I think you deserve more than this. Here is my key to trigonometry:

• Draw the dome, wall and ceiling
• Trigonometric functions is nothing more than the percentage of these three forms.

Metaphor for sine and cosine: dome

Instead of just looking at the triangles themselves, imagine them in action by finding a specific real-life example.

Imagine you are in the middle of a dome and want to hang a movie projector screen. You point your finger at the dome at a certain angle “x”, and the screen should be suspended from this point.

The angle you point to determines:

• sine(x) = sin(x) = screen height (from floor to dome mounting point)
• cosine(x) = cos(x) = distance from you to the screen (by floor)
• hypotenuse, the distance from you to the top of the screen, always the same, equal to the radius of the dome

Do you want the screen to be as large as possible? Hang it directly above you.

Do you want the screen to hang as far away from you as possible? Hang it straight perpendicular. The screen will have zero height in this position and will hang furthest away, as you asked.

Height and distance from the screen are inversely proportional: the closer the screen hangs, the greater its height.

Sine and cosine are percentages

No one during my years of study, alas, explained to me that the trigonometric functions sine and cosine are nothing more than percentages. Their values ​​range from +100% to 0 to -100%, or from a positive maximum to zero to a negative maximum.

Let's say I paid a tax of 14 rubles. You don't know how much it is. But if you say that I paid 95% in tax, you will understand that I was simply fleeced.

Absolute height doesn't mean anything. But if the sine value is 0.95, then I understand that the TV is hanging almost on the top of your dome. Very soon it will reach its maximum height in the center of the dome, and then begin to decline again.

How can we calculate this percentage? It's very simple: divide the current screen height by the maximum possible (the radius of the dome, also called the hypotenuse).

That's why we are told that “cosine = opposite side / hypotenuse.” It's all about getting interest! It is best to define sine as “the percentage of the current height from the maximum possible.” (The sine becomes negative if your angle points “underground.” The cosine becomes negative if the angle points toward the dome point behind you.)

Let's simplify the calculations by assuming we are at the center of the unit circle (radius = 1). We can skip the division and just take the sine equal to the height.

Each circle is essentially a single circle, scaled up or down to the desired size. So determine the unit circle connections and apply the results to your specific circle size.

Experiment: take any angle and see what percentage height to width it displays:

The graph of the growth of the sine value is not just a straight line. The first 45 degrees cover 70% of the height, but the last 10 degrees (from 80° to 90°) cover only 2%.

This will make it clearer to you: if you walk in a circle, at 0° you rise almost vertically, but as you approach the top of the dome, the height changes less and less.

Tangent and secant. Wall

One day a neighbor built a wall right next to each other to your dome. Cried your view from the window and a good price for resale!

But is it possible to somehow win in this situation?

Of course yes. What if we hung a movie screen right on our neighbor's wall? You target the angle (x) and get:

• tan(x) = tan(x) = screen height on the wall
• distance from you to the wall: 1 (this is the radius of your dome, the wall is not moving anywhere from you, right?)
• secant(x) = sec(x) = “length of the ladder” from you standing in the center of the dome to the top of the suspended screen

Let's clarify a couple of points regarding the tangent, or screen height.

• it starts at 0, and can go infinitely high. You can stretch the screen higher and higher on the wall to create an endless canvas for watching your favorite movie! (For such a huge one, of course, you will have to spend a lot of money).
• tangent is just a scaled up version of sine! And while the increase in sine slows down as you move towards the top of the dome, the tangent continues to grow!

Sekansu also has something to brag about:

• The secant starts at 1 (the ladder is on the floor, from you to the wall) and starts to rise from there
• The secant is always longer than the tangent. The slanted ladder you use to hang your screen should be longer than the screen itself, right? (With unrealistic sizes, when the screen is sooooo long and the ladder needs to be placed almost vertically, their sizes are almost the same. But even then the secant will be a little longer).

Remember, the values ​​are percent. If you decide to hang the screen at an angle of 50 degrees, tan(50)=1.19. Your screen is 19% larger than the distance to the wall (dome radius).

(Enter x=0 and check your intuition - tan(0) = 0 and sec(0) = 1.)

Cotangent and cosecant. Ceiling

Incredibly, your neighbor has now decided to build a roof over your dome. (What's wrong with him? Apparently he doesn't want you to spy on him while he's walking around the yard naked...)

