Trigonometric functions for dummies. Trigonometry is simple and clear. Trigonometric reduction formulas

As early as 1905, Russian readers could read in William James' Psychology, his reasoning about "why is cramming such a bad way of learning?"

“Knowledge acquired through mere cramming is almost inevitably forgotten completely without a trace. On the contrary, mental material, accumulated by memory gradually, day after day, in connection with various contexts, associated associatively with other external events and repeatedly subjected to discussion, forms such a system, enters into such a connection with other aspects of our intellect, is easily renewed in memory by a mass of external reasons that remain a long-term solid acquisition.

More than 100 years have passed since then, and these words amazingly remain topical. You see this every day when you work with schoolchildren. The mass gaps in knowledge are so great that it can be argued that the school mathematics course in didactic and psychological terms is not a system, but a kind of device that encourages short term memory and not at all care about long-term memory.

To know the school course of mathematics means to master the material of each of the areas of mathematics, to be able to update any of them at any time. To achieve this, you need to systematically address each of them, which is sometimes not always possible due to the heavy workload in the lesson.

There is another way of long-term memorization of facts and formulas - these are reference signals.

Trigonometry is one of the large sections of school mathematics studied in the course of geometry in grades 8, 9 and in the course of algebra in grade 9, algebra and the beginning of analysis in grade 10.

The largest amount of material studied in trigonometry falls on grade 10. Much of this trigonometry material can be learned and memorized on trigonometric circle(circle of unit radius centered at origin rectangular system coordinates). Application1.ppt

These are the following concepts of trigonometry:

definitions of sine, cosine, tangent and cotangent of an angle;
radian measurement of angles;
domain of definition and range of trigonometric functions
values of trigonometric functions for some values of numerical and angular argument;
periodicity of trigonometric functions;
even and odd trigonometric functions;
increase and decrease of trigonometric functions;
reduction formulas;
values of inverse trigonometric functions;
solution of the simplest trigonometric equations;
solution of the simplest inequalities;
basic formulas of trigonometry.

Consider the study of these concepts on a trigonometric circle.

1) Definition of sine, cosine, tangent and cotangent.

After introducing the concept of a trigonometric circle (a circle of unit radius centered at the origin), an initial radius (radius of a circle in the direction of the Ox axis), an angle of rotation, students independently receive definitions for sine, cosine, tangent and cotangent on a trigonometric circle, using definitions from the course geometry, that is, considering a right triangle with hypotenuse equal to 1.

The cosine of an angle is the abscissa of a point on a circle when the initial radius is rotated by a given angle.

The sine of an angle is the ordinate of a point on a circle when the initial radius is rotated by a given angle.

2) Radian measurement of angles on a trigonometric circle.

After introducing the radian measure of an angle (1 radian is the central angle, which corresponds to an arc length equal to the radius of the circle), students conclude that the radian angle measurement is the numerical value of the angle of rotation on the circle, equal to the length of the corresponding arc when the initial radius is rotated by given angle. .

The trigonometric circle is divided into 12 equal parts by the diameters of the circle. Knowing that an angle is a radian, one can determine the radian measurement for angles that are multiples of .

And radian measurements of angles that are multiples are obtained similarly:

3) Domain of definition and domain of values of trigonometric functions.

Will the correspondence of rotation angles and coordinate values of a point on a circle be a function?

Each angle of rotation corresponds to a single point on the circle, so this correspondence is a function.

Getting functions

It can be seen on the trigonometric circle that the domain of definition of functions is the set of all real numbers, and the domain of values is .

Let us introduce the concepts of lines of tangents and cotangents on a trigonometric circle.

1) Let We introduce an auxiliary straight line parallel to the Oy axis, on which the tangents are determined for any numerical argument.

2) Similarly, we obtain a line of cotangents. Let y=1, then . This means that the values of the cotangent are determined on a straight line parallel to the Ox axis.

On a trigonometric circle, one can easily determine the domain of definition and the range of values of trigonometric functions:

for tangent -

for cotangent -

4) Values of trigonometric functions on a trigonometric circle.

