In this lesson, we will learn the definitions trigonometric functions and their main properties, learn how to work with trigonometric circle, find out what is function period and remember the various ways to measure angles. In addition, let's look at using reduction formulas.
This lesson will help you prepare for one of the types of assignments. AT 7.
Preparation for the exam in mathematics
Experiment
Lesson 7Introduction to trigonometry.
Theory
Lesson summary
Today we are starting a section that has a frightening name for many, “Trigonometry”. Let's immediately find out that this is not a separate object, similar in name to geometry, as some people think. Although translated from Greek word"trigonometry" means "measurement of triangles" and is directly related to geometry. In addition, trigonometric calculations are widely used in physics and technology. But we will start with you precisely by considering how the basic trigonometric functions are introduced in geometry using a right triangle.
We have just used the term "trigonometric function" - this means that we will introduce whole class certain laws of correspondence to one variable from another.
For this, consider right triangle, which for convenience uses the standard designations of sides and corners, which you can see in the figure:
Consider, for example, the angleand enter the following actions for it:
The ratio of the opposite leg to the hypotenuse is called the sine, i.e.
The ratio of the adjacent leg to the hypotenuse is called the cosine, i.e. ;
The ratio of the opposite leg to the adjacent leg is called the tangent, i.e. ;
The ratio of the adjacent leg to the opposite leg will be called the cotangent, i.e. .
All these actions with an angle are called trigonometric functions. The angle itself, at the same time, is usually called argument of the trigonometric function and it can be denoted, for example, by x, as is customary in algebra.
It is important to understand right away that trigonometric functions depend on the angle in a right triangle, and not on its sides. This is easy to prove if we consider a triangle similar to this one, in which the lengths of the sides will be different, and all angles and ratios of the sides will not change, i.e. the trigonometric functions of the angles will also remain unchanged.
After such a definition of trigonometric functions, the question may arise: “Is there, for example,? After all, the cornercannot be in a right triangle» . Oddly enough, the answer to this question is yes, and the value of this expression is , which is even more surprising, since all trigonometric functions are the ratio of the sides of a right triangle, and the lengths of the sides are positive numbers.
But there is no paradox in this. The fact is that, for example, in physics, when describing some processes, it is necessary to use the trigonometric functions of angles not only large, but also large and even. To do this, it is necessary to introduce a more generalized rule for calculating trigonometric functions using the so-called "unit trigonometric circle".
It is a circle with unit radius drawn so that its center is at the origin of the Cartesian plane.
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To depict the angles in this circle, it is necessary to agree on where to put them. It is accepted for the angle reference beam to take the positive direction of the abscissa axis, i.e. x-axis. The direction of deposition of corners is considered to be the direction counterclockwise. Based on these agreements, we first set aside an acute angle. It is for such acute angles that we already know how to calculate the values of trigonometric functions in a right triangle. It turns out that with the help of the depicted circle it is also possible to calculate trigonometric functions, only more conveniently.
Sine and cosine values acute angle are the coordinates of the point of intersection of the side of this angle with the unit circle:
This can be written in this form:
:
Based on the fact that the coordinates on the abscissa show the value of the cosine, and the coordinates on the ordinate show the values of the sine of the angle, it is convenient to rename the names of the axes in the coordinate system with a unit circle as you see in the figure:
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The abscissa axis is renamed to the cosine axis, and the ordinate axis to the sine axis.
The indicated rule for determining the sine and cosine is generalized both to obtuse angles and to angles ranging from to. In this case, the sines and cosines can take both positive and negative values. Various the signs of the values of these trigonometric functions depending on which quarter the angle under consideration falls into, it is customary to depict it as follows:
As you can see, the signs of trigonometric functions are determined by the positive and negative directions of their respective axes.
In addition, it is worth paying attention to the fact that since the largest coordinate of a point on a unit circle and along the abscissa and along the ordinate axis is equal to one, and the smallest minus one, then sine and cosine values limited to these numbers:
These records are usually written in this form:
In order to introduce the functions of tangent and cotangent on a trigonometric circle, it is necessary to depict additional elements: the tangent to the circle at point A - the value of the tangent of the angle is determined from it, and the tangent to at point B - the value of the cotangent of the angle is determined from it.
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However, we will not delve into the definition of tangents and cotangents along a trigonometric circle, because. they can be easily calculated, knowing the values of the sine and cosine of a given angle, which we already know how to do. If you are interested in learning how to calculate tangent and cotangent in a trigonometric circle, repeat the program of the 10th grade algebra course.
Specify only the image on the circle signs of tangents and cotangents depending on the angle:
Note that similarly to the ranges of sine and cosine values, you can specify ranges of tangent and cotangent values. Based on their definition on a trigonometric circle, the values of these functions are not limited:
What else can be written like this:
In addition to angles in the range from to, the trigonometric circle allows you to work with angles that are larger and even with negative angles. Such angle values, although they seem meaningless for geometry, are used to describe some physical processes. For example, how would you answer the question: What angle will the clock hand turn in a day? During this time, it will complete two complete revolutions, and in one revolution it will pass, i.e. in a day will turn to . As you can see, such values have quite practical meaning. Angle signs are used to indicate the direction of rotation - one of the directions is agreed to be measured by positive angles, and the other by negative ones. How can this be taken into account in a trigonometric circle?
On a circle with such angles, they work as follows:
1) Angles that are greater than , are plotted counterclockwise with the passage of the reference point as many times as necessary. For example, to build an angle, you need to go through two full turns and more. For the final position and all trigonometric functions are calculated. It is easy to see that the value of all trigonometric functions for and for will be the same.
2) Negative angles are plotted exactly according to the same principle as positive ones, only clockwise.
Already by the method of constructing large angles, it can be concluded that the values of the sines and cosines of angles that differ by are the same. If we analyze the values of tangents and cotangents, then they will be the same for angles that differ by .
Such minimal non-zero numbers, when added to the argument, the value of the function does not change, is called period this function.
In this way, periodsine and cosine is, and the tangent and cotangent. And this means that no matter how much you add or subtract these periods from the angles under consideration, the values of trigonometric functions will not change.
For example, , and etc.
Later we will return to a more detailed explanation and application of this property of trigonometric functions.
There are certain relations between the trigonometric functions of the same argument, which are very often used and are called basic trigonometric identities.
They look like this:
1)
, the so-called "trigonometric unit"
3)
4)
5)
Note that, for example, the notation means that the entire trigonometric function is squared. Those. it can be represented in this form: 
. It is important to understand that this is not equal to such a notation as , in this case only the argument is squared, and not the entire function, moreover, expressions of this kind are extremely rare.
There are two very useful corollaries to the first identity that can be useful in solving many types of problems. After simple transformations, you can express the sine through the cosine of the same angle and vice versa:
The two possible signs of the expressions appear because extracting arithmetic square root gives only non-negative values, and the sine and cosine, as we have already seen, can have negative values. Moreover, the signs of these functions are most conveniently determined precisely with the help of a trigonometric circle, depending on which angles are present in them.
Now let's remember that the measurement of angles can be done in two ways: in degrees and in radians. Let us indicate the definitions of one degree and one radian.
one degree- this is the angle formed by two radii that subtend an arc equal to a circle.
One radian- this is the angle formed by two radii, which are contracted by an arc equal in length to the radii.
Those. they are just two different ways to measure angles that are absolutely equal. In the description of physical processes that are characterized by trigonometric functions, it is customary to use the radian measure of angles, so we will also have to get used to it.
It is customary to measure angles in radians in fractions of the number "pi", for example, or. In this case, the value of the number "pi", which is 3.14, can be substituted, but this is rarely done.
To convert the degree measure of angles to radians take advantage of the fact that the angle from which is easy to obtain general formula translation:
For example, let's convert to radians: 
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There is also an opposite formulaconversion from radians to degrees:
For example, let's convert to degrees: 
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We will use the radian measure of the angle in this topic quite often.
Now is the time to remember what specific values trigonometric functions of various angles can give. For some angles that are multiples of , there is table of values of trigonometric functions. For convenience, the angles are given in degree and radian measures.
These angles are often encountered in many problems, and it is desirable to be able to navigate confidently in this table. The values of the tangent and cotangent of some angles do not make sense, which is indicated in the table as dashes. Think for yourself why this is so, or read it in more detail in the insert to the lesson.
The last thing we need to be familiar with in our first trigonometry lesson is transformation of trigonometric functions according to the so-called reduction formulas.
It turns out that there is a certain kind of expression for trigonometric functions, which is quite common and conveniently simplified. For example, these are such expressions: etc.
Those. we will talk about functions that have an arbitrary angle as an argument, changed to a whole or half part. Such functions are simplified to an argument that is equal to an arbitrary angle of adding or subtracting parts. For example, 
, a 
. As we can see, the opposite function can become the result, and the function can change sign.
Therefore, the rules for transforming such functions can be divided into two stages. First, it is necessary to determine what function will be obtained after the transformation:
1) If an arbitrary argument is changed to an integer, then the function does not change. This is true for functions of type where any integer;
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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 activities. 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 a right triangle with 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 receives 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 the skater, who received the mark “unsatisfactory”, turn 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 deduced 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 used to solve problems in 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 more clearly, 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
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Usually, when they want to scare someone with TERRIBLE MATH, all sorts of sines and cosines are cited as an example, as something very complex and nasty. But in fact, this is a beautiful and interesting section that can be understood and solved.
The topic begins to take place in the 9th grade and everything is not always clear the first time, there are many subtleties and tricks. I tried to say something on the topic.
Introduction to the world of trigonometry:
Before throwing headlong into formulas, you need to understand from geometry what sine, cosine, etc. are.
Sine of an angle- the ratio of the opposite (angle) side to the hypotenuse.
Cosine is the ratio of the adjacent to the hypotenuse.
Tangent- opposite side in adjacent side
Cotangent- adjacent to the opposite.
Now consider a circle of unit radius on coordinate plane and mark some alpha angle on it: (pictures are clickable, at least some of them)
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Thin red lines are the perpendicular from the point of intersection of the circle and the right angle on the x and y axes. The red x and y are the value of the x and y coordinates on the axes (the gray x and y are just to indicate that these are coordinate axes and not just lines).
It should be noted that the angles are counted from the positive direction of the x-axis counterclockwise.
We find for it the sine, cosine, and so on.
sin a: opposite side is y, hypotenuse is 1.
sin a = y / 1 = y
To make it completely clear where I get y and 1 from, for clarity, let's arrange the letters and consider triangles.
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AF = AE = 1 - radius of the circle.
Therefore, AB = 1, as a radius. AB is the hypotenuse.
BD = CA = y - as value for oh.
AD \u003d CB \u003d x - as a value for oh.
sin a = BD / AB = y / 1 = y
Further cosine:
cos a: adjacent side - AD = x
cos a = AD / AB = x / 1 = x
We also deduce tangent and cotangent.
tg a = y / x = sin a / cos a
ctg a = x / y = cos a / sin a
Already suddenly we have derived the formula of tangent and cotangent.
Well, let's take a look at how it is solved with specific angles.
For example, a = 45 degrees.
We get a right triangle with one angle of 45 degrees. It is immediately clear to someone that this is a triangle with different sides, but I will sign it anyway.
Find the third corner of the triangle (first 90, second 5): b = 180 - 90 - 45 = 45
If two angles are equal, then the sides are equal, as it sounded like.
