometric functions (sine, cosine and tangent);
signs of trigonometric functions by quarters;
set of values of trigonometric functions;
basic formulas of trigonometry.
Understand:
that the basic trigonometric identity can only be used for the same argument;
algorithm for calculating one trigonometric function through another.
Apply:
the ability to correctly choose the right formula to solve a specific task;
ability to work with simple fractions;
ability to perform transformations of trigonometric expressions.
Analysis:
analyze errors in the logic of reasoning.
Synthesis:
suggest your own way of solving examples;
create a crossword using the knowledge you have gained.
Grade:
knowledge and skills on this topic for use in other sections of algebra.
Equipment: layout of a trigonometric circle, handouts with formulas and tables of values of trigonometric functions, computer, multimedia projector, presentation, sheets with tasks for independent work.
Sources used:
Algebra and beginnings of analysis: Textbook for grades 10-11. general education institutions / Sh.A.Alimov, Yu.V. Sidorov et al. Education, 2006.
Open Bank assignments for preparing for the Unified State Exam in mathematics, 2011.
INTERNET network resources.
Brief plan lesson:
Greetings. Communicating the purpose of the lesson and the lesson plan - 3-5 min.
Updating knowledge and skills.
Students are given lesson cards and given explanations on how to work with them.
Questions are displayed on the screen; students write down answers in a notebook; The teacher displays the correct answer on the screen. After completing the survey, students add points to the lesson card for Tasks No. 1 – 10 min.
Explanation of new material.
The teacher derives the formula for the main trigonometric identity – 5 minutes.
Students are asked to independently complete the recording of the examples displayed on the screen, check the correctness of the answers and add points to the lesson card for Tasks No. 2 – 5 minutes.
In the notebook, students are asked to independently express sine through cosine and cosine through sine from the basic trigonometric identity. The correct answer is displayed on the screen, students check and add points to the lesson card for Tasks No. 3 – 5-7 min.
The teacher solves examples on the blackboard using the basic trigonometric identity. Students answer the teacher’s questions during the explanation and write down examples in their notebooks - 15 minutes.
The teacher derives formulas showing the relationship between tangent and cotangent, students take an active part in deriving formulas, answer questions and make notes in a notebook - 5 minutes.
The teacher derives formulas showing the relationship between tangent and cosine, between sine and cotangent - 5 minutes.
Students are called to the board at will and, with the help of the teacher, solve examples using an algorithm. Everyone else writes down and answers questions as necessary - 10 min.
Reinforcing the material learned
At the end of the lesson, the correct answers are displayed on the screen, students check their answers and add points to the lesson card for Tasks No. 4 – 20 minutes.
Homework: Students write homework assignments in their notebooks - 3 min.
View document contents
"Reflection"
After attending seminars on RNS and conducting a lesson using technological map It became obvious to me that the rating system stimulates the maximum possible interest of students in a particular topic. In my case, these are the basic formulas of trigonometry.
Trigonometry is very often not perceived by students not so much because of its complexity, but because large quantity formulas that you need to be able to work with.
It is difficult to expect any incredible successes and results after one lesson conducted using a technological map, but it seems to me that the benefits rating system in the study of trigonometry and mathematics in general are as follows:
it became possible to organize and support both work in the classroom and independent, systematic work of students at home;
Attendance and level of discipline in lessons should increase;
motivation for educational activities increases;
decrease stressful situations upon receiving unsatisfactory grades;
creative attitude towards work is stimulated.
The only drawback of RNS (as it seems to me) is a large amount of work for the teacher, but this is work for results. After a single lesson taught using this system, students constantly ask whether we will continue to work this way. It means they were hooked by something. And we need to keep working.
View document contents
"Independent work"
INDEPENDENT WORK
Whatever level you choose, first carefully review all the tasks that I have given to you, and then complete the task corresponding to the level you have chosen (Before you are tasks of four options, the number of the option corresponds to the levels of self-esteem.)
1 option
Instructions:
Instructions:
Solve this example yourself: 
Option 2
Hint: To determine the cosine function, use formula (3) from today's lesson. Don't forget to determine the sign that will appear in front of the root. To calculate the values of tangent and cotangent, you can use the definition of these functions or use the formulas that we developed today in class.
Note. Group the first and third terms of the expression, take the common factor out of brackets....
Option 3
Option 4
View presentation content
"Presentation"
Repetition:
1. In which quarter is the angle in
1 radian and what is it approximately equal to?
In the first quarter, 1 rad. 57.3°
2. What word is missing in the definition of the sine function?
Sine of the angle called ………… points of the unit circle.
ORDINATE
3. What word is missing in the definition of the cosine function?
Cosine of the angle called
………… points of the unit circle.
ABSCISSA
4. Complete the formula:
tg
5. Determine the sign of the product:
tg
6. What value can sine take?
or
7. Calculate:
y
B(x;y)
R
Y=sin
O
x
x=cos
Complete the recording:
x
y
x
y
x
x
x
y
x
y
x
x
- I understood the topic and can solve examples using the algorithm, looking at the notebook, but with the help of leading questions (card - instructions).
- I understood the topic and can solve examples using the algorithm, looking at the notebook, using the teacher’s instructions.
- + I understood the topic and can solve examples using the algorithm, looking at the notebook, without leading questions or instructions.
- + I understood the topic and can solve examples using the algorithm without looking at the notebook.
Option 1:
Option 3:
Option 2:
Option 4:
Subject: Trigonometric formulas (25 hours)
Lesson 6 – 7: The relationship between sine, cosine and tangent of the same angle.