The leg opposite the angle at half the hypotenuse, that is, the other leg according to the Pythagorean theorem:

So by definition of sine, cosine, tangent, cotangent, you can determine values for angles that are multiples or radians. The sine values are determined along the Oy axis, the cosine values along the Ox axis, and the tangent and cotangent values can be determined from additional axes parallel to the Oy and Ox axes, respectively.

The tabular values of sine and cosine are located on the respective axes as follows:

Tabular values of tangent and cotangent -

5) Periodicity of trigonometric functions.

On the trigonometric circle, it can be seen that the values of the sine, cosine are repeated every radian, and the tangent and cotangent - every radian.

6) Even and odd trigonometric functions.

This property can be obtained by comparing the values of positive and opposite rotation angles of trigonometric functions. We get that

So the cosine is even function, all other functions are odd.

7) Increasing and decreasing trigonometric functions.

The trigonometric circle shows that the sine function increases and decreases

Arguing similarly, we obtain the intervals of increase and decrease of the cosine, tangent and cotangent functions.

8) Reduction formulas.

For the angle we take the smaller value of the angle on the trigonometric circle. All formulas are obtained by comparing the values of trigonometric functions on the legs of selected right triangles.

Algorithm for applying reduction formulas:

1) Determine the sign of the function when rotating through a given angle.

When turning a corner the function is preserved, when turning by an angle - an integer, an odd number, a cofunction is obtained (

9) Values of inverse trigonometric functions.

We introduce inverse functions for trigonometric functions using the definition of a function.

Each value of sine, cosine, tangent and cotangent on a trigonometric circle corresponds to only one value of the angle of rotation. So, for a function, the domain of definition is , the domain of values is - For the function, the domain of definition is , the domain of values is . Similarly, we obtain the domain of definition and the range of inverse functions for cosine and cotangent.

Algorithm for finding the values of inverse trigonometric functions:

1) finding on the corresponding axis the value of the argument of the inverse trigonometric function;

2) finding the angle of rotation of the initial radius, taking into account the range of values of the inverse trigonometric function.

For example:

10) Solution of the simplest equations on a trigonometric circle.

To solve an equation of the form , we find points on a circle whose ordinates are equal and write down the corresponding angles, taking into account the period of the function.

For the equation, we find points on the circle whose abscissas are equal and write down the corresponding angles, taking into account the period of the function.

Similarly for equations of the form The values are determined on the lines of tangents and cotangents and the corresponding angles of rotation are recorded.

All the concepts and formulas of trigonometry are received by the students themselves under the clear guidance of the teacher with the help of a trigonometric circle. In the future, this “circle” will serve as a reference signal for them or an external factor for reproducing in memory the concepts and formulas of trigonometry.

The study of trigonometry on a trigonometric circle contributes to:

choosing the style of communication that is optimal for this lesson, organizing educational cooperation;
lesson targets become personally significant for each student;
new material based on personal experience actions, thinking, feelings of the student;
lesson includes various forms work and methods of obtaining and assimilating knowledge; there are elements of mutual and self-learning; self- and mutual control;
occurs fast reaction on misunderstanding and error (joint discussion, support-hints, mutual consultations).

Back forward

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1. Introduction.

Approaching the school, I hear the voices of the guys from the gym, I go further - they sing, draw ... emotions, feelings are everywhere. My office, algebra lesson, tenth graders. Here is our textbook, in which the trigonometry course is half of its volume, and there are two bookmarks in it - these are the places where I found words that are not related to the theory of trigonometry.

Among the few are students who love mathematics, feel its beauty and do not ask why it is necessary to study trigonometry, where is the studied material applied? The majority are those who simply complete tasks so as not to get a bad grade. And we are firmly convinced that the applied value of mathematics is to gain knowledge sufficient for successful passing the exam and admission to the university (to enter and forget).