Well, it's time to build an exit to the roof and talk to your neighbor. You choose the angle of inclination and begin construction:

• the vertical distance between the roof outlet and the floor is always 1 (the radius of the dome)
• cotangent(x) = cot(x) = distance between the top of the dome and the exit point
• cosecant(x) = csc(x) = length of your path to the roof

Tangent and secant describe the wall, and COtangent and COsecant describe the ceiling.

Our intuitive conclusions this time are similar to the previous ones:

• If you take the angle equal to 0°, your exit to the roof will last forever, since it will never reach the ceiling. Problem.
• The shortest “ladder” to the roof will be obtained if you build it at an angle of 90 degrees to the floor. The cotangent will be equal to 0 (we do not move along the roof at all, we exit strictly perpendicularly), and the cosecant will be equal to 1 (“the length of the ladder” will be minimal).

Visualize connections

If all three cases are drawn in a dome-wall-ceiling combination, the result will be the following:

Well, it’s still the same triangle, increased in size to reach the wall and the ceiling. We have vertical sides (sine, tangent), horizontal sides (cosine, cotangent) and “hypotenuses” (secant, cosecant). (By the arrows you can see where each element reaches. The cosecant is the total distance from you to the roof).

A little bit of magic. All triangles share the same equalities:

From the Pythagorean theorem (a 2 + b 2 = c 2) we see how the sides of each triangle are connected. In addition, the “height to width” ratios should also be the same for all triangles. (Simply move from the largest triangle to the smaller one. Yes, the size has changed, but the proportions of the sides will remain the same).

Knowing which side in each triangle is equal to 1 (the radius of the dome), we can easily calculate that “sin/cos = tan/1”.

I have always tried to remember these facts through simple visualization. In the picture you clearly see these dependencies and understand where they come from. This technique is much better than memorizing dry formulas.

Don't forget about other angles

Psst... Don't get stuck on one graph, thinking that the tangent is always less than 1. If you increase the angle, you can reach the ceiling without reaching the wall:

Pythagorean connections always work, but the relative sizes may vary.

(You may have noticed that the sine and cosine ratios are always the smallest because they are contained within the dome).

To summarize: what do we need to remember?

For most of us, I'd say this will be enough:

• trigonometry explains the anatomy of mathematical objects such as circles and repeating intervals
• The dome/wall/roof analogy shows the relationship between different trigonometric functions
• The trigonometric functions result in percentages, which we apply to our scenario.

You don't need to memorize formulas like 1 2 + cot 2 = csc 2 . They are only suitable for stupid tests in which knowledge of a fact is passed off as understanding it. Take a minute to draw a semicircle in the form of a dome, a wall and a roof, label the elements, and all the formulas will come to you on paper.

Application: Inverse Functions

Any trigonometric function takes an angle as an input parameter and returns the result as a percentage. sin(30) = 0.5. This means that an angle of 30 degrees takes up 50% of the maximum height.

The inverse trigonometric function is written as sin -1 or arcsin. It is also often written asin in various languages programming.

If our height is 25% of the dome's height, what is our angle?

In our table of proportions you can find a ratio where the secant is divided by 1. For example, the secant by 1 (hypotenuse to the horizontal) will be equal to 1 divided by the cosine:

Let's say our secant is 3.5, i.e. 350% of the radius of a unit circle. What angle of inclination to the wall does this value correspond to?

Appendix: Some examples

Example: Find the sine of angle x.

A boring task. Let's complicate the banal “find the sine” to “What is the height as a percentage of the maximum (hypotenuse)?”

First, notice that the triangle is rotated. There's nothing wrong with that. The triangle also has a height, it is indicated in green in the figure.

What is the hypotenuse equal to? According to the Pythagorean theorem, we know that:

3 2 + 4 2 = hypotenuse 2 25 = hypotenuse 2 5 = hypotenuse

Fine! Sine is the percentage of the height of the triangle's longest side, or hypotenuse. In our example, the sine is 3/5 or 0.60.

Of course, we can go several ways. Now we know that the sine is 0.60, we can simply find the arcsine:

Asin(0.6)=36.9

Here's another approach. Note that the triangle is “facing the wall,” so we can use the tangent instead of the sine. The height is 3, the distance to the wall is 4, so the tangent is ¾ or 75%. We can use the arctangent to go from a percentage value back to an angle:

Tan = 3/4 = 0.75 atan(0.75) = 36.9 Example: Will you swim to the shore?

