cosine ratio. Right triangle

Lesson on the topic "Sine, cosine and tangent acute angle right triangle"

Lesson Objectives:

educational - to introduce the concept of sine, cosine, tangent of an acute angle in a right triangle, to explore the dependencies and relationships between these quantities;

developing - the formation of the concept of sine, cosine, tangent as functions of an angle, the domain of definition of trigonometric functions, development logical thinking, the development of correct mathematical speech;

educational - development of the skill of independent work, culture of behavior, accuracy in keeping records.

Course of the lesson:

1. Organizing time

“Education is not the number of lessons listened to, but the number of understood ones. So, if you want to go forward, then hurry slowly and be careful.

2. Lesson motivation.

One wise man said: “The highest manifestation of the spirit is the mind. The highest manifestation of the mind is geometry. The geometry cell is a triangle. It is as inexhaustible as the universe. The circle is the soul of geometry. Know the circumference, and you will not only know the soul of geometry, but you will elevate your soul.”

Together we will try to do a little research. Let's share any ideas that come to your mind, and don't be afraid to make a mistake, any thought can give us a new direction to search. Let our achievements not seem big to someone, but they will be our own achievements!

3. Actualization of basic knowledge.

What are the angles?

What are triangles?

What are the main elements that define a triangle?

What are triangles based on sides?

What are triangles based on angles?

What is a cathet?

What is a hypotenuse?

What are the sides of a right triangle called?

What are the relationships between the sides and angles of this triangle?

Why do you need to know the relationship between sides and angles?

What tasks from life can lead to the need to calculate unknown sides in a triangle?

The term "hypotenuse" comes from the Greek word "iponeinousa", meaning "stretching over something", "pulling". The word originates from the image of the ancient Greek harps, on which the strings are stretched at the ends of two mutually perpendicular stands. The term "katetos" comes from the Greek word "katetos", which means the beginning of "plumb line", "perpendicular".

Euclid said: "The legs are the sides that make a right angle."

AT Ancient Greece a method for constructing a right-angled triangle on the ground was already known. For this, a rope was used, on which 13 knots were tied, at the same distance from each other. During the construction of the pyramids in Egypt, this is how right triangles were made. This is probably why a right-angled triangle with sides 3,4,5 was called the Egyptian triangle.

4. Learning new material.

In ancient times, people followed the luminaries and, based on these observations, kept a calendar, calculated the sowing dates, the time of the flood of rivers; ships on the sea, caravans on land were guided along the way by the stars. All this led to the need to learn how to calculate the sides in a triangle, two of whose vertices are on the ground, and the third is represented by a point in the starry sky. Based on this need, a science arose - trigonometry - a science that studies the relationships between the sides in a triangle.

What do you think, are the relations already known to us sufficient for solving such problems?

The purpose of today's lesson is to explore new connections and dependencies, to derive relationships, using which in the following geometry lessons, you can solve such problems.

Let's feel ourselves in the role of scientists and, following the geniuses of antiquity Thales, Euclid, Pythagoras, we will follow the path of searching for truth.

For this we need a theoretical basis.

Highlight corner A and leg BC in red.

Highlight in green cathet AS.

Let us calculate what part is the opposite leg for an acute angle A to its hypotenuse, for this we compose the ratio of the opposite leg to the hypotenuse:

This ratio has a special name - such that every person at every point on the planet understands that we are talking about a number representing the ratio of the opposite leg of an acute angle to the hypotenuse. The word is sine. Write it down. Since the word sine without the name of the angle loses all meaning, the mathematical notation is as follows:

Now make the ratio of the adjacent leg to the hypotenuse for acute angle A:

This ratio is called cosine. His mathematical notation:

Consider one more relation for an acute angle A: the ratio of the opposite leg to the adjacent leg:

This ratio is called tangent. His mathematical notation:

5. Consolidation of new material.

Let's consolidate our intermediate discoveries.

Sinus is...

Cosine is...