The main purpose of the presented lesson is to show the applied value of trigonometry in various fields human activity. The examples given will help students to see the connection of this section of mathematics with other subjects studied at school. The content of this lesson is an element of student training.

Tell something new about a seemingly long-known fact. Show a logical connection between what we already know and what remains to be studied. Open the door a little and look beyond school curriculum. Unusual tasks, connection with the events of today - these are the techniques that I use to achieve my goals. After all, school mathematics as a subject contributes not so much to learning as to the development of the individual, his thinking, culture.

2. Summary of the lesson on algebra and the beginnings of analysis (Grade 10).

Organizing time: Arrange six tables in a semicircle (protractor model), worksheets for students on the tables (Attachment 1).

Announcement of the topic of the lesson: "Trigonometry is simple and clear."

In the course of algebra and the beginning of analysis, we begin to study trigonometry, I would like to talk about the applied significance of this branch of mathematics.

Thesis of the lesson:

“great book nature can only be read by those who know the language in which it is written, and that language is mathematics.”
(G. Galileo).

At the end of the lesson, we will think together whether we were able to look into this book and understand the language in which it is written.

Trigonometry of an acute angle.

Trigonometry is a Greek word and means “measurement of triangles”. The emergence of trigonometry is associated with measurements on the ground, construction, and astronomy. And the first acquaintance with her happened when you picked up a protractor. Did you pay attention to how the tables stand? Estimate in your mind: if you take one table for a chord, then what is the degree measure of the arc that it pulls together?

Recall the measure of angles: 1 ° = 1/360 part of the circle (“degree” - from the Latin grad - step). Do you know why the circle was divided into 360 parts, why not divided into 10, 100 or 1000 parts, as happens, for example, when measuring lengths? I will tell you one of the versions.

Previously, people believed that the Earth is the center of the Universe and it is motionless, and the Sun makes one revolution around the Earth per day, the geocentric system of the world, “geo” - the Earth ( Drawing No. 1). Babylonian priests, who made astronomical observations, discovered that on the day of the equinox, from sunrise to sunset, the Sun describes a semicircle in the firmament, in which the apparent diameter (diameter) of the Sun fits exactly 180 times, 1 ° - trace of the sun. ( Figure No. 2).

For a long time, trigonometry was purely geometric in nature. In you continue your acquaintance with trigonometry by solving right triangles. You learn that the sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse, the cosine is the ratio of the adjacent leg to the hypotenuse, the tangent is the ratio of the opposite leg to the adjacent leg, and the cotangent is the ratio of the adjacent leg to the opposite. And remember that in right triangle, which has a given angle, the ratio of the sides does not depend on the size of the triangle. Get acquainted with the sine and cosine theorems for solving arbitrary triangles.

In 2010, the Moscow Metro celebrated its 75th anniversary. Every day we go down to the subway and do not notice that ...

Task number 1. The angle of inclination of all escalators in the Moscow metro is 30 degrees. Knowing this, the number of lamps on the escalator and the approximate distance between the lamps, you can calculate the approximate depth of the station. There are 15 lamps on the escalator of the Tsvetnoy Bulvar station, and 2 lamps on the Prazhskaya station. Calculate the depth of these stations if the distances between the lamps, from the entrance of the escalator to the first lamp and from the last lamp to the exit from the escalator are 6 m ( Drawing No. 3). Answer: 48 m and 9 m

Homework. The deepest station of the Moscow metro is Park Pobedy. What is its depth? I suggest that you independently find the missing data to solve your homework problem.

I have a laser pointer in my hands, it is also a rangefinder. Let's measure, for example, the distance to the board.

Chinese designer Huan Qiaokong guessed to combine two laser rangefinders, a protractor into one device and got a tool that allows you to determine the distance between two points on a plane ( Drawing No. 4). How do you think, with the help of which theorem this problem is solved? Recall the formulation of the cosine theorem. Do you agree with me that your knowledge is already sufficient to make such an invention? Solve problems in geometry and make small discoveries every day!

Spherical trigonometry.

