Independent work:

Option 2:

Option 1:

Check the answers:

Option 2:

Option 1:

Cosine theorem:

The square of the side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them

• The oldest proof for the theorem of sines on a plane is described in the book by Nasir ad-Din At-Tusi "A Treatise on the Complete Quadripartite" written in the 13th century. The sine theorem for a spherical triangle was proved by mathematicians of the medieval East back in the 10th century. In the work of Al-Jayani of the 11th century "The Book of Unknown Arcs of a Sphere", a general proof of the theorem of sines on a sphere was given

Nasir ad-Din At-Tusi

Sine theorem :

The sides of a triangle are proportional to the sines of the opposite angles.

• Comment: It can be shown that the ratio of the side of a triangle to the sine of the opposite angle is equal to the diameter of the circumscribed circle. Therefore, for any triangle ABC with sides AB = c, BC = a, CA = b, the equalities
• Where R is the radius of the circumscribed circle.

1) Write down the sine theorem for a given triangle:

2) Write down the cosine theorem for calculating the MK side:

Find angle B.

Find the length of the BC side.

Find the length of the side AB.

Find MN.

Write down the formula to calculate:

• http://ppt4web.ru/geometrija/teoremy-sinusov-i-kosinusov0.html
• http://nsportal.ru/shkola/geometriya/library/2014/10/15/teorema-sinusov-i-kosinusov
• https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Johannes_Regiomontanus2.jpg/500px-Johannes_Regiomontanus2.jpg
• http://img1.liveinternet.ru/images/attach/c/10/110/217/110217775_Nesreddi_tusi.jpg
• http://www.biografguru.ru/about/evklid/?q=3117

Theorems of sines and cosines in problems with practical content

are true?

Exercise 1

works of these parties on sin angle between them.

The square of any side of the tr-ka is equal to the sum

squares of the other two sides without

works of these parties on cos angle between them.

The square of any side of the tr-ka is equal to the sum

squares of the other two sides without doubled

works of these parties on cos angle between them.

In a right triangle

the square of the leg is equal to the difference of the squares

hypotenuse and other leg.

Which of the following statements are true?

Assignment 2

sinuses of opposite angles.

The sides of the triangle are proportional

cosines of opposite angles.

The sides of the triangle are proportional

the sinuses of the adjacent angles.

The sides of the triangle are proportional

opposite corners.

Which of the following statements are true?

Assignment 3

area and perimeter.

Solving a triangle is measuring everything

its elements.

Solving a triangle is finding it

unknown elements for three known ones.

Solving a triangle means finding him

equal triangle.

Not true!

Not true!

Not true!

Establish a match?

Assignment 4

A) sine theorem

B) Heron's formula

C) the Pythagorean theorem

D) cosine theorem

A 1.7m man stands at a distance

8 steps from the post on which the lantern hangs.

A person's shadow is equal to four steps. Which one

height (in meters) is the lantern located?

Assignment 5

8 steps

4 steps

Tip (2)

Consider similar triangles

Δ ABC

Δ AKM

The soccer ball is at the Hedgehog, which is located at a distance of 23 m and 24 m from the goal posts. The width of the goal is 7 m. Find the angle of the ball into the goal?

Assignment 6

Assignment 7

Algorithm for solving practical problems

• Execute drawing
• Solve a geometric problem

Assignment 7

Find the distance to an inaccessible object

Algorithm for finding the distance to an inaccessible object

• Mark 2 points, the distance between which can be measured
• Measure angles
• Build a mathematical model (drawing)
• Solve a geometric problem using the sine theorem

Decide for yourself Option 1 To determine the width of the river (AC), 2 points C and B were marked at a distance of 50 m from each other. We measured the angles ACB and ABC, where A is a tree standing on the other side of the river at the water's edge. (<АCВ=550, <АВС=650) Option 2 To determine the width of the river (AC), 2 points B and C were marked at a distance of 40 m from each other. We measured the angles ACB and ABC, where A is a tree standing on the other side of the river at the water's edge. (<АCВ=600, <АВС=700) Проверьте друг друга <А=1800-600-700= 50 0 AВ = 49 м

Theme « Cosine theorem "

Lesson type : a lesson in the assimilation of new knowledge

Lesson location - the first lesson on this topic

Teaching purpose of the lesson :

knowledge of the formulation of the cosine theorem by students;

skill:

find the length of the third side by the known two others and the angle

between them;

determine the angle (cosine of the angle) of the triangle by three known

parties;

determine the type of triangle on three known sides.