Target: study the relationship between sine, cosine and tangent of the same angle. To achieve this goal it is necessary:
- Know:
- formulations of definitions of basic trigonometric functions (sine, cosine and tangent); signs of trigonometric functions by quarters; set of values of trigonometric functions; basic formulas of trigonometry.
- Understand:
- that the basic trigonometric identity can only be used for the same argument; algorithm for calculating one trigonometric function through another.
- Apply:
- the ability to correctly choose the right formula to solve a specific task; ability to work with simple fractions; ability to perform transformations of trigonometric expressions.
- Analysis:
- analyze errors in the logic of reasoning.
- Synthesis:
- suggest your own way of solving examples; create a crossword using the knowledge you have gained.
- Grade:
- knowledge and skills on this topic for use in other sections of algebra.
- Organizing time.
- Updating knowledge and skills.
- In which quarter is an angle of 1 radian located and what is it approximately equal to?
- What word is missing from the definition of the sine function?
- What word is missing in the definition of the cosine function?
- What values can sine take?
() - Explanation of new material.
Let us depict a unit circle with the center at point O. Let us assume that by rotating the radius OA, equal to R, by an angle , the radius OB is obtained (Fig. 5). Then by definition 
Where 
– abscissa of point B, 
– its ordinate. It follows that Point B belongs to the circle. Therefore, its coordinates satisfy the equation 
Taking advantage of what we get 
(1).
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. Equality (1) is called basic trigonometric identity. In equality (1) can take any value. Complete the recording yourself: 1.
Please check that your entry is correct. Add points to your lesson card for Tasks No. 2. Let's continue. We have derived the main trigonometric identity, but why do we need it? That's right - to find the cosine value from one known sine value and vice versa. Now you and I can always use the basic trigonometric identity, but the main thing is for the same argument. In the notebook, students are asked to independently express sine through cosine and cosine through sine from the basic trigonometric identity. Two students are called to the board to check. One is asked to express the sine through the cosine, the second - the cosine through the sine. The correct answer is displayed on the screen:
Students check their answers and add points to the lesson card for Tasks No. 3.
In these formulas, what does the sign in front of the root depend on? (It depends on which quadrant the angle of the trigonometric function we are defining is located in). Example 1 . Calculate
If 
Determine the quarter in which the angle is located 
. Quarter – III. Let us remember that the sine in the third quarter is negative, i.e. in formula (2) you need to put the “-” sign in front of the root: Example 2.
Calculate 
If 
We determine the quarter in which the angle is located. Quarter – IV, cosine in the fourth quarter is positive. Therefore, in formula (3) the “+” sign is needed before the root: Let's find out now relationship between tangent and cotangent. By definition of tangent and cotangent
Multiplying these equalities, we get:
From equality (4) we can express
through 
and vice versa: 
Equalities (4) – (6) are true for all values for which
make sense, i.e. when 
Let us now derive formulas expressing the relationship between tangent and cosine, as well as cotangent and sine of the same argument. Dividing both sides of equality (1) by 
, we get: 
those. 
If both sides of equality (1) are divided by
, then we will have: 
those. 
Let's look at examples of using the derived formulas to find the values of trigonometric functions according to known value one of them.
Example 1. Let's find if it is known that
Solution: 
- To find the cotangent of angle it is convenient to use formula (6):
Answer: Example 2. It is known that
. Let's find all the other trigonometric functions. Solution: - Let's use the formula (7).
We have:
, 
. According to the conditions of the problem, angle is the angle of 1 quarter, so its cosine is positive. Means 
Answer:
Established relationships between trigonometric functions of the same argument allow you to simplify trigonometric expressions. Example 3. Let's simplify the expression:
Solution: Let's use the formulas: 
. We get: - Consolidation.
And now the screen presents self-assessment rubrics on this topic. Mark what level you would like to reach today.
I understood the topic and can solve examples using the algorithm, looking at the notebook, but with the help of leading questions (card - instructions).
I understood the topic and can solve examples using the algorithm, looking at the notebook, using the teacher’s instructions.
I understood the topic and can solve examples using the algorithm, looking at the notebook, without leading questions or instructions.
I understood the topic and can solve examples using the algorithm without looking at the notebook.
Whatever level you choose, first carefully review all the tasks that I have given to you, and then complete the task corresponding to the level you have chosen (there are tasks in front of you in four options, the number of the option corresponds to the levels of self-esteem.)
1 option
Instructions:
Option 4
Now guys, let's check the answers. The correct answers are displayed on the screen, and students check their work and add points to the lesson card for Tasks No. 4. Evaluate yourself using the lesson map. Calculate your points and put them on the card.
- Homework.
- Write down all the derived formulas in the reference book. According to textbook No. 459 (3, 5), No. 460 (1)
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 this 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
Let's try to find the relationship between the main trigonometric functions of the same angle.
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 with the center at point O. This part is the arc of the unit circle. The unit circle is described by the equation
- x 2 +y 2 =1.
As is already known, the ordinate y and 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 is true for any value of angle a. It is called the basic trigonometric identity.
From the basic 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 example.
Calculate sin(a) if cos(a)=-3/5 and pi Let's use the formula given 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). Let's multiply these equalities and get tg(a)*ctg(a) =1. From this equality one can express one function through another. We get: It should be understood that these equalities are valid only when tg and ctg exist, that is, for any a, except 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. Let's divide the main trigonometric identity by (cos(a)) 2. (cos(a) is not equal to zero, otherwise the tangent would not exist. We obtain 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 correct if cos(a) is not equal to zero, that is, for all angles a, except a=pi/2 +pi*k, for any integer k.Relationship between tangent and cotangent of the same angle