Tangent is...

Solve verbally No. 88, 889, 892 (work in pairs).

Using the acquired knowledge to solve a practical problem:

“From the tower of the lighthouse, 70 m high, a ship is visible at an angle of 3 to the horizon. What is

distance from the lighthouse to the ship?

The task is solved frontally. During the discussion, we make a drawing and the necessary notes on the board and in notebooks.

When solving the problem, Bradis tables are used.

Consider the solution of the problem p.175.

Solve #902(1).

6. Fizminutka for the eyes.

Without turning your head, look around the classroom wall clockwise around the perimeter, the chalkboard around the perimeter counterclockwise, the triangle depicted on the stand clockwise and its equal triangle counterclockwise. Turn your head to the left and look at the horizon line, and now at the tip of your nose. Close your eyes, count to 5, open your eyes and...

We put our hands to our eyes,
Let's set our legs strong.
Turning to the right
Let's look majestic.
And to the left too
Look from under the palms.
And - to the right! And further
Over the left shoulder!
and now we will continue to work.

7. Independent work students.

Solve no.

8. The results of the lesson. Reflection. D/z.

What did you learn new? On the lesson:

have you considered...

did you analyze...

You received …

you concluded...

you have replenished your vocabulary with the following terms ...

World science began with geometry. A person cannot truly develop culturally and spiritually if he has not studied geometry at school. Geometry arose not only from the practical, but also the spiritual needs of man.

This is how she poetically explained her love for geometry

I love geometry...

I study geometry because I love

Geometry is needed, without it we are nowhere.

Sine, cosine, circle - everything is important here,

Everything is needed here

You just have to be very clear and understand everything.

Complete assignments and checklists on time.

Where the tasks for solving a right-angled triangle were considered, I promised to present a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which leg belongs to the hypotenuse (adjacent or opposite). I decided not to put it off indefinitely, necessary material below, please see

The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember very well that the leg refers to the hypotenuse, but which one- forget and confused. The price of a mistake, as you know in the exam, is a lost score.

The information that I will present directly to mathematics has nothing to do. It is associated with figurative thinking, and with the methods of verbal-logical connection. That's right, I myself, once and for all remembereddefinition data. If you still forget them, then with the help of the presented techniques it is always easy to remember.

Let me remind you the definitions of sine and cosine in a right triangle:

Cosine acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:

Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:

So, what associations does the word cosine evoke in you?

Probably everyone has their ownRemember the link:

Thus, you will immediately have an expression in your memory -

«… ratio of ADJACENT leg to hypotenuse».

The problem with the definition of cosine is solved.

If you need to remember the definition of the sine in a right triangle, then remembering the definition of the cosine, you can easily establish that the sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse. After all, there are only two legs, if the adjacent leg is “occupied” by the cosine, then only the opposite side remains for the sine.

What about tangent and cotangent? Same confusion. Students know that this is the ratio of legs, but the problem is to remember which one refers to which - either opposite to adjacent, or vice versa.

Definitions:

Tangent an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one:

Cotangent acute angle in a right triangle is the ratio of the adjacent leg to the opposite:

How to remember? There are two ways. One also uses a verbal-logical connection, the other - a mathematical one.

MATHEMATICAL METHOD

There is such a definition - the tangent of an acute angle is the ratio of the sine of an angle to its cosine:

* Remembering the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one.

Likewise.The cotangent of an acute angle is the ratio of the cosine of an angle to its sine:

So! Remembering these formulas, you can always determine that:

- the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent

- the cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite one.

VERBAL-LOGICAL METHOD

About tangent. Remember the link:

That is, if you need to remember the definition of the tangent, using this logical connection, you can easily remember what it is

"... the ratio of the opposite leg to the adjacent"

If it comes to cotangent, then remembering the definition of tangent, you can easily voice the definition of cotangent -

"... the ratio of the adjacent leg to the opposite"

There is an interesting technique for memorizing tangent and cotangent on the site " Mathematical tandem " , look.