Personal development objectives:

organize situations for:

self-determination of students for a predictable result

cognitive activity;

development of reflective abilities;

create conditions for:

development of communication skills of students;

development of students' thinking, ability to argue, prove.

Equipment and materials: multimedia installation, screen, board, chalk.

Short lesson plan

1. Organizing time.

2. Updating leading knowledge and methods of action.

3. Motivation and goal setting.

4. Main part. Proof of the cosine theorem. Performance

examples of the application of the cosine theorem in solving problems.

Self-application of knowledge. (Mini test).

5. Reflection. Summing up the lesson.

During the classes

Stage 1 Organizational. I greet the students and check the readiness of the schoolchildren's workplace for the class. I create a mood for work, I announce to the students that during the lesson they evaluate themselves by putting marks on the work card.

Stage 2Updating students' knowledge, proposing a hypothesis.

I propose to begin with a warm-up (test) according to the formulas "Casting Formulas", "Sine, cosine and tangent values ​​for angles from 0⁰ to 180⁰".

Write down the formula for finding the distance between points by their coordinates.

Stage 3 Creation of a problem situation, its solution.Motivation and goal setting.

The problematic task increases the motivation of students for further cognitive activity. A situation is organized for setting the goal of the lesson and predicting the results of the lesson, for example, it is necessary to find out a universal way of finding the length of the third side of a triangle by the known lengths of the other two sides and the angle between them.

Group work.

The solution of the problem . Task. Using the formula for the distance between the points, find the length of the BC side▲ ABC, if A (0; 0), B (c; 0), C (bcosA ; bsinA ).

Output: let us give a verbal formulation of the obtained equality. We obtain a theorem called cosine theorem:

the square of the side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them.

One of the most beautiful and simple proofs of the cosine theorem is the proof of it in the coordinate plane.

Can we say that the Pythagorean theorem is a special case of the cosine theorem? Yes, because cos 90 o = 0.

Stage 4. Physical minute.

6 stage. Problem statement: how many elements must be known for the problem to be solved? Build a model, define the type of problem, explore relationships and connections between triangle elements .

Question for discussion denia. What problem can be solved using the cosine theorem?

Knowing that has the form a 2 = b 2 + c 2 - 2bc × cosγ, transform this expression so that the desired value is the angle γ: b 2 + c 2 = 2bc × cosγ + a 2 .
Then give the shown equation to a slightly different form: b 2 + c 2- a 2 = 2bc × cosγ. Then the given expression follows into the one below:

cosγ = √b 2 + c 2 -a2 / 2bc.
Question for discussion denia. What can be found by this formula?

The value of the cosine of an angle in a triangle.

Students are asked to calculate the cosine of a larger angle in a triangle with known lengths of three sides and determine the type of this triangle.

Calculate the cosine of a larger angle in a triangle if its sides are equal:

Option number 1

Option number 2

Option number 3

c = 6, b = 8, a = 9

c = 6, b = 8, a = 10

c = 6, b = 8, a = 11

cos 19/96

cos 0

cos 0

79 0

90 0

103 0

The results of the calculations of each group are entered into a table, discussed, and conclusions are drawn:

To determine the type of triangle (acute, rectangular, obtuse)

necessary:

Calculate the cosine of the angle opposite the larger side;

If cos 0, acute-angled triangle;

If cos 0, rectangular triangle;

If cos 0, the triangle is obtuse.

Question for discussion denia.How can you answer this question without calculating the cosine of the largest angle? I recall the theorem on the relationship between the sides and angles of a triangle. (In a triangle opposite the larger side there is a larger angle and, conversely, the larger side lies opposite the larger angle).

OUTPUT.

Let c be the largest side
- if with 2 < a 2 + b 2, then the triangle is acute-angled;
- if with 2 = a 2 + b 2, then the triangle is rectangular;
- if with 2 > a 2 + b 2, then the triangle is obtuse.

