The general formula of the sine in trigonometry. Sine, cosine, tangent and cotangent - everything you need to know at the OGE and the USE

The ratios between the main trigonometric functions - sine, cosine, tangent and cotangent - are given trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this also explains the abundance of trigonometric formulas. Some formulas connect the trigonometric functions of the same angle, others - the functions of a multiple angle, others - allow you to lower the degree, the fourth - to express all functions through the tangent of a half angle, etc.

In this article, we list in order all the basic trigonometric formulas, which are enough to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them according to their purpose, and enter them into tables.

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Basic trigonometric identities

Basic trigonometric identities set the relationship between the sine, cosine, tangent and cotangent of one angle. They follow from the definition of sine, cosine, tangent and cotangent, as well as the concept of the unit circle. They allow you to express one trigonometric function through any other.

For a detailed description of these trigonometry formulas, their derivation and application examples, see the article.

Cast formulas

Cast formulas follow from the properties of sine, cosine, tangent and cotangent, that is, they reflect the property of periodicity of trigonometric functions, the property of symmetry, and also the property of shift by a given angle. These trigonometric formulas allow you to move from working with arbitrary angles to working with angles ranging from zero to 90 degrees.

The rationale for these formulas, a mnemonic rule for memorizing them, and examples of their application can be studied in the article.

Addition Formulas

Trigonometric addition formulas show how the trigonometric functions of the sum or difference of two angles are expressed in terms of the trigonometric functions of these angles. These formulas serve as the basis for the derivation of the following trigonometric formulas.

Formulas for double, triple, etc. angle

Formulas for double, triple, etc. angle (they are also called multiple angle formulas) show how the trigonometric functions of double, triple, etc. angles () are expressed in terms of trigonometric functions of a single angle. Their derivation is based on addition formulas.

More detailed information is collected in the article formulas for double, triple, etc. angle .

Half Angle Formulas

Half Angle Formulas show how the trigonometric functions of a half angle are expressed in terms of the cosine of an integer angle. These trigonometric formulas follow from the double angle formulas.

Their conclusion and examples of application can be found in the article.

Reduction Formulas

Trigonometric formulas for decreasing degrees are designed to facilitate the transition from natural powers of trigonometric functions to sines and cosines in the first degree, but multiple angles. In other words, they allow one to reduce the powers of trigonometric functions to the first.

Formulas for the sum and difference of trigonometric functions

The main purpose sum and difference formulas for trigonometric functions consists in the transition to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equations, as they allow factoring the sum and difference of sines and cosines.

Formulas for the product of sines, cosines and sine by cosine

The transition from the product of trigonometric functions to the sum or difference is carried out through the formulas for the product of sines, cosines and sine by cosine.

Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.

Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.

Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

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We begin our study of trigonometry with a right triangle. Let's define what the sine and cosine are, as well as the tangent and cotangent of an acute angle. These are the basics of trigonometry.

Recall that right angle is an angle equal to 90 degrees. In other words, half of the unfolded corner.

Sharp corner- less than 90 degrees.

Obtuse angle- greater than 90 degrees. In relation to such an angle, "blunt" is not an insult, but a mathematical term :-)

Let's draw a right triangle. A right angle is usually denoted . Note that the side opposite the corner is denoted by the same letter, only small. So, the side lying opposite the angle A is denoted.

An angle is denoted by the corresponding Greek letter.

Hypotenuse A right triangle is the side opposite the right angle.

Legs- sides opposite sharp corners.

The leg opposite the corner is called opposite(relative to angle). The other leg, which lies on one side of the corner, is called adjacent.

Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:

Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:

Tangent acute angle in a right triangle - the ratio of the opposite leg to the adjacent:

Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of an angle to its cosine:

Cotangent acute angle in a right triangle - the ratio of the adjacent leg to the opposite (or, equivalently, the ratio of cosine to sine):

Pay attention to the basic ratios for sine, cosine, tangent and cotangent, which are given below. They will be useful to us in solving problems.

Let's prove some of them.

Okay, we have given definitions and written formulas. But why do we need sine, cosine, tangent and cotangent?

We know that the sum of the angles of any triangle is.

We know the relationship between parties right triangle. This is the Pythagorean theorem: .

It turns out that knowing two angles in a triangle, you can find the third one. Knowing two sides in a right triangle, you can find the third. So, for angles - their ratio, for sides - their own. But what to do if in a right triangle one angle (except for a right one) and one side are known, but you need to find other sides?

This is what people faced in the past, making maps of the area and the starry sky. After all, it is not always possible to directly measure all the sides of a triangle.

Sine, cosine and tangent - they are also called trigonometric functions of the angle- give the ratio between parties and corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.

We will also draw a table of sine, cosine, tangent and cotangent values ​​for "good" angles from to.

Notice the two red dashes in the table. For the corresponding values ​​of the angles, the tangent and cotangent do not exist.

Let's analyze several problems in trigonometry from the Bank of FIPI tasks.