So, it turns out as if, if we add two such triangles on top of each other, we get a square with a diagonal equal to radius \u003d 1. By the Pythagorean theorem, we know that the diagonal of a square with side a is equal to the roots of two.
Now we think. If 1 (the hypotenuse aka the diagonal) is equal to the side of the square multiplied by the root of two, then the side of the square must equal 1/sqrt(2), and if we multiply the numerator and denominator of this fraction by the root of two, we get sqrt(2)/2 . And since the triangle is isosceles, then AD = AC => x = y
Finding our trigonometric functions:
sin 45 = sqrt(2)/2 / 1 = sqrt(2)/2
cos 45 = sqrt(2)/2 / 1 = sqrt(2)/2
tg 45 = sqrt(2)/2 / sqrt(2)/2 = 1
ctg 45 = sqrt(2)/2 / sqrt(2)/2 = 1
With the rest of the angles, you need to work in the same way. Only the triangles will not be isosceles, but the sides are just as easy to find using the Pythagorean theorem.
In this way, we get a table of values of trigonometric functions from different angles:
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Moreover, this table is cheating and very convenient.
How to make it yourself without any hassle: you draw such a table and write the numbers 1 2 3 in the cells.
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Now from these 1 2 3 you extract the root and divide by 2. It turns out like this:
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Now we cross out the sine and write the cosine. Its values are the mirrored sine:
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It is just as easy to derive the tangent - you need to divide the value of the sine line by the value of the cosine line:
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The value of the cotangent is the inverted value of the tangent. As a result, we get something like this:
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note that the tangent does not exist in P/2, for example. Think why. (You can't divide by zero.)
What to remember here: sine is the y value, cosine is the x value. The tangent is the ratio of y to x, and the cotangent is the other way around. so, in order to determine the values of sines / cosines, it is enough to draw a plate, which I described above and a circle with coordinate axes (it is convenient to look at the values \u200b\u200bat angles 0, 90, 180, 360).
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Well, I hope you can tell quarters:
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The sign of its sine, cosine, etc. depends on which quarter the angle is in. Although, absolutely primitive logical thinking will lead you to the correct answer, if you take into account that in the second and third quarters x is negative, and y is negative in the third and fourth. Nothing terrible or frightening.
I think it would not be superfluous to mention reduction formulas ala ghosts, as everyone hears, which has a grain of truth. There are no formulas as such, for uselessness. The very meaning of all this action: We easily find the values of the angles only for the first quarter (30 degrees, 45, 60). Trigonometric functions are periodic, so we can drag any large angle to the first quadrant. Then we will immediately find its meaning. But just dragging is not enough - you need to remember about the sign. That's what casting formulas are for.
So, we have a large angle, or rather more than 90 degrees: a \u003d 120. And you need to find its sine and cosine. To do this, we decompose 120 into such angles that we can work with:
sin a = sin 120 = sin (90 + 30)
We see that this angle lies in the second quarter, the sine is positive there, therefore the + sign in front of the sine is preserved.
To get rid of 90 degrees, we change the sine to cosine. Well, here's a rule to remember:
sin (90 + 30) = cos 30 = sqrt(3) / 2
And you can imagine it in another way:
sin 120 = sin (180 - 60)
To get rid of 180 degrees, we do not change the function.
sin (180 - 60) = sin 60 = sqrt(3) / 2
We got the same value, so everything is correct. Now cosine:
cos 120 = cos (90 + 30)
The cosine in the second quarter is negative, so we put a minus sign. And we change the function to the opposite, since we need to remove 90 degrees.
cos (90 + 30) = - sin 30 = - 1 / 2
Or:
cos 120 = cos (180 - 60) = - cos 60 = - 1 / 2
What you need to know, be able to do and do in order to translate corners in the first quarter:
-decompose the angle into digestible terms;
- take into account in which quarter the angle is located, and put the appropriate sign if the function in this quarter is negative or positive;
-get rid of excess
*if you need to get rid of 90, 270, 450 and the rest 90+180n, where n is any integer, then the function is reversed (sine to cosine, tangent to cotangent and vice versa);
*if you need to get rid of 180 and the remaining 180+180n, where n is any integer, then the function does not change. (There is one feature here, but it is difficult to explain it in words, well, okay).
