1. Trigonometric functions are elementary functions whose argument is corner. By using trigonometric functions describes the relationship between the parties and sharp corners in a right triangle. The areas of application of trigonometric functions are extremely diverse. For example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear when solving differential and functional equations.
2. Trigonometric functions include the following 6 functions: sinus, cosine, tangent,cotangent, secant And cosecant. For each of these functions there is an inverse trigonometric function.
3. It is convenient to introduce the geometric definition of trigonometric functions using unit circle. The figure below shows a circle with radius r=1. The point M(x,y) is marked on the circle. The angle between the radius vector OM and the positive direction of the Ox axis is equal to α.
4. Sinus angle α is the ratio of the ordinate y of the point M(x,y) to the radius r:
sinα=y/r.
Since r=1, then the sine is equal to the ordinate of the point M(x,y).
5. Cosine angle α is the ratio of the abscissa x of the point M(x,y) to the radius r:
cosα=x/r
6. Tangent angle α is the ratio of the ordinate y of a point M(x,y) to its abscissa x:
tanα=y/x,x≠0
7. Cotangent angle α is the ratio of the abscissa x of a point M(x,y) to its ordinate y:
cotα=x/y,y≠0
8. Secant angle α is the ratio of the radius r to the abscissa x of the point M(x,y):
secα=r/x=1/x,x≠0
9. Cosecant angle α is the ratio of the radius r to the ordinate y of the point M(x,y):
cscα=r/y=1/y,y≠0
10. In the unit circle, the projections x, y of the points M(x,y) and radius r form a right triangle, in where x,y are legs, and r is the hypotenuse. Therefore, the above definitions of trigonometric functions in the appendix to right triangle are formulated as follows:
Sinus angle α is the ratio of the opposite side to the hypotenuse.
Cosine angle α is the ratio of the adjacent leg to the hypotenuse.
Tangent angle α is called the opposite leg to the adjacent one.
Cotangent angle α is called the adjacent side to the opposite side.
Secant angle α is the ratio of the hypotenuse to the adjacent leg.
Cosecant angle α is the ratio of the hypotenuse to the opposite leg.
11. Graph of the sine function
y=sinx, domain of definition: x∈R, range of values: −1≤sinx≤1
12. Graph of the cosine function
y=cosx, domain: x∈R, range: −1≤cosx≤1
13. Graph of the tangent function 14. Graph of the cotangent function 15. Graph of the secant function The figure below shows several unit circles, which indicate the signs of sine, cosine, tangent and cotangent in various coordinate quarters. Lecture: Sine, cosine, tangent, cotangent of an arbitrary angle
Sine, cosine of an arbitrary angle
To understand what trigonometric functions are, let's look at a circle with unit radius. This circle has a center at the origin on the coordinate plane. To determine the given functions we will use the radius vector OR, which starts at the center of the circle, and the point R is a point on the circle. This radius vector forms an angle alpha with the axis OH. Since the circle has a radius equal to one, then OR = R = 1. If from the point R lower the perpendicular to the axis OH, then we get a right triangle with a hypotenuse equal to one. If the radius vector moves clockwise, then this direction is called negative, if it moves counterclockwise - positive.
Sine of the angle OR, is the ordinate of the point R vector on a circle. That is, to obtain the value of the sine of a given angle alpha, it is necessary to determine the coordinate U on surface. How was this value obtained? Since we know that the sine of an arbitrary angle in a right triangle is the ratio of the opposite side to the hypotenuse, we get that And since R=1, That sin(α) = y 0
. In a unit circle, the ordinate value cannot be less than -1 and greater than 1, which means The sine takes a positive value in the first and second quarters of the unit circle, and negative in the third and fourth. Cosine of the angle given circle formed by the radius vector OR, is the abscissa of the point R vector on a circle. That is, to obtain the cosine value of a given angle alpha, it is necessary to determine the coordinate X on surface. The cosine of an arbitrary angle in a right triangle is the ratio of the adjacent leg to the hypotenuse, we get that And since R=1, That cos(α) = x 0
. In the unit circle, the abscissa value cannot be less than -1 and greater than 1, which means The cosine takes a positive value in the first and fourth quarters of the unit circle, and negative in the second and third. Tangentarbitrary angle The ratio of sine to cosine is calculated. If we consider a right triangle, then this is the ratio of the opposite side to the adjacent side. If we are talking about the unit circle, then this is the ratio of the ordinate to the abscissa. Judging by these relationships, it can be understood that the tangent cannot exist if the abscissa value is zero, that is, at an angle of 90 degrees. The tangent can take all other values. The tangent is positive in the first and third quarters of the unit circle, and negative in the second and fourth. The question, as they say, is interesting... It is possible, it is possible to pass with a 4! And at the same time not to burst... The main condition is to exercise regularly. Here is the basic preparation for the Unified State Exam in