Trigonometry, as a science, originated in the Ancient East. First trigonometric ratios were deduced by astronomers to create accurate calendar and navigation by the stars. These calculations related to spherical trigonometry, while in school course study the ratios of sides and angles of a plane triangle.

Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationship between the sides and angles of triangles.

During the heyday of culture and science in the 1st millennium AD, knowledge spread from Ancient East to Greece. But the main discoveries of trigonometry are the merit of husbands Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values ​​for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

Basic trigonometric functions numeric argument– these are sine, cosine, tangent and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values ​​of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants are equal in all directions,” since the proof is given using the example of an isosceles right triangle.

Sine, cosine and other relationships establish the relationship between the acute angles and sides of any right triangle. Let us present formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:

As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b in cos form A * c, then we get following formulas for tangent and cotangent:

Trigonometric circle

Graphically, the relationship between the mentioned quantities can be represented as follows:

Circumference, in in this case, represents everything possible values angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the size of the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For α from 180° to 360° (III and IV quarters), sin α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the value of the quantities.

Values ​​of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values ​​of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.

Angles in tables for trigonometric functions correspond to radian values:

So, it’s not difficult to guess that 2π is full circle or 360°.

Properties of trigonometric functions: sine and cosine

In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located at two-dimensional system coordinates

Consider comparison table properties for sine and cosine:

Sine wave	Cosine
y = sin x	y = cos x
ODZ [-1; 1]	ODZ [-1; 1]
sin x = 0, for x = πk, where k ϵ Z	cos x = 0, for x = π/2 + πk, where k ϵ Z
sin x = 1, for x = π/2 + 2πk, where k ϵ Z	cos x = 1, at x = 2πk, where k ϵ Z
sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z	cos x = - 1, for x = π + 2πk, where k ϵ Z
sin (-x) = - sin x, i.e. the function is odd	cos (-x) = cos x, i.e. the function is even
	the function is periodic, shortest period- 2π
sin x › 0, with x belonging to the 1st and 2nd quarters or from 0° to 180° (2πk, π + 2πk)	cos x › 0, with x belonging to the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk)
sin x ‹ 0, with x belonging to the third and fourth quarters or from 180° to 360° (π + 2πk, 2π + 2πk)	cos x ‹ 0, with x belonging to the 2nd and 3rd quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk)
increases in the interval [- π/2 + 2πk, π/2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases on intervals [π/2 + 2πk, 3π/2 + 2πk]	decreases on intervals
derivative (sin x)’ = cos x	derivative (cos x)’ = - sin x

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs coincide, the function is even, in otherwise- odd.

The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:

It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.

Properties of tangentsoids and cotangentsoids

The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values ​​tg and ctg are reciprocals of each other.

1. Y = tan x.
2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
3. Least positive period tangents is equal to π.
4. Tg (- x) = - tg x, i.e. the function is odd.
5. Tg x = 0, for x = πk.
6. The function is increasing.
7. Tg x › 0, for x ϵ (πk, π/2 + πk).
8. Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
9. Derivative (tg x)’ = 1/cos 2 ⁡x.

Let's consider graphic image cotangentoids below in the text.

Main properties of cotangentoids:

1. Y = cot x.
2. Unlike the sine and cosine functions, in the tangentoid Y can take on the values ​​of the set of all real numbers.
3. The cotangentoid tends to the values ​​of y at x = πk, but never reaches them.
4. The smallest positive period of a cotangentoid is π.
5. Ctg (- x) = - ctg x, i.e. the function is odd.
6. Ctg x = 0, for x = π/2 + πk.
7. The function is decreasing.
8. Ctg x › 0, for x ϵ (πk, π/2 + πk).
9. Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
10. Derivative (ctg x)’ = - 1/sin 2 ⁡x Correct

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 coordinate plane. For determining specified 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 a circle has a radius, equal to one, That 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 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 sine value given angle alpha it is necessary to determine the coordinate U on surface.

How given value was received? Since we know that the sine of an arbitrary angle in a right triangle is the ratio opposite leg to the hypotenuse, we get that

And since R=1, That sin(α) = y 0 .

IN 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 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're talking about 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.

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, decide 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's happened number circle? As soon as you ask these harmless questions, the person turns pale and tries to divert the conversation... But in vain. This 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's the point in 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.

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:

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 memorize 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...)

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 first three. But, in Hard time... You understand.)

IN standard tasks, like the ones below, there is a way to do without 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 sinx value, 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

Substitute here known quantity, 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 advice:

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! 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 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.