In addition to the plane geometry of Euclid (planimetry), there may be other geometries in which the properties of figures are considered not on the plane, but on other surfaces, for example, on the surface of a ball ( Drawing No. 5). The first mathematician who laid the foundation for the development of non-Euclidean geometries was N.I. Lobachevsky - "Copernicus of Geometry". From 1827, for 19 years, he was the rector of the Kazan University.

Spherical trigonometry, which is part of spherical geometry, considers the relationships between the sides and angles of triangles on a sphere formed by arcs of great circles on a sphere ( Drawing No. 6).

Historically, spherical trigonometry and geometry arose from the needs of astronomy, geodesy, navigation, and cartography. Consider which of these directions last years has received such rapid development that its result is already used in modern communicators. ... A modern application of navigation is a satellite navigation system that allows you to determine the location and speed of an object from the signal of its receiver.

Global Navigation System (GPS). To determine the latitude and longitude of the receiver, it is necessary to receive signals from at least three satellites. Reception of a signal from the fourth satellite also makes it possible to determine the height of the object above the surface ( Drawing No. 7).

The receiver computer solves four equations in four unknowns until a solution is found that draws all circles through one point ( Drawing No. 8).

Knowledge from the trigonometry of an acute angle turned out to be insufficient for solving more complex practical problems. When studying rotational and circular motions, the value of the angle and circular arc are not limited. There was a necessity of transition to trigonometry of the generalized argument.

Trigonometry of the generalized argument.

The circle ( Drawing No. 9). Positive angles are plotted counterclockwise, negative angles are plotted clockwise. Are you familiar with the history of such an agreement?

As you know, mechanical and sundials are designed in such a way that their hands rotate “according to the sun”, i.e. in the same direction in which we see the apparent movement of the Sun around the Earth. (Remember the beginning of the lesson - the geocentric system of the world). But with the discovery by Copernicus of the true (positive) movement of the Earth around the Sun, the apparent (ie apparent) movement of the Sun around the Earth is fictitious (negative). Heliocentric system of the world (helio - Sun) ( Drawing No. 10).

Warm up.

Pull out right hand in front of you, parallel to the surface of the table and perform a circular rotation of 720 degrees.
Pull out left hand in front of you, parallel to the surface of the table and perform a circular turn by (-1080) degrees.
Place your hands on your shoulders and do 4 circular motions back and forth. What is the sum of the angles of rotation?

In 2010 the Winter Olympic Games in Vancouver, we will find out the criteria for grading a skater's exercise by solving the problem.

Task number 2. If a skater makes a 10,800-degree turn while performing the screw exercise in 12 seconds, then he gets an “excellent” mark. Determine how many revolutions the skater will make during this time and the speed of his rotation (revolutions per second). Answer: 2.5 revolutions / sec.

Homework. At what angle does a skater turn, who received an “unsatisfactory” rating, if, with the same rotation time, his speed was 2 revolutions per second.

The most convenient measure of arcs and angles associated with rotational movements turned out to be radian (radius) measure, as a larger unit of measurement of angle or arc ( Drawing No. 11). This measure of angle measurement entered science through the remarkable works of Leonhard Euler. Swiss by birth, he lived in Russia for 30 years, was a member of the St. Petersburg Academy of Sciences. It is to him that we owe the “analytical” interpretation of all trigonometry, he derived the formulas that you are now studying, introduced uniform signs:. sin x, cos x, tg x.ctg x.

If until the 17th century the development of the doctrine of trigonometric functions was built on a geometric basis, then, starting from the 17th century, trigonometric functions began to be applied to solving problems of mechanics, optics, electricity, to describe oscillatory processes, wave propagation. Wherever one has to deal with periodic processes and oscillations, trigonometric functions have found application. Functions expressing the laws of periodic processes have a special property inherent only to them: they repeat their values through the same interval of change of the argument. Changes of any function are most clearly transmitted on its graph ( Drawing No. 12).

We have already turned to our body for help in solving rotation problems. Let's listen to the beating of our heart. The heart is an independent organ. The brain controls every muscle in our body except the heart. She has her own control center - the sinus node. With each contraction of the heart throughout the body - starting from the sinus node (the size of a millet grain) - spreads electricity. It can be recorded using an electrocardiograph. It draws an electrocardiogram (sinusoid) ( Drawing No. 13).

Now let's talk about music. Mathematics is music, it is the union of mind and beauty.
Music is mathematics by calculation, algebra by abstraction, trigonometry by beauty. harmonic oscillation(harmonic) is a sine wave. The graph shows how the air pressure on the listener's eardrum changes: up and down in an arc, periodically. The air pushes harder, then weaker. The impact force is quite small and the oscillations occur very quickly: hundreds and thousands of shocks every second. We perceive such periodic vibrations as sound. Adding two different harmonics produces a more complex waveform. The sum of three harmonics is even more complicated, and natural sounds and the sounds of musical instruments are made up of a large number of harmonics. ( Drawing No. 14.)

Each harmonic is characterized by three parameters: amplitude, frequency and phase. The oscillation frequency indicates how many shocks of air pressure occur in one second. Large frequencies are perceived as "high", "thin" sounds. Above 10 kHz - squeak, whistle. Small frequencies are perceived as "low", "bass" sounds, rumble. Amplitude is the range of oscillation. The larger the span, the stronger the impact on the eardrum, and the louder sound which we hear Drawing No. 15). Phase is the displacement of oscillations in time. Phase can be measured in degrees or radians. Depending on the phase, the zero count is shifted on the graph. To specify the harmonic, it is enough to specify the phase from -180 to +180 degrees, since the oscillation repeats at large values. Two sinusoidal signals with the same amplitude and frequency but different phases are added algebraically ( Drawing No. 16).

Summary of the lesson. Do you think we were able to read a few pages from the Great Book of Nature? Having learned about the applied meaning of trigonometry, did you understand its role in various fields of human activity, did you understand the material presented? Then remember and list the areas of application of trigonometry that you met today or knew before. I hope that each of you found something new and interesting for yourself in today's lesson. Perhaps this new one will show you the way to choose future profession, but no matter who you become, your mathematical education will help you become a professional in your field and an intellectually developed person.

Homework. Read the lesson outline

Once at school, a separate course was allocated for the study of trigonometry. The certificate was given grades in three mathematical disciplines: algebra, geometry and trigonometry.

Then, as part of the reform school education trigonometry ceased to exist as a separate subject. AT modern school the first acquaintance with trigonometry occurs in the geometry course of the 8th grade. A deeper study of the subject continues in the 10th grade algebra course.

The definitions of sine, cosine, tangent and cotangent are first given in geometry through the relationship of the sides of a right triangle.

The acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse.

cosine acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.

tangent 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 called the ratio of the adjacent leg to the opposite.

These definitions apply only to acute angles (from 0º to 90°).

For example,

in triangle ABC, where ∠C=90°, BC is the leg opposite to angle A, AC is the leg adjacent to angle A, AB is the hypotenuse.

In the 10th grade algebra course, the definitions of sine, cosine, tangent and cotangent for any angle (including negative ones) are introduced.

Consider a circle of radius R centered at the origin, the point O(0;0). The intersection point of the circle with the positive direction of the x-axis will be denoted by P 0 .

In geometry, an angle is considered as a part of a plane bounded by two rays. With this definition, the value of the angle varies from 0° to 180°.

In trigonometry, the angle is considered as the result of the rotation of the ray OP 0 around the starting point O.

At the same time, they agreed to consider the rotation of the beam counterclockwise as the positive direction of the bypass, and clockwise as negative (this agreement is associated with the true movement of the Sun around the Earth).

For example, when the beam OP 0 rotates around the point O at an angle α counterclockwise, the point P 0 will go to the point P α ,

when turning through the angle α clockwise - to the point F.

With this definition, the angle can take any value.

If we continue to rotate the beam OP 0 counterclockwise, when turning through the angle α°+360°, α°+360° 2,…,α°+360° n, where n is an integer (n∈Ζ), again we get to the point P α:

Angles are measured in degrees and radians.

1° is an angle equal to 1/180 of the degree measure of a straight angle.

1 radian is a central angle whose arc length is equal to the radius of the circle:

∠AOB=1 rad.

The radian notation is usually not written. The designation of the degree in the record must not be omitted.

For example,

The point P α , obtained from the point P 0 by turning the beam OP 0 around the point O at an angle α counterclockwise, has the coordinates P α (x;y).

Let us drop the perpendicular P α A from the point P α to the x-axis.

In a right triangle OP α A:

P α A is the leg opposite the angle α,

OA is the leg adjacent to the angle α,

OP α is the hypotenuse.

P α A=y, OA=x, OP α =R.

By definition of sine, cosine, tangent and cotangent in a right triangle we have:

Thus, in the case of a circle centered at the origin of arbitrary radius sinus angle α is the ratio of the ordinate of the point P α to the length of the radius.

cosine angle α is the ratio of the abscissa of the point P α to the length of the radius.

tangent angle α is the ratio of the ordinate of the point P α to its abscissa.

Cotangent angle α is the ratio of the abscissa of the point P α to its ordinate.

The values of sine, cosine, tangent and cotangent depend only on the value of α and do not depend on the length of the radius R (this follows from the similarity of circles).

Therefore, it is convenient to choose R=1.

A circle centered at the origin and radius R=1 is called a unit circle.

Definitions

1) sinus the angle α is the ordinate of the point P α (x; y) of the unit circle:

2) cosine the angle α is called the abscissa of the point P α (x; y) of the unit circle:

3) tangent angle α is the ratio of the ordinate of the point P α (x; y) to its abscissa, that is, the ratio of sin α to cos α (where cos α≠ 0):

4) Cotangent angle α is the ratio of the abscissa of the point P α (x; y) to its ordinate, that is, the ratio of cosα to sinα (where sinα≠0):

The definitions introduced in this way allow us to consider not only the trigonometric functions of angles, but also the trigonometric functions of numerical arguments (if we consider sinα, cosα, tgα and ctgα as the corresponding trigonometric functions of an angle in α radians, that is, the sine of the number α is the sine of the angle in α radians, the cosine of α is the cosine of the angle in α radians, etc.).

The properties of trigonometric functions are studied in the course of algebra in the 10th or 11th grade as a separate topic. Trigonometric functions widely used in physics.

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In this lesson, we will talk about how the need arises for the introduction of trigonometric functions and why they are studied, what you need to understand in this topic, and where you just need to fill your hand (which is a technique). Note that technique and understanding are two different things. Agree, there is a difference: to learn to ride a bike, that is, to understand how to do it, or to become a professional cyclist. We will talk about understanding, about why we need trigonometric functions.

There are four trigonometric functions, but they can all be expressed in terms of one using identities (equalities that connect them).

Formal definitions of trigonometric functions for acute angles in right triangles (Fig. 1).

sinus The acute angle of a right triangle is called the ratio of the opposite leg to the hypotenuse.

cosine The acute angle of a right triangle is called the ratio of the adjacent leg to the hypotenuse.

tangent The acute angle of a right triangle is called the ratio of the opposite leg to the adjacent leg.

Cotangent The acute angle of a right triangle is called the ratio of the adjacent leg to the opposite leg.

Rice. 1. Definition of trigonometric functions of an acute angle of a right triangle

These definitions are formal. It is more correct to say that there is only one function, for example, sine. If they were not so needed (not so often used) in technology, so many different trigonometric functions would not be introduced.

For example, the cosine of an angle is equal to the sine of the same angle with the addition of (). In addition, the cosine of an angle can always be expressed in terms of the sine of the same angle, up to a sign, using the basic trigonometric identity(). The tangent of an angle is the ratio of sine to cosine or inverted cotangent (Fig. 2). Some do not use the cotangent at all, replacing it with . Therefore, it is important to understand and be able to work with one trigonometric function.