mathematics. With all the secrets and mysteries of the Unified State Exam, which you will not read about in textbooks... Study this section, solve more tasks from various sources - and everything will work out! It is assumed that the basic section "A C is enough for you!" it doesn't cause you any problems. But if suddenly... Follow the links, don’t be lazy! And we will start with a great and terrible topic. Attention! This topic causes a lot of problems for students. It is considered one of the most severe. What are sine and cosine? What are tangent and cotangent? What is a number circle? As soon as you ask these harmless questions, the person turns pale and tries to divert the conversation... But in vain. These are simple concepts. And this topic is no more difficult than others. You just need to clearly understand the answers to these very questions from the very beginning. It is very important. If you understand, you will like trigonometry. So, Let's start with ancient times. Don’t worry, we’ll go through all 20 centuries of trigonometry in about 15 minutes. And, without noticing it, we’ll repeat a piece of geometry from 8th grade. Let's draw a right triangle with sides a, b, c and angle X. Here it is. Let me remind you that the sides that form a right angle are called legs. a and c– legs. There are two of them. The remaining side is called the hypotenuse. With– hypotenuse. Triangle and triangle, just think! What to do with him? But the ancient people knew what to do! Let's repeat their actions. Let's measure the side V. In the figure, the cells are specially drawn, as happens in Unified State Examination tasks. Side V equal to four cells. OK. Let's measure the side A. Three cells. Now let's divide the length of the side A per side length V. Or, as they also say, let’s take the attitude A To V. a/v= 3/4. On the contrary, you can divide V on A. We get 4/3. Can V divide by With. Hypotenuse With It’s impossible to count by cells, but it is equal to 5. We get high quality= 4/5. In short, you can divide the lengths of the sides by each other and get some numbers. So what? What is the point of this interesting activity? None yet. A pointless exercise, to put it bluntly.) Now let's do this. Let's enlarge the triangle. Let's extend the sides in and with, but so that the triangle remains rectangular. Corner X, of course, does not change. To see this, hover your mouse over the picture, or touch it (if you have a tablet). Parties a, b and c will turn into m, n, k, and, of course, the lengths of the sides will change. But their relationship is not! Attitude a/v was: a/v= 3/4, became m/n= 6/8 = 3/4. The relationships of other relevant parties are also won't change
. You can change the lengths of the sides in a right triangle as you like, increase, decrease, without changing the angle x – the relationship between the relevant parties will not change
. You can check it, or you can take the ancient people’s word for it. But this is already very important! The ratios of the sides in a right triangle do not depend in any way on the lengths of the sides (at the same angle). This is so important that the relationship between the parties has earned its own special name. Your names, so to speak.) Meet me. What is the sine of angle x
? This is the ratio of the opposite side to the hypotenuse: sinx = a/c What is the cosine of the angle x
? This is the ratio of the adjacent leg to the hypotenuse: Withosx=
high quality What is tangent x
? This is the ratio of the opposite side to the adjacent side: tgx =a/v What is the cotangent of angle x
? This is the ratio of the adjacent side to the opposite: ctgx = v/a Everything is very simple. Sine, cosine, tangent and cotangent are some numbers. Dimensionless. Just numbers. Each angle has its own. Why am I repeating everything so boringly? Then what is this need to remember. It's important to remember. Memorization can be made easier. Is the phrase “Let’s start from afar…” familiar? So start from afar. Sinus angle is a ratio distant from the leg angle to the hypotenuse. Cosine– the ratio of the neighbor to the hypotenuse. Tangent angle is a ratio distant from the leg angle to the near one. Cotangent- vice versa. It's easier, right? Well, if you remember that in tangent and cotangent there are only legs, and in sine and cosine the hypotenuse appears, then everything will become quite simple. This whole glorious family - sine, cosine, tangent and cotangent is also called trigonometric functions. Now a question for consideration. Why do we say sine, cosine, tangent and cotangent corner? We are talking about the relationship between the parties, like... What does it have to do with it? corner? Let's look at the second picture. Exactly the same as the first one. Hover your mouse over the picture. I changed the angle X. Increased it from x to x. All relationships have changed! Attitude a/v was 3/4, and the corresponding ratio t/v became 6/4. And all other relationships became different! Therefore, the ratios of the sides do not depend in any way on their lengths (at one angle x), but depend sharply on this very angle! And only from him. Therefore, the terms sine, cosine, tangent and cotangent refer to corner. The angle here is the main one. It must be clearly understood that the angle is inextricably linked with its trigonometric functions. Each angle has its own sine and cosine. And almost everyone has their own tangent and cotangent. It is important. It is believed that if we are given an angle, then its sine, cosine, tangent and cotangent we know