Check the output on completed tasks (at home).

Stage 7. Building a long-term plan for further work.

- teacher's question : Question for discussion... What problems can be solved using the cosine theorem?

- students' answers

find the length of the third side by the known two others and the angle between them;

determine the angle (cosine of the angle) of a triangle by three known sides

determine the shape of a triangle by three known sides

5 stage. Anchoring. Mini - tes

Mini test

Condition

Answer options

In a triangle with sides m , n , p against the side

p lies the angle α ... Then the following is true.

formula:

A) m 2 n 2 p 2 2 np cosα

B) m n 2 p 2 2 np cos α

V) p 2 m 2 n 2 mn cos α ;

G) p m 2 n 2 mn cos α ;

If the cosine of the larger angle of the triangle

is negative, then this triangle:

A) acute-angled; B) rectangular;

V) obtuse.

The lengths of the two sides of the triangle are equal and 3, and the angle

between them 45 0. Then the length of the third side is:

A) 2; B) 3; B) √ 5; G) 5

In a triangle, the lengths of the sides are equal to √3; 4; √7. Determine the type of triangle

A) acute-angled; B) rectangular;

V) obtuse.

Examination.

Answer options

1

V) p 2 m 2 n 2 mn cosα ;

2

V) obtuse.

3

B) √ 5

4

V) obtuse

What else needs to be done to complete the lesson? "

Pupils: "Set homework."

Teacher: "If you were a teacher, what homework would you give?"

Stage 8. Homework. P.98, No. 1025 (d).

I propose to put a final mark in the working cards and carry out reflection on filling out the table.

Discussion filling out the table. Evaluations

Appendices No. 1. Warm-up. Test

"Casting Formulas", "Sine, Cosine and Tangent Values ​​for Angles from 0⁰ to 180⁰"

1. sin (90 ⁰ - α ) =

2. cos (90 ⁰ - α ) =

3. sin (180 ⁰ - α ) = 1.cosα 2.sinα 3. - cosα 4. - sinα

4. cos (180 ⁰ - α ) 1) cosα 2) sinα 3) - cosα 4) - sinα

5. cos 60 ⁰ = 1) 2) 3)

6.cos 30 ⁰ = 1) 2) 3)

Slide 3

In the 10th century. Baghdad scholar Muhammad from Bujan, known as Abu al-Vefa, formulated the theorem of sines. Nasir-ed-Din of Tus (1201-1274) systematically considered all cases of solving oblique spherical triangles and indicated a number of new solutions. In the 12th century. a number of astronomical works were translated from Arabic into Latin, which made it possible for Europeans to familiarize themselves with them. But, unfortunately, much remained untranslated, and the outstanding German astronomer and mathematician Johann Müller (1436-1476), whom his contemporaries knew as Regiomontana (this is how the name of his hometown of Konigsberg is translated into Latin), 200 years after Nasir-ed- Dean rediscovered his theorems. A bit of history

Slide 4

FRANCOIS VIET (1540 - 1603) Viet stood at the origins of the creation of a new science - trigonometry. Many trigonometric formulas were first written by Viet. In 1593, he was the first to formulate the cosine theorem in verbal form. Cosine is a contraction of the Latin expression completelysinus, that is, “complementary sine” (or otherwise “sine of the complementary arc”; cosa = sin (90 ° - a)).

Slide 5

The modern notation of sine and cosine by the signs sinx and cos x was first introduced in 1739 by I. Bernoulli in a letter to the St. Petersburg mathematician L. Euler. Having come to the conclusion that these designations are very convenient, he began to use them in his mathematical works. In addition, Euler introduces the following abbreviations for trigonometric functions of the angle x: tang x, cot x, sec x, cosec x.

Slide 6

State the area theorem for a triangle

The area of ​​a triangle is half the product of its two sides by the sine of the angle between them. Write down the area of ​​the triangle ABC A B C

Slide 7

Sine theorem

The sides of the triangle are proportional to the sines of the opposite angles M F N А В С Write the theorem of sines for the triangle MNF

Slide 8

Write down the sine theorem for triangles: