Open lesson cosine theorem presentation. Cosine theorem presentation for a lesson in geometry (grade 9) on the topic

15.02.2022

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Slides captions:

Cosine theorem

Theorem 12.1 (Cosine Theorem) The square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them.

a 2 \u003d B a A C c b The square of a side of a triangle is equal to the sum of the squares of the other two sides by the cosine of the angle between them. minus twice the product of these sides b 2 + c 2 - 2bc cosA

AB 2 \u003d The square of a side of a triangle is equal to the sum of the squares of the other two sides by the cosine of the angle between them. minus twice the product of these sides BC 2 + CA 2 cos The cosine theorem (∆ABC is rectangular) A C B – 2 BC CA 90 0 C 0 AB 2 = BC 2 + CA 2 The cosine theorem is sometimes called the generalized Pythagorean theorem.

XR 2 = The square of a side of a triangle is equal to the sum of the squares of the other two sides times the cosine of the angle between them. minus twice the product of these sides RO 2 + XO 2 cosO O X R - 2 RO XO RO 2 = RX 2 + XO 2 cosX - 2 RX XO XO 2 = RX 2 + RO 2 cosR - 2 RX RO

F D С Write the cosine theorem for each side of the given triangle.

Corollary of the cosine theorem The square of any side of a triangle is equal to the sum of the squares of the other two sides, twice the product of one of these sides and the projection of the other. The "+" sign is placed when the opposite angle is obtuse, the "̶" sign when it is acute.

A C B H The square of any side of a triangle is equal to the sum of the squares of the other two sides, twice the product of one of these sides and the projection of the other.

In practice, it is convenient to compare the square of the larger side and the sum of the squares of the other two.

Determine the type of triangle with sides 5, 6, 7 cm. > Determine the type of triangle with sides 2, 3, 4 cm. > Oral work

4 4 5 AB 2 = The square of a side of a triangle is the sum of the squares of the other two sides times the cosine of the angle between them. minus twice the product of these sides BC 2 + AC 2 cosC C A B - 2 BC AC 5 AB 2 \u003d 41 - 40 3 2 AB \u003d 41 - 20 3 2 2 5 30 0 30 0 2? 4 Find AB

4 C A B? Find angle B 2 2 3

4 C A B? Find angle B 2 2 3 = 30 0 60 0

6 0 0 5 5 3 3 3 5 V D 2 = AB 2 + AD 2 cos - 2 AB AD V D 2 = 34 - 30 1 2 V D 2 = 19 2 2 V D = 19? А 6 0 0 D A B C AB С D is a parallelogram. Find B D . 6 0 0

Homework 161-162, p. 109; According to workbook No. 93, 95, 96, 98

On the topic: methodological developments, presentations and notes

Lesson - Solving problems in geometry 9 cells. "The area of a triangle. The sine theorem. The cosine theorem."

Problem solving involves the ability to apply knowledge under standard conditions or with small deviations from them. It also considers tasks in which it is required to be able to apply knowledge in a complicated ...

The purpose of the lesson is to study the cosine theorem and its consequences, to develop students' skills in solving problems on this topic.

The lesson establishes a personal contact between the teacher and the students through the formation of the objectives of the lesson, their mutual acceptance and the inclusion of a motive for joint work. Positive motivation reached...

Independent work:

Option 2:

1 option:

Check answers:

Option 2:

1 option:

Cosine theorem:

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

The oldest proof for the sine theorem on the plane is described in the book by Nasir ad-Din At-Tusi "Treatise on the Complete Quadrilateral" written in the 13th century. The sine theorem for a spherical triangle was proven by mathematicians of the medieval East as early as the 10th century. In the work of Al-Jayani of the XI century "The Book of the Unknown Arcs of the Sphere" a general proof of the theorem of sines on the 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 proved 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 hold
Where R is the radius of the circumscribed circle.

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

2) Write down the cosine theorem to calculate the side of the MK:

Find angle B.

Find the length of side BC.

Find the length of side AB.

Find M.N.

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

slide 3

In the 10th century Baghdad scientist Muhammad from Bujan, known as Abu-l-Vefa, formulated the sine theorem. Nasir-ed-Din of Tus (1201-1274) systematically considered all cases of solving oblique spherical triangles and pointed out 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 under the name of Regiomontanus (this is how the name of his hometown of Koenigsberg is translated into Latin), 200 years after Nasir-ed- Dina 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 down by Viet. In 1593, he was the first to verbally formulate the cosine theorem. Cosine is an abbreviation of the Latin expression completelysinus, i.e. “additional sine” (or otherwise “sine of an additional arc”; cosa \u003d sin (90 ° - a)).

slide 5

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

slide 6

Formulate the triangle area theorem

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

Slide 7

Sine theorem

The sides of a triangle are proportional to the sines of the opposite angles M F N A B C Write the sine theorem for triangle MNF

Slide 8

Write down the sine theorem for triangles:

Slide 9

Note The ratio of a side of a triangle to the sine of the opposite angle is equal to the diameter of the circumscribed circle.