! And vice versa. Given a sine, or any other trigonometric function, it means we know the angle. There are special tables where for each angle its trigonometric functions are described. They are called Bradis tables. They were compiled a very long time ago. When there were no calculators or computers yet... Of course, it is impossible to remember the trigonometric functions of all angles. You are required to know them only for a few angles, more on this later. But the spell I know an angle, which means I know its trigonometric functions” - always works! So we repeated a piece of geometry from 8th grade. Do we need it for the Unified State Exam? Necessary. Here is a typical problem from the Unified State Exam. To solve this problem, 8th grade is enough. Given picture: All. There is no more data. We need to find the length of the side of the aircraft. The cells do not help much, the triangle is somehow incorrectly positioned.... On purpose, I guess... From the information there is the length of the hypotenuse. 8 cells. For some reason, the angle was given. This is where you need to immediately remember about trigonometry. There is an angle, which means we know all its trigonometric functions. Which of the four functions should we use? Let's see, what do we know? We know the hypotenuse and the angle, but we need to find adjacent catheter to this corner! It’s clear, the cosine needs to be put into action! Here we go. We simply write, by the definition of cosine (the ratio adjacent leg to hypotenuse): cosC = BC/8 Our angle C is 60 degrees, its cosine is 1/2. You need to know this, without any tables! That is: 1/2 = BC/8 Elementary linear equation. Unknown – Sun. Those who have forgotten how to solve equations, take a look at the link, the rest solve: BC = 4 When ancient people realized that each angle has its own set of trigonometric functions, they had a reasonable question. Are sine, cosine, tangent and cotangent somehow related to each other? So that knowing one angle function, you can find the others? Without calculating the angle itself? They were so restless...) Of course, sine, cosine, tangent and cotangent of the same angle are related to each other. Any connection between expressions is given in mathematics by formulas. In trigonometry there are a colossal number of formulas. But here we will look at the most basic ones. These formulas are called: basic trigonometric identities. Here they are: You need to know these formulas thoroughly. Without them, there is generally nothing to do in trigonometry. Three more auxiliary identities follow from these basic identities: I warn you right away that the last three formulas quickly fall out of your memory. For some reason.) You can, of course, derive these formulas from the first three. But, in difficult times... You understand.) In standard problems, like the ones below, there is a way to avoid these forgettable formulas. AND dramatically reduce errors due to forgetfulness, and in calculations too. This practice is in Section 555, lesson "Relationships between trigonometric functions of the same angle." In what tasks and how are the basic trigonometric identities used? The most popular task is to find some angle function if another is given. In the Unified State Examination such a task is present from year to year.) For example: Find the value of sinx if x is an acute angle and cosx=0.8. The task is almost elementary. We are looking for a formula that contains sine and cosine. Here is the formula: sin 2 x + cos 2 x = 1 We substitute here a known value, namely 0.8 instead of cosine: sin 2 x + 0.8 2 = 1 Well, we count as usual: sin 2 x + 0.64 = 1 sin 2 x = 1 - 0.64 That's practically all. We have calculated the square of the sine, all that remains is to extract the square root and the answer is ready! The root of 0.36 is 0.6. The task is almost elementary. But the word “almost” is there for a reason... The fact is that the answer sinx= - 0.6 is also suitable... (-0.6) 2 will also be 0.36. There are two different answers. And you need one. The second one is wrong. How to be!? Yes, as usual.) Read the assignment carefully. For some reason it says:... if x is an acute angle... And in tasks, every word has a meaning, yes... This phrase is additional information for the solution. An acute angle is an angle less than 90°. And at such corners All trigonometric functions - sine, cosine, and tangent with cotangent - positive. Those. We simply discard the negative answer here. We have the right. Actually, eighth graders don’t need such subtleties. They only work with right triangles, where the corners can only be acute. And they don’t know, happy ones, that there are both negative angles and angles of 1000°... And all these terrible angles have their own trigonometric functions, both plus and minus... But for high school students, without taking into account the sign - no way. Much knowledge multiplies sorrows, yes...) And for the correct solution, additional information is necessarily present in the task (if it is necessary). For example, it can be given by the following entry: Or some other way. You will see in the examples below.) To solve such examples you need to know Which quarter does the given angle x fall into and what sign does the desired trigonometric function have in this quarter? These basics of trigonometry are discussed in the lessons on what a trigonometric circle is, the measurement of angles on this circle, the radian measure of an angle. Sometimes you need to know the table of sines, cosines of tangents and cotangents. So, let's note the most important thing: Practical tips: 1. Remember the definitions of sine, cosine, tangent and cotangent. It will be very useful. 