One of the areas of mathematics that students struggle with the most is trigonometry. It is not surprising: in order to freely master this area of ​​​​knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.

Origins of trigonometry

Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.

Historically, the main object of study in this section mathematical science were right triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values ​​of all parameters of the figure in question using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy and even in art.

First stage

Initially, people talked about the relationship between angles and sides solely using the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in Everyday life this branch of mathematics.

The study of trigonometry in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract problems. trigonometric equations, work with which begins in high school.

Spherical trigonometry

Later, when science reached the next level of development, formulas with sine, cosine, tangent, and cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence at least because earth's surface, and the surface of any other planet is convex, which means that any surface marking will be in three-dimensional space"arc-shaped".

Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.

Right triangle

Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.

The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. It is the longest. We remember that according to the Pythagorean theorem, its numerical value equal to the root of the sum of the squares of the other two sides.

For example, if the two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.

The two remaining sides, which form a right angle, are called legs. In addition, we must remember that the sum of the angles in a triangle is rectangular system coordinates is 180 degrees.

Definition

Finally, with a firm understanding of the geometric basis, one can turn to the definition of sine, cosine and tangent of an angle.

The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse.

Remember that neither sine nor cosine can be more than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.

Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.

Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.

So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.

The simplest formulas

In trigonometry you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.

The first formula that you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you need to know the size of the angle rather than the side.

Many students cannot remember the second formula, which is also very popular when solving school tasks: the sum of one and the square of the tangent of the angle is equal to one divided by the square of the cosine of the angle. Take a closer look: this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation does trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, conversion rules and several basic formulas you can at any time withdraw the required more complex formulas on a piece of paper.

Formulas for double angles and addition of arguments

Two more formulas that you need to learn are related to the values ​​of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of sine and cosine is added.

There are also formulas associated with arguments in the form double angle. They are completely derived from the previous ones - as a training try to get them yourself by taking the alpha angle equal to the angle beta.

Finally, note that double angle formulas can be rearranged to reduce the power of sine, cosine, tangent alpha.

Theorems

The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of ​​the figure, and the size of each side, etc.

The sine theorem states that by dividing the length of each side of a triangle by the opposite angle, we get same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a given triangle.

The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the adjacent angle - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.

Careless mistakes

Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.

Firstly, you shouldn't convert fractions to decimals until you get the final result - you can leave the answer as common fraction, unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will be wasting your time on unnecessary mathematical operations. This is especially true for values ​​such as the root of three or the root of two, because they are found in problems at every step. The same goes for rounding “ugly” numbers.

Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but you will also demonstrate a complete lack of understanding of the subject. This is worse than a careless mistake.

Thirdly, do not confuse the values ​​for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because sine is 30 degrees equal to cosine 60, and vice versa. It is easy to confuse them, as a result of which you will inevitably get an erroneous result.

Application

Many students are in no hurry to start studying trigonometry because they do not understand its practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.

Finally

So you're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.

The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: length three sides and sizes three corners. The only difference in the tasks lies in the fact that different input data are given.

You now know how to find sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a ratio, and a ratio is a fraction, main goal trigonometric problem is finding the roots of an ordinary equation or a system of equations. And here regular school mathematics will help 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. The sine of an 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 the side of a triangle equal to the sum the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them.”

There are two interpretations of the sine theorem: small and extended. According to the small one: “In a triangle, the angles are proportional opposing parties». This theorem 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. It was easy to evaluate angles and sides using the theorems of cosines and sines, breaking down complex figures and objects into “simple” triangles. Engineers often deal with aspect ratio calculations and degree measures, spent a lot of time and effort to calculate 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 different angles. IN Soviet time some teachers forced their students to memorize pages of Bradis tables.

Radian - angular magnitude arcs, length 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).

Cosine table for angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°.

Angle x (in degrees)	0°	30°	45°	60°	90°	120°	135°	150°	180°	210°	225°	240°	270°	300°	315°	330°	360°
Angle x (in radians)	0	π/6	π/4	π/3	π/2	2 x π/3	3 x π/4	5 x π/6	π	7 x π/6	5 x π/4	4 x π/3	3 x π/2	5 x π/3	7 x π/4	11 x π/6	2 x π
cos x	1	√3/2 (0,8660)	√2/2 (0,7071)	1/2 (0,5)	0	-1/2 (-0,5)	-√2/2 (-0,7071)	-√3/2 (-0,8660)	-1	-√3/2 (-0,8660)	-√2/2 (-0,7071)	-1/2 (-0,5)	0	1/2 (0,5)	√2/2 (0,7071)	√3/2 (0,8660)	1