Why the most important properties of trigonometric operations are fulfilled. Trigonometry is simple and clear

Sine, cosine, tangent - when pronouncing these words in the presence of high school students, you can be sure that two-thirds of them will lose interest in further conversation. The reason lies in the fact that the basics of trigonometry at school are taught in complete isolation from reality, and therefore students do not see the point in studying formulas and theorems.

In fact, this field of knowledge, upon closer examination, turns out to be very interesting, as well as applied - trigonometry is used in astronomy, construction, physics, music and many other areas.

Let's get acquainted with the basic concepts and name several reasons to study this branch of mathematical science.

Story

It is not known at what point in time humanity began to create future trigonometry from scratch. However, it is documented that already in the second millennium BC, the Egyptians were familiar with the basics of this science: archaeologists found a papyrus with a task in which it is required to find the angle of inclination of the pyramid on two known sides.

Scientists of Ancient Babylon achieved more serious successes. Being engaged in astronomy for centuries, they mastered a number of theorems, introduced special methods of measuring angles, which, by the way, we use today: degrees, minutes and seconds were borrowed by European science in Greco-Roman culture, in which these units came from the Babylonians.

It is assumed that the famous Pythagorean theorem, relating to the basics of trigonometry, was known to the Babylonians almost four thousand years ago.

Name

Literally, the term "trigonometry" can be translated as "measurement of triangles." The main object of study within this section of science for many centuries has been a right-angled triangle, or rather, the relationship between the magnitudes of the angles and the lengths of its sides (today, the study of trigonometry begins from this section from scratch). In life, situations are not uncommon when it is impossible to practically measure all the required parameters of an object (or the distance to the object), and then it becomes necessary to obtain the missing data through calculations.

For example, in the past, a person could not measure the distance to space objects, but attempts to calculate these distances occur long before our era. Trigonometry also played an important role in navigation: with some knowledge, the captain could always navigate by the stars at night and correct the course.

Basic concepts

To master trigonometry from scratch, you need to understand and remember a few basic terms.

The sine of an angle is the ratio of the opposite leg to the hypotenuse. Let us clarify that the opposite leg is the side lying opposite the angle we are considering. Thus, if the angle is 30 degrees, the sine of this angle will always, for any size of the triangle, be equal to ½. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.

Tangent is the ratio of the opposite leg to the adjacent one (or, equivalently, the ratio of sine to cosine). The cotangent is the unit divided by the tangent.

It is worth mentioning the famous number Pi (3.14…), which is half the length of a circle with a radius of one unit.

Popular Mistakes

People who learn trigonometry from scratch make a number of mistakes - mostly due to inattention.

First, when solving problems in geometry, it must be remembered that the use of sines and cosines is possible only in a right triangle. It happens that the student “on the machine” takes the longest side of the triangle as the hypotenuse and receives incorrect calculation results.

Secondly, at first it is easy to confuse the values ​​of sine and cosine for the chosen angle: recall that the sine of 30 degrees is numerically equal to the cosine of 60, and vice versa. If you substitute the wrong number, all further calculations will be wrong.

Thirdly, until the problem is completely solved, it is not worth rounding off any values, extracting roots, writing an ordinary fraction as a decimal. Often, students strive to get a “beautiful” number in a trigonometry problem and immediately extract the root of three, although after exactly one action this root can be reduced.

Etymology of the word "sine"

The history of the word "sine" is truly unusual. The fact is that the literal translation of this word from Latin means "hollow". This is because the correct understanding of the word was lost when translating from one language to another.

The names of the basic trigonometric functions originated from India, where the concept of sine was denoted by the word "string" in Sanskrit - the fact is that the segment, together with the arc of a circle on which it rested, looked like a bow. During the heyday of the Arab civilization, Indian achievements in the field of trigonometry were borrowed, and the term passed into the Arabic language in the form of a transcription. It so happened that this language already had a similar word for a depression, and if the Arabs understood the phonetic difference between a native and a borrowed word, then the Europeans, translating scientific treatises into Latin, by mistake literally translated the Arabic word, which had nothing to do with the concept of sine . We use them to this day.

Tables of values

There are tables that contain numerical values ​​​​for sines, cosines and tangents of all possible angles. Below we present data for angles of 0, 30, 45, 60 and 90 degrees, which must be learned as a mandatory section of trigonometry for "dummies", since it is quite easy to remember them.

If it so happened that the numerical value of the sine or cosine of the angle "flew out of my head", there is a way to derive it yourself.

Geometric representation

Let's draw a circle, draw the abscissa and ordinate axes through its center. The abscissa axis is horizontal, the ordinate axis is vertical. They are usually signed as "X" and "Y" respectively. Now we draw a straight line from the center of the circle in such a way that we get the angle we need between it and the X axis. Finally, from the point where the straight line intersects the circle, we lower the perpendicular to the X axis. The length of the resulting segment will be equal to the numerical value of the sine of our angle.

This method is very relevant if you forgot the desired value, for example, in an exam, and there is no trigonometry textbook at hand. You won’t get the exact figure this way, but you will definitely see the difference between ½ and 1.73 / 2 (sine and cosine of an angle of 30 degrees).

Application

One of the first specialists to use trigonometry were sailors who had no other reference point on the high seas than the sky above their heads. Today, captains of ships (aircraft and other modes of transport) do not look for the shortest path through the stars, but actively resort to the help of GPS navigation, which would be impossible without the use of trigonometry.

In almost every section of physics, you will find calculations using sines and cosines: whether it is the application of force in mechanics, calculations of the path of objects in kinematics, vibrations, wave propagation, light refraction - you simply cannot do without basic trigonometry in formulas.

Another profession that is unthinkable without trigonometry is a surveyor. Using a theodolite and a level, or a more sophisticated device - a tachometer, these people measure the difference in height between different points on the earth's surface.

Repeatability

Trigonometry deals not only with the angles and sides of a triangle, although this is where it began its existence. In all areas where cyclicity is present (biology, medicine, physics, music, etc.), you will encounter a graph whose name you probably know - this is a sinusoid.

Such a graph is a circle unfolded along the time axis and looks like a wave. If you've ever worked with an oscilloscope in a physics class, you know what I'm talking about. Both the music equalizer and the heart rate monitor use trigonometry formulas in their work.

Finally

When thinking about how to learn trigonometry, most middle and high school students begin to consider it a difficult and impractical science, because they get acquainted only with boring textbook information.

As for impracticality, we have already seen that, to one degree or another, the ability to handle sines and tangents is required in almost any field of activity. And as for the complexity ... Think: if people used this knowledge more than two thousand years ago, when an adult had less knowledge than today's high school student, is it really possible for you personally to study this area of ​​\u200b\u200bscience at a basic level? A few hours of thoughtful practice with problem solving - and you will achieve your goal by studying the basic course, the so-called trigonometry for "dummies".

When performing trigonometric transformations, follow these tips:

1. Do not try to immediately come up with a scheme for solving an example from start to finish.
2. Don't try to convert the whole example at once. Move forward in small steps.
3. Remember that in addition to trigonometric formulas in trigonometry, you can still apply all the fair algebraic transformations (bracketing, reducing fractions, abbreviated multiplication formulas, and so on).
4. Believe that everything will be fine.

Basic trigonometric formulas

Most formulas in trigonometry are often applied both from right to left and from left to right, so you need to learn these formulas so well that you can easily apply some formula in both directions. To begin with, we write down the definitions of trigonometric functions. Let there be a right triangle:

Then, the definition of sine is:

Definition of cosine:

Definition of tangent:

Definition of cotangent:

Basic trigonometric identity:

The simplest corollaries from the basic trigonometric identity:

Double angle formulas. Sine of a double angle:

Cosine of a double angle:

Double angle tangent:

Double angle cotangent:

Additional trigonometric formulas

Trigonometric addition formulas. Sine of sum:

Sine of difference:

Cosine of the sum:

Cosine of difference:

Tangent of the sum:

Difference tangent:

Cotangent of the sum:

Difference cotangent:

Trigonometric formulas for converting a sum to a product. The sum of the sines:

Sine Difference:

Sum of cosines:

Cosine difference:

sum of tangents:

Tangent difference:

Sum of cotangents:

Cotangent difference:

Trigonometric formulas for converting a product into a sum. The product of sines:

The product of sine and cosine:

Product of cosines:

Degree reduction formulas.