2. We clearly understand: sine, cosine, tangent and cotangent are tightly connected with angles. We know one thing, which means we know another. 3. We clearly understand: sine, cosine, tangent and cotangent of one angle are related to each other by basic trigonometric identities. We know one function, which means we can (if we have the necessary additional information) calculate all the others. Now let’s decide, as usual. First, tasks in the scope of 8th grade. But high school students can do it too...) 1. Calculate the value of tgA if ctgA = 0.4. 2. β is an angle in a right triangle. Find the value of tanβ if sinβ = 12/13. 3. Determine the sine of the acute angle x if tgх = 4/3. 4. Find the meaning of the expression: 6sin 2 5° - 3 + 6cos 2 5° 5. Find the meaning of the expression: (1-cosx)(1+cosx), if sinx = 0.3 Answers (separated by semicolons, in disarray): 0,09; 3; 0,8; 2,4; 2,5 Happened? Great! Eighth graders can already go get their A's.) Didn't everything work out? Tasks 2 and 3 are somehow not very good...? No problem! There is one beautiful technique for such tasks. Everything can be solved practically without formulas at all! And, therefore, without errors. This technique is described in the lesson: “Relationships between trigonometric functions of one angle” in Section 555. All other tasks are also dealt with there. These were problems like the Unified State Exam, but in a stripped-down version. Unified State Exam - light). And now almost the same tasks, but in a full-fledged format. For knowledge-burdened high school students.) 6. Find the value of tanβ if sinβ = 12/13, and 7. Determine sinх if tgх = 4/3, and x belongs to the interval (- 540°; - 450°). 8. Find the value of the expression sinβ cosβ if ctgβ = 1. Answers (in disarray): 0,8; 0,5; -2,4. Here in problem 6 the angle is not specified very clearly... But in problem 8 it is not specified at all! This is on purpose). Additional information is taken not only from the task, but also from the head.) But if you decide, one correct task is guaranteed! What if you haven't decided? Hmm... Well, Section 555 will help here. There the solutions to all these tasks are described in detail, it is difficult not to understand. This lesson provides a very limited understanding of trigonometric functions. Within 8th grade. And the elders still have questions... For example, if the angle X(look at the second picture on this page) - make it stupid!? The triangle will completely fall apart! So what should we do? There will be no leg, no hypotenuse... The sine has disappeared... If ancient people had not found a way out of this situation, we would not have cell phones, TV, or electricity now. Yes Yes! The theoretical basis for all these things without trigonometric functions is zero without a stick. But the ancient people did not disappoint. How they got out is in the next lesson. By the way, I have a couple more interesting sites for you.) You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!) You can get acquainted with functions and derivatives.
y=tanx, range of definition: x∈R,x≠(2k+1)π/2, range of values: −∞ 
y=cotx, domain: x∈R,x≠kπ, range: −∞ 
y=secx, domain: x∈R,x≠(2k+1)π/2, range: secx∈(−∞,−1]∪∪Properties of cosine
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Properties of tangent
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Properties of cotangent
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Unified State Exam for 4? Won't you burst with happiness?
Trigonometry
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)What are sine and cosine? What are tangent and cotangent?
Relationship between trigonometric functions of one angle.
If you like this site...
Allows you to establish a number of characteristic results - properties of sine, cosine, tangent and cotangent. In this article we will look at three main properties. The first of them indicates the signs of the sine, cosine, tangent and cotangent of the angle α depending on the angle of which coordinate quarter is α. Next we will consider the property of periodicity, which establishes the invariance of the values of sine, cosine, tangent and cotangent of the angle α when this angle changes by an integer number of revolutions. The third property expresses the relationship between the values of sine, cosine, tangent and cotangent of opposite angles α and −α.
If you are interested in the properties of the functions sine, cosine, tangent and cotangent, then you can study them in the corresponding section of the article.
Page navigation.
Signs of sine, cosine, tangent and cotangent by quarters
Below in this paragraph the phrase “angle of I, II, III and IV coordinate quarter” will appear. Let's explain what these angles are.
Let's take a unit circle, mark the starting point A(1, 0) on it, and rotate it around the point O by an angle α, and we will assume that we will get to the point A 1 (x, y).