That's all. I do not consider it necessary to memorize the formulas themselves, when you can remember a couple of rules and use them easily. By the way, these formulas are very easy to prove:
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And they make up bulky tables, then we know:
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Basic trigonometry equations: they need to be known very, very well, by heart.
Basic trigonometric identity(equality):
sin^2(a) + cos^2(a) = 1
If you don't believe me, check it out yourself and see for yourself. Substitute the values of the different angles.
This formula is very, very useful, always remember it. with it, you can express the sine through the cosine and vice versa, which is sometimes very useful. But, like with any other formula, you need to be able to handle it. Always remember that the sign of the trigonometric function depends on the quarter in which the angle is located. That's why when extracting the root, you need to know a quarter.
Tangent and cotangent: we have already derived these formulas at the very beginning.
tg a = sin a / cos a
ctg a = cos a / sin a
Product of tangent and cotangent:
tg a * ctg a = 1
Because:
tg a * ctg a = (sin a / cos a) * (cos a / sin a) = 1 - fractions cancel.
As you can see, all formulas are a game and a combination.
Here are two more, obtained from dividing by the cosine square and sine square of the first formula:
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Please note that the last two formulas can be used with a restriction on the value of the angle a, since you cannot divide by zero.
Addition formulas: are proved using vector algebra.
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They are used rarely, but aptly. There are formulas on the scan, but it may be illegible or the digital form is easier to perceive:
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Double angle formulas:
They are obtained based on addition formulas, for example: the cosine of a double angle is cos 2a = cos (a + a) - does it remind you of anything? They just replaced beta with alpha.
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The two following formulas are derived from the first substitution sin^2(a) = 1 - cos^2(a) and cos^2(a) = 1 - sin^2(a).
With the sine of a double angle, it is simpler and is used much more often:
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And special perverts can derive the tangent and cotangent of a double angle, given that tg a \u003d sin a / cos a, and so on.
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For the above persons Triple angle formulas: they are derived by adding the angles 2a and a, since we already know the formulas for the double angle.
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Half angle formulas:
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I don’t know how they are derived, or rather how to explain it ... If you write these formulas, substituting the basic trigonometric identity with a / 2, then the answer will converge.
Formulas for adding and subtracting trigonometric functions:
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They are obtained from addition formulas, but no one cares. Meet not often.
As you understand, there are still a lot of formulas, the enumeration of which is simply meaningless, because I won’t be able to write something adequate about them, and dry formulas can be found anywhere, and they are a game with the previous existing formulas. Everything is terribly logical and accurate. I'll just tell you last about the auxiliary angle method:
Converting the expression a cosx + b sinx to the form Acos(x+) or Asin(x+) is called the method of introducing an auxiliary angle (or additional argument). The method is used in solving trigonometric equations, in estimating the values of functions, in extremum problems, and what is important to note, some problems cannot be solved without introducing an auxiliary angle.
As you, I did not try to explain this method, nothing came of it, so you have to do it yourself:
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It's scary, but useful. If you solve problems, it should work.
From here for example: mschool.kubsu.ru/cdo/shabitur/kniga/trigonom/metod/metod2/met2/met2.htm
Next on the course are graphs of trigonometric functions. But one lesson is enough. Considering that this is taught at school for six months.
Write your questions, solve problems, ask for scans of some tasks, figure it out, try it.
Always yours, Dan Faraday.
When performing trigonometric transformations, follow these tips:
- Do not try to immediately come up with a scheme for solving an example from start to finish.
- Don't try to convert the whole example at once. Move forward in small steps.
- 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).
- 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:
- Can be applied 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:
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.
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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.
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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 values 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).