You are in a boat and you have enough fuel to travel 2 km. You are now 0.25 km from the coast. At what maximum angle to the shore can you swim to it so that you have enough fuel? Addition to the problem statement: we only have a table of arc cosine values.

What we have? The coastline can be represented as a “wall” in our famous triangle, and the “length of the ladder” attached to the wall is the maximum possible distance to be covered by boat to the shore (2 km). A secant appears.

First, you need to go to percentages. We have 2 / 0.25 = 8, that is, we can swim a distance that is 8 times the straight distance to the shore (or to the wall).

The question arises: “What is the secant of 8?” But we cannot answer it, since we only have arc cosines.

We use our previously derived dependencies to relate the secant to the cosine: “sec/1 = 1/cos”

The secant of 8 is equal to the cosine of ⅛. An angle whose cosine is ⅛ is equal to acos(1/8) = 82.8. And this is the largest angle we can afford on a boat with the specified amount of fuel.

Not bad, right? Without the dome-wall-ceiling analogy, I would have gotten lost in a bunch of formulas and calculations. Visualizing the problem greatly simplifies the search for a solution, and it is also interesting to see which trigonometric function will ultimately help.

For each problem, think like this: Am I interested in the dome (sin/cos), the wall (tan/sec), or the ceiling (cot/csc)?

And trigonometry will become much more enjoyable. Easy calculations for you!

Sine and cosine originally arose from the need to calculate quantities in right triangles. It was noticed that if the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how much these sides change in length, always remains the same.

This is how the concepts of sine and cosine were introduced. Sinus acute angle in a right triangle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the side adjacent to the hypotenuse.

Theorems of cosines and sines

But cosines and sines can be used for more than just right triangles. To find the value of an obtuse or acute angle or side of any triangle, it is enough to apply the theorem of cosines and sines.

The cosine theorem is quite simple: “The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them.”

There are two interpretations of the sine theorem: small and extended. According to the minor: “In a triangle, the angles are proportional to the opposite sides.” This theorem is often expanded due to the property of the circumscribed circle of a triangle: “In a triangle, the angles are proportional to the opposite sides, and their ratio is equal to the diameter of the circumscribed circle.”

Derivatives

The derivative is a mathematical tool that shows how quickly a function changes relative to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.

When solving problems, you need to know the tabular values ​​of the derivatives of trigonometric functions: sine and cosine. The derivative of a sine is a cosine, and a cosine is a sine, but with a minus sign.

Application in mathematics

Sines and cosines are especially often used in solving right triangles and problems related to them.

The convenience of sines and cosines is also reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking down complex shapes and objects into “simple” triangles. Engineers who often deal with calculations of aspect ratios and degree measures spent a lot of time and effort calculating the cosines and sines of non-tabular angles.

Then Bradis tables came to the rescue, containing thousands of values ​​of sines, cosines, tangents and cotangents of different angles. IN Soviet time some teachers forced their students to memorize pages of Bradis tables.

Radian is the angular value of an arc whose length is equal to the radius or 57.295779513° degrees.

Degree (in geometry) - 1/360th part of a circle or 1/90th part right angle.

π = 3.141592653589793238462… ( approximate value Pi numbers).

Teachers believe that every student should be able to carry out calculations, know trigonometric formulas, but not every teacher explains what sine and cosine are. What is their meaning, where are they used? Why are we talking about triangles, but the textbook shows a circle? Let's try to connect all the facts together.

School subject

The study of trigonometry usually begins in grades 7-8 high school. At this time, students are explained what sine and cosine are and are asked to solve geometric problems using these functions. Later, more complex formulas and expressions appear that need to be transformed algebraically (formulas of double and half angle, power functions), work is carried out with a trigonometric circle.

However, teachers are not always able to clearly explain the meaning of the concepts used and the applicability of the formulas. Therefore, the student often does not see the point in this subject, and the memorized information is quickly forgotten. However, once you explain to a high school student, for example, the connection between a function and oscillatory motion, the logical connection will be remembered for many years, and jokes about the uselessness of the subject will become a thing of the past.

