The video course “Get an A” includes all the topics necessary for successful passing the Unified State Exam in mathematics for 60-65 points. Completely all problems 1-13 Profile Unified State Examination in mathematics. Also suitable for passing the Basic Unified State Examination in mathematics. If you want to pass the Unified State Exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!
Preparation course for the Unified State Exam for grades 10-11, as well as for teachers. Everything you need to solve Part 1 of the Unified State Exam in mathematics (the first 12 problems) and Problem 13 (trigonometry). And this is more than 70 points on the Unified State Exam, and neither a 100-point student nor a humanities student can do without them.
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Sine, cosine, tangent - when you pronounce 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, upon closer examination, this area of knowledge turns out to be very interesting, as well as applied - trigonometry is used in astronomy, construction, physics, music and many other fields.
Let's get acquainted with the basic concepts and name several reasons to study this branch of mathematical science.
Story
It is unknown at what point in time humanity began to create the 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 was necessary to find the angle of inclination of the pyramid on two known sides.
The scientists of Ancient Babylon achieved more serious successes. Over the centuries, studying astronomy, they mastered a number of theorems, introduced special ways measurements of angles, which, by the way, we use today: degrees, minutes and seconds were borrowed European science in Greco-Roman culture, into 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 was the right triangle, or more precisely, the relationship between the magnitudes of the angles and the lengths of its sides (today, the study of trigonometry from scratch begins with this section). There are often situations in life when it is practically impossible to 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, people could not measure the distance to space objects, but attempts to calculate these distances occurred long before the advent of our era. The most important role Trigonometry also played a role in navigation: with some knowledge, the captain could always navigate by the stars at night and adjust the course.
Basic Concepts
Mastering trigonometry from scratch requires understanding and remembering several basic terms.
The sine of a certain angle is the ratio of the opposite side to the hypotenuse. Let us clarify that the opposite leg is the side lying opposite the angle we are considering. Thus, if an 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 side to the adjacent side (or, which is the same, the ratio of sine to cosine). 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 learning trigonometry from scratch make a number of mistakes - mostly due to inattention.
Firstly, when solving problems in geometry, it is necessary to remember that the use of sines and cosines is possible only in right triangle. It happens that a student “automatically” takes the longest side of a triangle as the hypotenuse and gets incorrect calculation results.
Secondly, at first it is easy to confuse the values of sine and cosine for the selected angle: recall that the sine of 30 degrees is numerically equal to cosine 60, and vice versa. If you substitute an incorrect number, all further calculations will be incorrect.
Thirdly, until the problem is completely solved, you should not round any values, extract roots, write common 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 during translation from one language to another.
Basic names 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 the circle on which it rested, looked like a bow. During the heyday of Arab civilization, Indian achievements in the field of trigonometry were borrowed, and the term passed into Arabic in the form of a transcription. It so happened that in this language there was already similar word, denoting 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, mistakenly translated literally Arabic word, which has nothing to do with the concept of sine. We still use it 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”; fortunately, they are quite easy to remember.
If it happens that the numerical value of the sine or cosine of an angle “got out of your head,” there is a way to derive it yourself.
Geometric representation
Let's draw a circle and 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 will draw a straight line from the center of the circle so that the angle we need is obtained between it and the X axis. Finally, from the point where the straight line intersects the circle, we drop a 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 have forgotten desired value, for example, during an exam, and there is no textbook on trigonometry at hand. You won’t get an exact number 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
Some of the first experts to use trigonometry were sailors who had no other reference point on the open sea except the sky above their heads. Today, captains of ships (airplanes and other modes of transport) do not look for the shortest path using the stars, but actively resort to 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: be it the application of force in mechanics, calculations of the path of objects in kinematics, vibrations, wave propagation, refraction of light - you simply cannot do without basic trigonometry in the formulas.
Another profession that is unthinkable without trigonometry is a surveyor. Using a theodolite and a level or a more complex device - a tachometer, these people measure the difference in height between various 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 is probably familiar to you - this is a sine wave.
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 physics class, you know what we're talking about. Both the music equalizer and the heart rate monitor use trigonometry formulas in their work.
In conclusion
When thinking about how to learn trigonometry, most secondary and high school they begin to consider it a complex and impractical science, since they only get acquainted with boring information from a textbook.
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. 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 realistic for you personally to study this field of science at a basic level? A few hours of thoughtful practice solving problems - and you will achieve your goal by studying basic course, so-called trigonometry for dummies.
Back in 1905, Russian readers could read in William James’s book “Psychology” his reasoning about “why is rote learning such a bad way of learning?”
“Knowledge acquired through simple rote learning is almost inevitably forgotten completely without a trace. On the contrary, mental material, acquired 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 the other aspects of our intellect, is easily restored in memory by a mass of external occasions, which remains a durable acquisition for a long time.”
More than 100 years have passed since then, and these words remain amazingly topical. You become convinced of this every day when working with schoolchildren. The massive gaps in knowledge are so great that it can be argued: 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 care at all about long-term memory.
Know school course Mathematics means mastering the material of each of the areas of mathematics, being able to update any of them at any time. To achieve this, you need to systematically contact 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 elementary analysis in grade 10.
The largest volume of material studied in trigonometry falls on the 10th grade. Most of this trigonometry material can be learned and memorized on trigonometric circle(a circle of unit radius with its center at the origin of the rectangular coordinate system). Appendix1.ppt
This the following concepts trigonometry:
- definitions of sine, cosine, tangent and cotangent of an angle;
- radian angle measurement;
- domain of definition and range of values of trigonometric functions
- values of trigonometric functions for some values of the numerical and angular argument;
- periodicity of trigonometric functions;
- evenness and oddity of trigonometric functions;
- increasing and decreasing trigonometric functions;
- reduction formulas;
- values of inverse trigonometric functions;
- solving simple trigonometric equations;
- solving simple inequalities;
- basic formulas of trigonometry.
Let's consider studying these concepts on the trigonometric circle.
1) Definition of sine, cosine, tangent and cotangent.
After introducing the concept of a trigonometric circle (a circle of unit radius with a center at the origin), the initial radius (the radius of the circle in the direction of the Ox axis), and the angle of rotation, students independently obtain definitions for sine, cosine, tangent and cotangent on a trigonometric circle, using the definitions from the course geometry, that is, considering a right triangle with a 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 angle (1 radian is central angle, which corresponds to an arc length equal to the length of the radius of the circle), students conclude that the radian measurement of an angle is the numerical value of the angle of rotation on a circle, equal to the length of the corresponding arc when rotating the initial radius by a given angle. .
The trigonometric circle is divided into 12 equal parts by the diameters of the circle. Knowing that the angle is in radians, you can determine the radian measurement for angles that are multiples of .
And radian measurements of angles, multiples, are obtained similarly:
3) Domain of definition and range of values of trigonometric functions.
Will the correspondence between 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, which means this correspondence is a function.
Getting the functions
On the trigonometric circle you can see that the domain of definition of functions is the set of all real numbers, and the range of values is .
Let us introduce the concepts of lines of tangents and cotangents on a trigonometric circle.
1) Let 
Let us introduce an auxiliary line parallel to the Oy axis, on which tangents are determined for any numerical argument.
2) Similarly, we obtain a line of cotangents. Let y=1, then . This means that the cotangent values are determined on a straight line parallel to the Ox axis.
On a trigonometric circle you can easily determine the domain of definition and range of values of trigonometric functions:
for tangent -
for cotangent -
4) Values of trigonometric functions on a trigonometric circle.
The leg opposite the angle in is equal to half the hypotenuse, that is, the other leg according to the Pythagorean theorem:
This means that by defining 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 along the Ox axis, and the tangent and cotangent values can be determined using additional axes parallel to the Oy and Ox axes, respectively.
The tabulated values of sine and cosine are located on the corresponding axes as follows: 
Table values of tangent and cotangent - 
5) Periodicity of trigonometric functions.
On the trigonometric circle you can see that the values of sine and cosine are repeated every radian, and tangent and cotangent - every radian.
6) Evenness and oddness of trigonometric functions.
This property can be obtained by comparing the values of positive and opposite angles of rotation of trigonometric functions. We get that
So, cosine - even function, all other functions are odd.
 |
7) Increasing and decreasing trigonometric functions.
The trigonometric circle shows that the sine function increases 
and decreases 
Reasoning similarly, we obtain the intervals of increasing and decreasing functions of cosine, tangent and cotangent.
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 rotated by an angle - an integer, odd number, the cofunction (
9) Values of inverse trigonometric functions.
Let us introduce inverse functions for trigonometric functions using the definition of a function.