METHOD UNIVERSAL

You can just grind.But as practice shows, thanks to verbal-logical connections, a person remembers information for a long time, and not only mathematical.

I hope the material was useful to you.

Sincerely, Alexander Krutitskikh

P.S: I would be grateful if you tell about the site in social networks.

Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were derived by astronomers to create accurate calendar and orientation by the stars. These calculations were related to spherical trigonometry, while in school course study the ratio of the sides and angle of a flat triangle.

Trigonometry is a branch of mathematics dealing with the properties of trigonometric functions and the relationship between the sides and angles of triangles.

During the heyday of culture and science of the 1st millennium AD, knowledge spread from ancient east to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazvi introduced such functions as tangent and cotangent, compiled the first tables of values ​​for sines, tangents and cotangents. The concept of sine and cosine was introduced by Indian scientists. A lot of attention is devoted to trigonometry in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

The basic trigonometric functions of a numerical argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values ​​of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the wording: “ Pythagorean pants, are equal in all directions, since the proof is given on the example of an isosceles right triangle.

Sine, cosine and other dependencies establish a relationship between acute angles and sides of any right triangle. We give formulas for calculating these quantities for angle A and trace the relationship of trigonometric functions:

As you can see, tg and ctg are inverse functions. If we represent leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, then we get the following formulas for tangent and cotangent:

trigonometric circle

Graphically, the ratio of the mentioned quantities can be represented as follows:

circle, in this case, represents all possible values ​​of the angle α — from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will be with a “+” sign if α belongs to the I and II quarters of the circle, that is, it is in the range from 0 ° to 180 °. With α from 180° to 360° (III and IV quarters), sin α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the meaning of the quantities.

The values ​​of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values ​​of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen by chance. The designation π in the tables is for radians. Rad is the angle at which the length of a circular arc corresponds to its radius. This value was introduced in order to establish a universal relationship; when calculating in radians, the actual length of the radius in cm does not matter.

The angles in the tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a full circle or 360°.

Properties of trigonometric functions: sine and cosine

In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.

Consider comparison table properties for sinusoid and cosine wave:

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs are the same, the function is even; otherwise, it is odd.

Introduction to radians and enumeration basic properties sinusoids and cosine waves allow us to bring the following regularity:

It is very easy to verify the correctness of the formula. For example, for x = π/2, the sine is equal to 1, as is the cosine of x = 0. Checking can be done by looking at tables or by tracing function curves for given values.

Properties of tangentoid and cotangentoid

The graphs of the tangent and cotangent functions differ significantly from the sinusoid and cosine wave. The values ​​tg and ctg are inverse to each other.

1. Y = tgx.
2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
3. The smallest positive period of the tangentoid is π.
4. Tg (- x) \u003d - tg x, i.e., the function is odd.
5. Tg x = 0, for x = πk.
6. The function is increasing.
7. Tg x › 0, for x ϵ (πk, π/2 + πk).
8. Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
9. Derivative (tg x)' = 1/cos 2 ⁡x .

Consider the graphical representation of the cotangentoid below in the text.

The main properties of the cotangentoid:

1. Y = ctgx.
2. Unlike the sine and cosine functions, in the tangentoid Y can take on the values ​​of the set of all real numbers.
3. The cotangentoid tends to the values ​​of y at x = πk, but never reaches them.
4. The smallest positive period of the cotangentoid is π.
5. Ctg (- x) \u003d - ctg x, i.e., the function is odd.
6. Ctg x = 0, for x = π/2 + πk.
7. The function is decreasing.
8. Ctg x › 0, for x ϵ (πk, π/2 + πk).
9. Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
10. Derivative (ctg x)' = - 1/sin 2 ⁡x Fix

The sine is one of the basic trigonometric functions, the application of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and the calculation of the sine is sometimes necessary to solve various tasks. In general, the calculation of the sine will help to consolidate drawing skills and knowledge of trigonometric identities.