They say that angle α is the angle of the I, II, III, IV coordinate quadrant, if point A 1 lies in the I, II, III, IV quarters, respectively; if the angle α is such that point A 1 lies on any of the coordinate lines Ox or Oy, then this angle does not belong to any of the four quarters.
For clarity, here is a graphic illustration. The drawings below show rotation angles of 30, −210, 585, and −45 degrees, which are the angles of the I, II, III, and IV coordinate quarters, respectively.
Angles 0, ±90, ±180, ±270, ±360, … degrees do not belong to any of the coordinate quarters.
Now let's figure out what signs have the values of sine, cosine, tangent and cotangent of the angle of rotation α, depending on which quadrant angle α is.
For sine and cosine this is easy to do.
By definition, the sine of angle α is the ordinate of point A 1. Obviously, in the I and II coordinate quarters it is positive, and in the III and IV quarters it is negative. Thus, the sine of angle α has a plus sign in the 1st and 2nd quarters, and a minus sign in the 3rd and 6th quarters.
In turn, the cosine of the angle α is the abscissa of point A 1. In the I and IV quarters it is positive, and in the II and III quarters it is negative. Consequently, the values of the cosine of the angle α in the I and IV quarters are positive, and in the II and III quarters they are negative.
To determine the signs of the quarters of tangent and cotangent, you need to remember their definitions: tangent is the ratio of the ordinate of point A 1 to the abscissa, and cotangent is the ratio of the abscissa of point A 1 to the ordinate. Then from rules for dividing numbers with the same and different signs it follows that tangent and cotangent have a plus sign when the abscissa and ordinate signs of point A 1 are the same, and have a minus sign when the abscissa and ordinate signs of point A 1 are different. Consequently, the tangent and cotangent of the angle have a + sign in the I and III coordinate quarters, and a minus sign in the II and IV quarters.
Indeed, for example, in the first quarter both the abscissa x and the ordinate y of point A 1 are positive, then both the quotient x/y and the quotient y/x are positive, therefore, tangent and cotangent have + signs. And in the second quarter, the abscissa x is negative, and the ordinate y is positive, therefore both x/y and y/x are negative, hence the tangent and cotangent have a minus sign.
Let's move on to the next property of sine, cosine, tangent and cotangent.
Periodicity property
Now we will look at perhaps the most obvious property of sine, cosine, tangent and cotangent of an angle. It is as follows: when the angle changes by an integer number of full revolutions, the values of the sine, cosine, tangent and cotangent of this angle do not change.
This is understandable: when the angle changes by an integer number of revolutions, we will always get from the starting point A to point A 1 on the unit circle, therefore, the values of sine, cosine, tangent and cotangent remain unchanged, since the coordinates of point A 1 are unchanged.
Using formulas, the considered property of sine, cosine, tangent and cotangent can be written as follows: sin(α+2·π·z)=sinα, cos(α+2·π·z)=cosα, tan(α+2·π· z)=tgα, ctg(α+2·π·z)=ctgα, where α is the angle of rotation in radians, z is any, the absolute value of which indicates the number of full revolutions by which the angle α changes, and the sign of the number z indicates the direction turn.
If the rotation angle α is specified in degrees, then the indicated formulas will be rewritten as sin(α+360° z)=sinα , cos(α+360° z)=cosα , tg(α+360° z)=tgα , ctg(α+360°·z)=ctgα .
Let's give examples of using this property. For example, 
, because 
, A 
. Here's another example: or .
This property, together with reduction formulas, is very often used when calculating the values of sine, cosine, tangent and cotangent of “large” angles.
The considered property of sine, cosine, tangent and cotangent is sometimes called the property of periodicity.
Properties of sines, cosines, tangents and cotangents of opposite angles
Let A 1 be the point obtained by rotating the initial point A(1, 0) around point O by an angle α, and point A 2 be the result of rotating point A by an angle −α, opposite to angle α.
The property of sines, cosines, tangents and cotangents of opposite angles is based on a fairly obvious fact: the points A 1 and A 2 mentioned above either coincide (at) or are located symmetrically relative to the Ox axis. That is, if point A 1 has coordinates (x, y), then point A 2 will have coordinates (x, −y). From here, using the definitions of sine, cosine, tangent and cotangent, we write the equalities and .
Comparing them, we come to relationships between sines, cosines, tangents and cotangents of opposite angles α and −α of the form.
This is the property under consideration in the form of formulas.
Let's give examples of using this property. For example, the equalities and 
.
It only remains to note that the property of sines, cosines, tangents and cotangents of opposite angles, like the previous property, is often used when calculating the values of sine, cosine, tangent and cotangent, and allows you to completely avoid negative angles.
Bibliography.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