Usage

For the sake of curiosity, let's look into various branches of physics. Do you want to determine the range of a projectile? Or are you calculating the friction force between an object and a certain surface? Swinging the pendulum, watching the rays passing through the glass, calculating the induction? Trigonometric concepts appear in almost any formula. So what are sine and cosine?

Definitions

The sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the same hypotenuse. There is absolutely nothing complicated here. Perhaps students are usually confused by the values ​​they see on the trigonometry table because it involves square roots. Yes, getting decimals from them is not very convenient, but who said that all numbers in mathematics must be equal?

In fact, you can find a funny hint in trigonometry problem books: most of the answers here are even and, in the worst case, contain the root of two or three. The conclusion is simple: if your answer turns out to be a “multi-story” fraction, double-check the solution for errors in calculations or reasoning. And you will most likely find them.

What to remember

Like any science, trigonometry has data that needs to be learned.

First, you should memorize the numerical values ​​for right triangle sines, cosines 0 and 90, as well as 30, 45 and 60 degrees. These indicators are found in nine out of ten school problems. By looking at these values ​​in a textbook, you will lose a lot of time, and there will be nowhere to look at them at all during a test or exam.

It must be remembered that the value of both functions cannot exceed one. If anywhere in your calculations you get a value outside the 0-1 range, stop and try the problem again.

The sum of the squares of sine and cosine is equal to one. If you have already found one of the values, use this formula to find the remaining one.

Theorems

There are two basic theorems in basic trigonometry: sines and cosines.

The first states that the ratio of each side of a triangle to the sine of the opposite angle is the same. The second is that the square of any side can be obtained by adding the squares of the two remaining sides and subtracting their double product multiplied by the cosine of the angle lying between them.

Thus, if we substitute the value of an angle of 90 degrees into the cosine theorem, we get... the Pythagorean theorem. Now, if you need to calculate the area of ​​a figure that is not a right triangle, you don’t have to worry anymore - the two theorems discussed will significantly simplify the solution of the problem.

Goals and objectives

Learning trigonometry will become much easier when you realize one simple fact: all the actions you perform are aimed at achieving just one goal. Any parameters of a triangle can be found if you know the bare minimum of information about it - this could be the value of one angle and the length of two sides or, for example, three sides.

To determine the sine, cosine, tangent of any angle, these data are sufficient, and with their help you can easily calculate the area of ​​the figure. Almost always, the answer requires one of the mentioned values, and they can be found using the same formulas.

Inconsistencies in learning trigonometry

One of the confusing questions that students prefer to avoid is discovering the connections between different concepts in trigonometry. It would seem that triangles are used to study the sines and cosines of angles, but for some reason the symbols are often found in the figure with a circle. In addition, there is a completely incomprehensible wave-like graph called a sine wave, which has no external resemblance to either a circle or triangles.

Moreover, angles are measured either in degrees or in radians, and the number Pi, written simply as 3.14 (without units), for some reason appears in the formulas, corresponding to 180 degrees. How is all this connected?

Units

Why is Pi exactly 3.14? Do you remember what this meaning is? This is the number of radii that fit in an arc on half a circle. If the diameter of the circle is 2 centimeters, the circumference will be 3.14 * 2, or 6.28.

Second point: you may have noticed the similarity between the words “radian” and “radius”. The fact is that one radian is numerically equal to the angle taken from the center of the circle to an arc one radius long.

Now we will combine the acquired knowledge and understand why “Pi in half” is written on top of the coordinate axis in trigonometry, and “Pi” is written on the left. This is an angular value measured in radians, because a semicircle is 180 degrees, or 3.14 radians. And where there are degrees, there are sines and cosines. It is easy to draw a triangle from the desired point, setting aside segments to the center and to the coordinate axis.

Let's look into the future

Trigonometry, studied in school, deals with a rectilinear coordinate system, where, no matter how strange it may sound, a straight line is a straight line.

But there is more complex ways working with space: the sum of the angles of the triangle here will be more than 180 degrees, and the straight line in our view will look like a real arc.

Let's move from words to action! Take an apple. Make three cuts with a knife so that when viewed from above you get a triangle. Take out the resulting piece of apple and look at the “ribs” where the peel ends. They are not straight at all. The fruit in your hands can be conventionally called round, but now imagine how complex the formulas must be with which you can find the area of ​​the cut piece. But some specialists solve such problems every day.