Rice. 2. Connection of various trigonometric functions

But why do you need such functions at all? What practical problems are they used for? Let's look at a few examples.

Two people ( BUT and AT) push the car out of the puddle (Fig. 3). Human AT can push the car sideways, while it is unlikely to help BUT. On the other hand, the direction of his efforts may gradually shift (Fig. 4).

Rice. 3. AT pushes the car to the side

Rice. four. AT begins to change direction

It is clear that their efforts will be most effective when they push the car in one direction (Fig. 5).

Rice. 5. The most effective joint direction of efforts

How much AT helps pushing the machine, as far as the direction of its force is close to the direction of the force with which it acts BUT, is a function of the angle and is expressed in terms of its cosine (Fig. 6).

Rice. 6. Cosine as a characteristic of the effectiveness of efforts AT

If we multiply the magnitude of the force with which AT, on the cosine of the angle, we get the projection of its force on the direction of the force with which it acts BUT. The closer the angle between the directions of forces to , the more effective the result will be. joint action BUT and AT(Fig. 7). If they push the car with the same force in opposite directions, the car will stay in place (Fig. 8).

Rice. 7. The effectiveness of joint efforts BUT and AT

Rice. 8. Opposite direction of forces BUT and AT

It is important to understand why we can replace the angle (its contribution to the final result) with the cosine (or other trigonometric function of the angle). In fact, this follows from such a property of similar triangles. Since in fact we are saying the following: the angle can be replaced by the ratio of two numbers (leg-hypotenuse or leg-leg). This would be impossible if, for example, for the same angle of different right-angled triangles, these ratios would be different (Fig. 9).

Rice. 9. Equal ratios of sides in similar triangles

For example, if the ratio and the ratio were different, then we would not be able to introduce the tangent function, since for the same angle in different right triangles the tangent would be different. But due to the fact that the ratios of the lengths of the legs of similar right-angled triangles are the same, the value of the function will not depend on the triangle, which means that the acute angle and the values of its trigonometric functions are one-to-one.

Suppose we know the height of a certain tree (Fig. 10). How to measure the height of a nearby building?

Rice. 10. Illustration of the condition of example 2

We find a point such that the line drawn through this point and the top of the house will pass through the top of the tree (Fig. 11).

Rice. 11. Illustration of the solution of the problem of example 2

We can measure the distance from this point to the tree, the distance from it to the house, and we know the height of the tree. From the proportion you can find the height of the house:.

Proportion is the ratio of two numbers. AT this case equality of the ratio of the lengths of the legs of similar right triangles. Moreover, these ratios are equal to some measure of the angle, which is expressed in terms of a trigonometric function (by definition, this is a tangent). We get that for each acute angle the value of its trigonometric function is unique. That is, sine, cosine, tangent, cotangent are really functions, since each acute angle corresponds to exactly one value of each of them. Therefore, they can be further explored and their properties can be used. The values of the trigonometric functions for all angles have already been calculated, they can be used (they can be found from the Bradis tables or using any engineering calculator). But to solve the inverse problem (for example, by the value of the sine to restore the measure of the angle that corresponds to it), we can not always.

Let the sine of some angle be equal to or approximately (Fig. 12). What angle will correspond to this value of the sine? Of course, we can again use the Bradis table and find some value, but it turns out that it will not be the only one (Fig. 13).

Rice. 12. Finding an angle by the value of its sine

Rice. 13. Polyvalence of inverse trigonometric functions

Therefore, when restoring the value of the trigonometric function of the angle, there is a polysemy of inverse trigonometric functions. It may seem complicated, but in fact we face similar situations every day.