1. In a triangle, the angle is , . Find .

The problem is solved in four seconds.

Insofar as , .

2. In a triangle, the angle is , , . Find .

Let's find by the Pythagorean theorem.

Problem solved.

Often in problems there are triangles with angles and or with angles and . Memorize the basic ratios for them by heart!

For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.

A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.

We considered problems for solving right triangles - that is, for finding unknown sides or angles. But that's not all! In the variants of the exam in mathematics, there are many tasks where the sine, cosine, tangent or cotangent of the outer angle of the triangle appears. More on this in the next article.

I will not convince you not to write cheat sheets. Write! Including cheat sheets on trigonometry. Later I plan to explain why cheat sheets are needed and how cheat sheets are useful. And here - information on how not to learn, but to remember some trigonometric formulas. So - trigonometry without a cheat sheet! We use associations for memorization.

1. Addition formulas:

cosines always "go in pairs": cosine-cosine, sine-sine. And one more thing: cosines are “inadequate”. They “everything is wrong”, so they change the signs: “-” to “+”, and vice versa.

Sinuses - "mix": sine-cosine, cosine-sine.

2. Sum and difference formulas:

cosines always "go in pairs". Having added two cosines - "buns", we get a pair of cosines - "koloboks". And subtracting, we definitely won’t get koloboks. We get a couple of sines. Still with a minus ahead.

Sinuses - "mix" :

3. Formulas for converting a product into a sum and a difference.

When do we get a pair of cosines? When adding the cosines. So

When do we get a pair of sines? When subtracting cosines. From here:

"Mixing" is obtained both by adding and subtracting sines. Which is more fun: adding or subtracting? That's right, fold. And for the formula take addition:

In the first and third formulas in brackets - the amount. From the rearrangement of the places of the terms, the sum does not change. The order is important only for the second formula. But, in order not to get confused, for ease of remembering, in all three formulas in the first brackets we take the difference

and secondly, the sum

Crib sheets in your pocket give peace of mind: if you forget the formula, you can write it off. And they give confidence: if you fail to use the cheat sheet, the formulas can be easily remembered.

Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were developed by astronomers to create an accurate calendar and orientate by the stars. These calculations related to spherical trigonometry, while in the school course they study the ratio of the sides and angle of a flat triangle.

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

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

Basic quantities of trigonometry

The basic trigonometric functions of a numerical argument 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, equal in all directions,” since the proof is given on the example of an isosceles right triangle.

Sine, cosine and other dependencies establish a relationship between acute angles and sides of any right triangle. We give formulas for calculating these quantities for angle A and trace the relationship of trigonometric functions:

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

trigonometric circle

Graphically, the ratio of the mentioned quantities can be represented as follows:

The circle, in this case, represents all possible values ​​of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will be with a “+” sign if α belongs to the I and II quarters of the circle, that is, it is in the range from 0 ° to 180 °. With 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 meaning of the quantities.

The 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 by chance. The designation π in the tables is for radians. Rad is the angle at which the length of a circular arc corresponds to its radius. This value was introduced in order to establish a universal relationship; when calculating in radians, the actual length of the radius in cm does not matter.

The angles in the tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a 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 in a two-dimensional coordinate system.

Consider a comparative table of properties for a sine wave and a cosine wave:

sinusoid	cosine wave
y = sin x	y = cos x
ODZ [-1; one]	ODZ [-1; one]
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, for 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. odd function	cos (-x) = cos x, i.e. the function is even
	the function is periodic, the smallest period is 2π
sin x › 0, with x belonging to quarters I and II or from 0° to 180° (2πk, π + 2πk)	cos x › 0, with x belonging to quarters I and IV or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk)
sin x ‹ 0, with x belonging to quarters III and IV or from 180° to 360° (π + 2πk, 2π + 2πk)	cos x ‹ 0, with x belonging to quarters II and III or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk)
increases on the interval [- π/2 + 2πk, π/2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases on the intervals [ π/2 + 2πk, 3π/2 + 2πk]	decreases in 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 signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs are the same, the function is even; otherwise, it is odd.

The introduction of radians and the enumeration of the main properties of the sinusoid and cosine wave allow us to bring the following pattern:

It is very easy to verify the correctness of the formula. For example, for x = π/2, the sine is equal to 1, as is the cosine of x = 0. Verification can be done by looking at tables or by tracing function curves for given values.

Properties of tangentoid and cotangentoid

The graphs of the tangent and cotangent functions differ significantly from the sinusoid and cosine wave. The values ​​tg and ctg are inverse to each other.

1. Y = tgx.
2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
3. The smallest positive period of the tangentoid is π.
4. Tg (- x) \u003d - 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 .

Consider the graphical representation of the cotangentoid below in the text.

The main properties of the cotangentoid:

1. Y = ctgx.
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 the cotangentoid is π.
5. Ctg (- x) \u003d - 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 Fix