Trigonometric functions in life

Have you noticed that the shortest route for an airplane from point A to point B on the surface of our planet has a pronounced arc shape? The reason is simple: the Earth is spherical, which means you can’t calculate much using triangles - you have to use more complex formulas.

You cannot do without the sine/cosine of an acute angle in any questions related to space. It’s interesting that a whole lot of factors come together here: trigonometric functions are required when calculating the motion of planets along circles, ellipses and various trajectories more complex shapes; the process of launching rockets, satellites, shuttles, undocking research vehicles; observing distant stars and studying galaxies that humans will not be able to reach in the foreseeable future.

In general, the field of activity for a person who knows trigonometry is very wide and, apparently, will only expand over time.

Conclusion

Today we learned, or at least repeated, what sine and cosine are. These are concepts that you don’t need to be afraid of - just want them and you will understand their meaning. Remember that trigonometry is not a goal, but only a tool that can be used to satisfy real human needs: build houses, ensure traffic safety, even explore the vastness of the universe.

Indeed, science itself may seem boring, but as soon as you find in it a way to achieve your own goals and self-realization, the learning process will become interesting, and your personal motivation will increase.

As homework Try to find ways to apply trigonometric functions in an area of ​​activity that interests you personally. Imagine, use your imagination, and then you will probably find that new knowledge will be useful to you in the future. And besides, mathematics is useful for general development thinking.

Sinus acute angle α right triangle is an attitude opposite leg to hypotenuse.
It is denoted as follows: sin α.

Cosine The acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is designated as follows: cos α.

Tangent acute angle α is the ratio of the opposite side to the adjacent side.
It is designated as follows: tg α.

Cotangent acute angle α is the ratio of the adjacent side to the opposite side.
It is designated as follows: ctg α.

The sine, cosine, tangent and cotangent of an angle depend only on the size of the angle.

Rules:

Basic trigonometric identities in a right triangle:

(α - acute angle opposite to the leg b and adjacent to the leg a . Side With – hypotenuse. β – second acute angle).

b sin α = - c	sin 2 α + cos 2 α = 1
a cos α = - c	1 1 + tan 2 α = -- cos 2 α
b tan α = - a	1 1 + cotg 2 α = -- sin 2 α
a ctg α = - b	1 1 1 + -- = -- tan 2 α sin 2 α
sinα tg α = -- cos α

As the acute angle increasessin α andtg α increase, andcos α decreases.

For any acute angle α:

sin (90° – α) = cos α

cos (90° – α) = sin α

Example-explanation:

Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.

Let's find out the sine of angle A and the cosine of angle B.

Solution .

1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of the acute angles is 90º, then angle B = 60º:

B = 90º – 30º = 60º.

2) Let's calculate sin A. We know that the sine is equal to the ratio of the opposite side to the hypotenuse. For angle A, the opposite side is side BC. So:

BC 3 1
sin A = -- = - = -
AB 6 2

3) Now let's calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC by AB - that is, perform the same actions as when calculating the sine of angle A:

BC 3 1
cos B = -- = - = -
AB 6 2

The result is:
sin A = cos B = 1/2.

sin 30º = cos 60º = 1/2.

It follows from this that in a right triangle, the sine of one acute angle is equal to the cosine of the other acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° – α) = cos α
cos (90° – α) = sin α

Let's make sure of this again:

1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º – 60º) = cos 60º.
sin 30º = cos 60º.

2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° – 30º) = sin 30º.
cos 60° = sin 30º.

(For more information about trigonometry, see the Algebra section)

Unified State Exam for 4? Won't you burst with happiness?

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.

Trigonometry

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

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,

What are sine and cosine? What are tangent and cotangent?

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 in Unified State Exam assignments It happens. 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 are also called trigonometric functions.

And 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

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. For those who have forgotten how to solve equations, follow 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...)

Relationship between trigonometric functions of one angle.

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 right decision The task must contain additional information (if 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 trick for similar tasks. Everything can be solved practically without formulas at all! Well, 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 Unified State Exam type, 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 not only taken 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! Theoretical basis all these things without trigonometric functions are zero without a stick. But the ancient people did not disappoint. How they got out is in the next lesson.

If you like this site...

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.