Half Angle Formulas.

Trigonometric reduction formulas

The cosine function is called cofunction sine function and vice versa. Similarly, the functions tangent and cotangent are cofunctions. The reduction formulas can be formulated as the following rule:

• If in the reduction formula the angle is subtracted (added) from 90 degrees or 270 degrees, then the reducible function changes to a cofunction;
• If in the reduction formula the angle is subtracted (added) from 180 degrees or 360 degrees, then the name of the reduced function is preserved;
• In this case, the reduced function is preceded by the sign that the reduced (i.e., original) function has in the corresponding quarter, if we consider the subtracted (added) angle to be acute.

Cast formulas are given in the form of a table:

By trigonometric circle it is easy to determine tabular values ​​of trigonometric functions:

Trigonometric equations

To solve a certain trigonometric equation, it must be reduced to one of the simplest trigonometric equations, which will be discussed below. For this:

• You can apply the trigonometric formulas above. In this case, you do not need to try to convert the entire example at once, but you need to move forward in small steps.
• We must not forget about the possibility of transforming some expression with the help of algebraic methods, i.e. for example, put something out of the bracket or, conversely, open the brackets, reduce the fraction, apply the abbreviated multiplication formula, reduce fractions to a common denominator, and so on.
• When solving trigonometric equations, you can apply grouping method. It must be remembered that in order for the product of several factors to be equal to zero, it is enough that any of them be equal to zero, and the rest existed.
• Applying variable replacement method, as usual, the equation after the introduction of the replacement should become simpler and not contain the original variable. You also need to remember to do the reverse substitution.
• Remember that homogeneous equations often occur in trigonometry as well.
• When opening modules or solving irrational equations with trigonometric functions, one must remember and take into account all the subtleties of solving the corresponding equations with ordinary functions.
• Remember about the ODZ (in trigonometric equations, the restrictions on the ODZ basically boil down to the fact that you cannot divide by zero, but do not forget about other restrictions, especially about the positivity of expressions in rational powers and under roots of even degrees). Also remember that sine and cosine values ​​can only lie between minus one and plus one, inclusive.

The main thing is, if you don’t know what to do, do at least something, while the main thing is to use trigonometric formulas correctly. If what you get is getting better and better, then continue with the solution, and if it gets worse, then go back to the beginning and try applying other formulas, so do until you stumble upon the correct solution.

Formulas for solving the simplest trigonometric equations. For the sine, there are two equivalent forms of writing the solution:

For other trigonometric functions, the notation is unique. For cosine:

For tangent:

For cotangent:

Solution of trigonometric equations in some special cases:

Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty, solve most of the digital transformation at the right time. After that, you will only have to think about the most difficult tasks.

Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to solve both options. Again, on the DT, in addition to the ability to quickly and efficiently solve problems, and the knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and tasks, or your own surname. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of.

Found an error?

If you, as it seems to you, found an error in the training materials, then please write about it by mail. You can also write about the error on the social network (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.

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 a school mathematics course in didactic and psychological terms is not a system, but a kind of device that encourages short-term memory and does not care at all 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 the origin of the rectangular coordinate system). 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

Hence, the cosine is an 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 is based on the personal experience of action, thinking, feeling of the student;
• the lesson includes various forms of work and ways of obtaining and assimilating knowledge; there are elements of mutual and self-learning; self- and mutual control;
• there is a quick response to misunderstanding and error (joint discussion, support-hints, mutual consultations).

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). Man 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. 4. 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 will be the result of joint actions 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. In this case, the 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 in 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, the 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. 20. Trigonometric Formulas

The double angle formula is just as easy to derive from the sine of the sum by substituting (similarly for the 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.

3. Cable-stayed bridge (Fig. 23). Trigonometric functions help to calculate the number of shrouds, how they should be directed and tensioned.

Rice. 23. Cable-stayed bridge

Rice. 24. "String Bridge"

Rice. 25. Big Obukhovsky bridge

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