If you curtain the windows and do not know whether it is light or dark outside, or if you find yourself in a cave, then, upon waking up, it is difficult to say whether it is now the hour of the day, night, or the next day (Fig. 14). In fact, if you ask us "What time is it?", we should honestly answer: "Hour plus multiply by where"

Rice. 14. Illustration of polysemy on the example of a clock

We can conclude that - this is the period (the interval after which the clock will show the same time as now). Trigonometric functions also have periods: sine, cosine, etc. That is, their values are repeated after some change in the argument.

If the planet did not have a change of day and night or a change of seasons, then we could not use periodic time. After all, we only number the years in ascending order, and there are hours in the day, and every new day the count starts anew. The situation is the same with months: if it is January now, then in months January will come again, and so on. External reference points help us to use the periodic counting of time (hours, months), for example, the rotation of the Earth around its axis and the change in the position of the Sun and Moon in the sky. If the Sun always hung in the same position, then to calculate the time we would count the number of seconds (minutes) since the occurrence of this very calculation. Date and time could then sound like this: a billion seconds.

Conclusion: there are no difficulties in terms of the ambiguity of inverse functions. Indeed, there may be options when for the same sine there are different angle values (Fig. 15).

Rice. 15. Restoration of an angle by the value of its sine

Usually, when solving practical problems, we always work in the standard range from to . In this range, for each value of the trigonometric function, there are only two corresponding values of the measure of the angle.

Consider a moving belt and a pendulum in the form of a bucket with a hole from which sand falls out. The pendulum swings, the tape moves (Fig. 16). As a result, the sand will leave a trace in the form of a graph of the sine (or cosine) function, which is called a sine wave.

In fact, the graphs of the sine and cosine differ from each other only in the reference point (if you draw one of them and then erase the coordinate axes, then you won’t be able to determine which graph was drawn). Therefore, it makes no sense to call the cosine graph (why come up with a separate name for the same graph)?

Rice. 16. Illustration of the problem statement in example 4

From the graph of the function, you can also understand why the inverse functions will have many values. If the value of the sine is fixed, i.e. draw a straight line parallel to the x-axis, then at the intersection we get all the points at which the sine of the angle is equal to the given one. It is clear that there will be infinitely many such points. As in the example with the clock, where the time value differed by , only here the angle value will differ by an amount (Fig. 17).

Rice. 17. Illustration of polysemy for sine

If we consider the clock example, then the point (the end of the hour hand) moves around the circle. In the same way, trigonometric functions can be defined - consider not the angles in a right triangle, but the angle between the radius of the circle and the positive direction of the axis. The number of circles that the point will pass (we agreed to count the movement clockwise with a minus sign, and counter-clockwise with a plus sign), this is the period (Fig. 18).

Rice. 18. The value of the sine on the circle

So, inverse function is uniquely defined on some interval. For this interval, we can calculate its values, and get all the rest from the found values by adding and subtracting the period of the function.

Consider another example of a period. The car is moving along the road. Imagine that her wheel drove into the paint or into a puddle. You can see occasional paint marks or puddles on the road (Figure 19).

Rice. 19. Period illustration

There are a lot of trigonometric formulas in the school course, but by and large it is enough to remember just one (Fig. 20).

Rice. twenty. Trigonometric formulas

Formula double angle it is also easy to derive sums from the sine by substituting (similarly for cosine). You can also derive product formulas.

In fact, you need to remember very little, since with the solution of problems these formulas will be remembered by themselves. Of course, someone will be too lazy to decide a lot, but then he will not need this technique, and hence the formulas themselves.

And since the formulas are not needed, then there is no need to memorize them. You just need to understand the idea that trigonometric functions are functions with which, for example, bridges are calculated. Almost no mechanism can do without their use and calculation.

1. The question often arises as to whether wires can be absolutely parallel to ground. Answer: no, they cannot, since one force acts downward, while the others act in parallel - they will never balance (Fig. 21).

2. Swan, crayfish and pike pull the cart in the same plane. The swan flies in one direction, the crayfish pulls in the other, and the pike in the third (Fig. 22). Their powers can balance. You can calculate this balancing just with the help of trigonometric functions.