Each value of sine, cosine, tangent and cotangent on the trigonometric circle corresponds to only one value of the angle of rotation. This means that for a function the domain of definition is , the range of values is - For the function the domain of definition is , the range of values is . Similarly, we obtain the domain of definition and range of values inverse functions for cosine and cotangent.
Algorithm for finding the values of inverse trigonometric functions:
1) finding the value of the argument of the inverse trigonometric function on the corresponding axis;
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) Solving simple equations on a trigonometric circle.
To solve an equation of the form , we find points on the 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 concepts and formulas of trigonometry are learned by the students themselves under the clear guidance of the teacher using a trigonometric circle. In the future, this “circle” will serve as a reference signal for them or external factor to reproduce trigonometry concepts and formulas in memory.
Studying trigonometry on a trigonometric circle helps:
- choosing the optimal communication style for a given lesson, organizing educational cooperation;
- lesson targets become personally significant for each student;
- new material relies on personal experience actions, thinking, sensations of the student;
- lesson includes various shapes work and methods 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 tips, mutual consultations).
When executing trigonometric transformations follow these tips:
- Don't try to immediately come up with a solution to the example from start to finish.
- Don't try to convert the entire example at once. Take small steps forward.
- Remember that in addition to trigonometric formulas in trigonometry, you can still use all fair algebraic transformations (bracketing, abbreviating fractions, abbreviated multiplication formulas, and so on).
- Believe that everything will be fine.
Basic trigonometric formulas
Most formulas in trigonometry are often used 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. Let us first write down the definitions of trigonometric functions. Let there be a right triangle:
Then, the definition of sine:
Definition of cosine:
Tangent definition:
Definition of cotangent:
Basic trigonometric identity:
The simplest corollaries from the basic trigonometric identity:
Formulas double angle. Sine of double angle:
Cosine of double angle:
Tangent of double angle:
Cotangent of double angle:
Additional trigonometric formulas
Trigonometric formulas addition. Sine of the sum:
Sine of the difference:
Cosine of the sum:
Cosine of the difference:
Tangent of the sum:
Tangent of difference:
Cotangent of the amount:
Cotangent of the difference:
Trigonometric formulas for converting a sum into a product. Sum of sines:
Sine difference:
Sum of cosines:
Difference of cosines:
Sum of tangents:
Tangent difference:
Sum of cotangents:
Cotangent difference:
Trigonometric formulas for converting a product into a sum. Product of sines:
Product of sine and cosine:
Product of cosines:
Degree reduction formulas.
Half angle formulas.
Trigonometric reduction formulas
The cosine function is called cofunction sine functions and vice versa. Similarly, the tangent and cotangent functions are cofunctions. Reduction formulas can be formulated as the following rule:
- If in the reduction formula an angle is subtracted (added) from 90 degrees or 270 degrees, then the reduced 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 retained;
- In this case, the sign that the reduced (i.e., original) function has in the corresponding quadrant is placed in front of the reduced function, if we consider the subtracted (added) angle to be acute.
Reduction formulas are given in table form:
By trigonometric circle 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. To do this:
- You can use the trigonometric formulas given above. At the same time, you don’t need to try to transform 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 using algebraic methods, i.e. for example, take something out of brackets or, conversely, open brackets, reduce a fraction, apply an abbreviated multiplication formula, bring fractions to a common denominator, and so on.
- When solving trigonometric equations, you can use grouping method. It must be remembered that in order for the product of several factors to be equal to zero, it is sufficient that any of them be equal to zero, and the rest existed.
- Applying variable replacement method, as usual, the equation after introducing the replacement should become simpler and not contain the original variable. You also need to remember to perform a reverse replacement.
- Remember that homogeneous equations often appear in trigonometry.
- When opening modules or solving irrational equations with trigonometric functions, you need to remember and take into account all the subtleties of solving the corresponding equations with ordinary functions.
- Remember about ODZ (in trigonometric equations, restrictions on ODZ mainly come 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 the values of sine and cosine can only lie in the range from minus one to plus one, inclusive.
The main thing is, if you don’t know what to do, do at least something, and the main thing is to use trigonometric formulas correctly. If what you get gets better and better, then continue the solution, and if it gets worse, then go back to the beginning and try to apply other formulas, do this until you come across the correct solution.
Formulas for solutions of the simplest trigonometric equations. For sine there are two equivalent forms of writing the solution:
For other trigonometric functions, the notation is unambiguous. For cosine:
For tangent:
For cotangent:
Solving trigonometric equations in some special cases:
Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.
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