Slide 10

Proof: Draw a diameter. Consider, C - rectangular => BC \u003d × sin. If t lies on the BAC arc, then A1 \u003d A, if on the BDC arc, then A1 \u003d 180 ° - A. In both cases sin \u003d sin A \u003d BC \u003d *sin A, BC \u003d 2RsinA or Given: R is the radius of the circumscribed circle, BC = a, is the diameter Prove: (BC=2RsinA)

slide 11

Cosine theorem

The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the angle between them. M F N

Theorems of sines and cosines in problems with practical content

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 double

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 true?

Task 2

sines of opposite angles.

The sides of a triangle are proportional

cosines of opposite angles.

The sides of a triangle are proportional

sines of adjacent angles.

The sides of a triangle are proportional

opposite corners.

Which of the following statements true?

Task 3

area and perimeter.

Solving a triangle means measuring everything

its elements.

Solving a triangle means finding it

unknown elements by three known ones.

To solve a triangle means to find it

equal triangle.

Not true!

Match?

Task 4

A) the sine theorem

B) Heron's formula

C) the Pythagorean theorem

D) cosine theorem

A man 1.7 m tall stands at a distance

8 steps from the pole on which the lantern hangs.

The shadow of a person is equal to four steps. Which

height (in meters) is the lantern located?

Task 5

8 steps

4 steps

Hint (2)

Consider similar triangles

Δ ABC

Δ AKM

The soccer ball is at the Hedgehog, which is located at distances of 23 m and 24 m from the goalposts. The width of the goal is 7 m. Find the angle at which the ball hits the goal?

Task 6

Task 7

Algorithm for solving practical problems

Run drawing
Solve a geometric problem

Task 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
Take an angle measurement
Build a mathematical model (drawing)
Solve a geometric problem using the sine theorem

Using the data shown in the figure, find the width AB of the lake. Give your answer as a whole number of meters.

Decide for yourself 1 option To determine the width of the river (AC), 2 points C and B were marked at a distance of 50m from each other. We measured the angles DAB 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 DAB 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

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Slides captions:

Sine theorem

Theorem 12.2 (sine theorem) The sides of a triangle are proportional to the sines of the opposite angles.

C B A a sinA b sinB = = c sinC a b c The sides of a triangle are proportional to the sines of the opposite angles.

M O X MO sinX MX sinO = = OX sinC The sides of a triangle are proportional to the sines of the opposite angles.

C D E CD sinE EC sinD = = DE sinC The sides of a triangle are proportional to the sines of the opposite angles.

Consequence from the sine theorem where R is the radius of a circle circumscribed about ∆ ABC

Problem Find the radius of a circle circumscribed about ∆ ABC, if AC = 2 cm, ABC = 45° A С В 45 0 2 As a consequence of the sine theorem R = R = 2: (2 ) R =

Trigonometric table No. 1 No. 2 No. 3 No. 4 No. 5

AB sinC AC sinB = C A B 75 0 60 0 60 0 4 4 ? 45 0 45 0 Find AB Task No. 1 Table

AB sinC BC sinA = C A B 60 0 60 0 ? 2 3 3 2 Task No. 2 Table

2 AB sinC AC sinB = C A B ? 2 2 2 2 2 13 5 0 13 5 0 Find the angle A Task No. 3 Table

120 0 AC sinD AD sinC = AB C D is a parallelogram. Find AC. D A B C 30 0 30 0 6 0 0 5 5 ? 120 0 30 0 Task No. 4 Table

45 0 2 45 0 BC sinA AB sinC = AB C D is a parallelogram. Find BC. D A B C 30 0 30 0 2 ? 105 0 30 0 Task No. 5 Table

Homework 162-163, item 110; prove Theorem 12.2; according to workbook No. 99 - 104

On the topic: methodological developments, presentations and notes

An interactive test that contains 5 tasks with the choice of one correct answer out of four proposed, taking into account the time spent on passing the test; The test was created in PowerPoint-2007 with...

Lesson - Solving problems in geometry 9 cells. "The area of a triangle. The sine theorem. The cosine theorem."