Ruler and pencil games

A simple task: how to find the sine of an angle drawn on paper? To solve, you need a regular ruler, a triangle (or a compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, first you need to complete the acute angle to the figure of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. It will be necessary to observe an angle of exactly 90 °, for which we need a clerical triangle.

Using a compass is a bit more precise, but will take longer. On one of the rays, you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until these lines intersect. By connecting the points of intersection of our circles with each other, we will get a strict perpendicular to the ray of our angle, it remains only to extend the line until it intersects with another ray.

In the resulting triangle, you need to measure the side opposite the corner and the long side on one of the rays with a ruler. The ratio of the first measurement to the second will be the desired value of the sine of the acute angle.

Find the sine for an angle greater than 90°

For obtuse angle the task is not much more difficult. Draw a ray from a vertex to opposite side using a ruler to form a straight line with one of the rays of the angle of interest to us. With the resulting acute angle, you should proceed as described above, the sines of adjacent angles, forming together a developed angle of 180 °, are equal.

Calculating the sine from other trigonometric functions

Also, the calculation of the sine is possible if the values ​​​​of other trigonometric functions of the angle or at least the length of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.

How to find the sine with a known cosine of an angle? The first trigonometric identity, coming from the Pythagorean theorem, says that the sum of the squares of the sine and cosine of the same angle is equal to one.

How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far leg by the near one or by dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the squared sine according to the first trigonometric identity and by simple manipulations we bring the equation to the calculation of the square sine through the tangent, respectively, to calculate the sine, you will have to extract the root from the result obtained.

How to find the sine with a known cotangent of an angle? The cotangent value can be calculated by dividing the length of the near leg from the leg angle by the length of the far one, as well as dividing the cosine by the sine, that is, the cotangent is the inverse function of the tangent with respect to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α \u003d 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with the tangent, which will look like this.

How to find the sine of the three sides of a triangle

There is a formula for finding the length of the unknown side of any triangle, not just a right triangle, given two known sides using the trigonometric function of the cosine of the opposite angle. She looks like this.

Initially, sine and cosine arose due to the need to calculate quantities in right triangles. It was noticed that if the value of the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how much these sides change in length, always remains the same.

This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse, and the cosine is the ratio of the adjacent leg to the hypotenuse.

Theorems of cosines and sines

But cosines and sines can be used not only in right triangles. To find the value of an obtuse or acute angle, the side of any triangle, it is enough to apply the cosine and sine theorem.

The cosine theorem is quite simple: "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 these sides by the cosine of the angle between them."

There are two interpretations of the sine theorem: small and extended. According to the small: "In a triangle, the angles are proportional to the opposite sides." This theorem is often extended due to the property of the circumcircle of a triangle: "In a triangle, the angles are proportional to opposite sides, and their ratio is equal to the diameter of the circumscribed circle."

Derivatives

A derivative is a mathematical tool that shows how quickly a function changes with respect to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.

When solving problems, you need to know the tabular values ​​\u200b\u200bof the derivatives of trigonometric functions: sine and cosine. The derivative of the sine is the cosine, and the derivative of the cosine is the sine, but with a minus sign.

Application in mathematics

Especially often, sines and cosines are used in solving right triangles and problems related to them.

The convenience of sines and cosines is also reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking complex shapes and objects into "simple" triangles. Engineers and, often dealing with calculations of aspect ratios and degree measures, spent a lot of time and effort calculating cosines and sines of non-table angles.

Then Bradis tables came to the rescue, containing thousands of values ​​​​of sines, cosines, tangents and cotangents of different angles. AT Soviet time some teachers forced their wards to memorize the pages of the Bradys tables.

Radian - the angular value of the arc, along the length equal to the radius or 57.295779513 ° degrees.

Degree (in geometry) - 1/360th part of a circle or 1/90th part right angle.

π = 3.141592653589793238462… (approximate value of pi).

Cosine table for angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°.