All major trigonometric formulas. The most necessary trigonometric formulas

This is the last and most important lesson necessary for solving problems B11. We already know how to translate angles from the radian measure to a degree (see the lesson "Radian and degree of the corner"), and also know how to identify the sign of a trigonometric function, focusing on the coordinate quarters (see the lesson "Signs of trigonometric functions").

The point is left behind: to calculate the value of the function itself - the same number that is written in response. Here, the main trigonometric identity comes to the rescue.

Basic trigonometric identity. For any angle α, the assertion is true:

sIN 2 α + COS 2 α \u003d 1.

This formula binds sinus and cosine of one angle. Now, knowing sinus, we will easily find cosine - and vice versa. It is enough to remove the square root:

Pay attention to the "±" sign before roots. The fact is that from the main trigonometric identity it is not clear what the original sinus and cosine was: positive or negative. After all, the construction of a square is an even function that "burns" all cons (if they were).

That is why in all tasks of B11, which are found in the exam in mathematics, there are necessarily additional conditions that help get rid of uncertainty with signs. Usually this is an indication of a coordinate quarter for which you can define a sign.

Attentive reader will certainly ask: "And what about Tangent and Kotangent?" Directly calculate these functions from the above formulas can not. However, there are important consequences from the main trigonometric identity that already contain tangents and catanges. Namely:

Important: For any angle α, you can rewrite the main trigonometric identity as follows:

These equations are easily derived from the main identity - it is sufficient to divide both sides on COS 2 α (to obtain a tangent) or SIN 2 α (for Cotangent).

Consider all this on specific examples. Below are the true B11 tasks that are taken from the trial options of the EGE in mathematics 2012.

We are known to cosine, but is unknown sinus. The main trigonometric identity (in the "pure" form) connects just these functions, so we will work with it. We have:

sIN 2 α + COS 2 α \u003d 1 ⇒ SIN 2 α + 99/100 \u003d 1 ⇒ SIN 2 α \u003d 1/100 ⇒ Sin α \u003d ± 1/10 \u003d ± 0.1.

To solve the problem, it remains to find a sinus sign. Since the angle α ∈ (π / 2; π), then this is written to the degree: α ∈ (90 °; 180 °).

Consequently, the angle α lies in the II coordinate quarter - all sines are positive there. Therefore, sin α \u003d 0.1.

So, we are famous for sinus, and you need to find a cosine. Both of these functions are mainly trigonometric identity. We substitute:

sIN 2 α + COS 2 α \u003d 1 ⇒ 3/4 + COS 2 α \u003d 1 ⇒ COS 2 α \u003d 1/4 ⇒ COS α \u003d ± 1/2 \u003d ± 0.5.

It remains to deal with the sign before the fraction. What to choose: plus or minus? By condition, the angle α belongs to the gap (π 3π / 2). We translate the angles from the radian measure to the degree - we obtain: α ∈ (180 °; 270 °).

Obviously, this is the III coordinate quarter, where all cosines are negative. Therefore, Cos α \u003d -0.5.

A task. Find TG α if the following is known:

Tangent and cosine are associated with the equation as follows from the main trigonometric identity:

We obtain: TG α \u003d ± 3. The tangent sign is determined by the angle α. It is known that α ∈ (3π / 2; 2π). We translate angles from the radian measure to the degree - we obtain α ∈ (270 °; 360 °).

Obviously, this is an IV coordinate quarter, where all tangents are negative. Therefore, Tg α \u003d -3.

A task. Find COS α if the following is known:

Again the sinus is known and unknown cosine. We write the main trigonometric identity:

sIN 2 α + COS 2 α \u003d 1 ⇒ 0,64 + COS 2 α \u003d 1 ⇒ COS 2 α \u003d 0.36 ⇒ COS α \u003d ± 0.6.

Sign determining the corner. We have: α ∈ (3π / 2; 2π). We translate the angles from the degree measure to the radian: α ∈ (270 °; 360 °) is the IV coordinate quarter, the cosines are positive there. Consequently, Cos α \u003d 0.6.

A task. Find SIN α if the following is known:

We write down the formula that follows from the main trigonometric identity and directly connects the sinus and Kotangent:

From here we obtain that Sin 2 α \u003d 1/25, i.e. Sin α \u003d ± 1/5 \u003d ± 0.2. It is known that the angle α ∈ (0; π / 2). In degree, this is written as follows: α ∈ (0 °; 90 °) - I coordinate a quarter.

So, the angle is in the first coordinate quarter - all trigonometric functions are positive there, therefore Sin α \u003d 0.2.

At the very beginning of this article, we looked at the concept of trigonometric functions. The main purpose of their purpose is the study of the foundations of trigonometry and the study of periodic processes. And we did not paint trigonometric circle, because in most cases trigonometric functions are defined as the ratio of the parties of the triangle or its certain segments in a single circle. I also mentioned the indisputably huge value of trigonometry in modern life. But science does not stand still, as a result we can significantly expand the scope of trigonometry and transfer its position to real, and sometimes to complex numbers.

Trigonometry formulas There are several types. Consider them in order.

The ratio of the trigonometric functions of the same angle

Here we approached the consideration of such a thing as basic trigonometric identities.

Trigonometric identity is equality that consists of trigonometric ratios and which is performed for all values \u200b\u200bof the values \u200b\u200bof the corners that are included in it.

Consider the most important trigonometric identities and their evidence:

The first identity follows from the very determination of the tangent.

Take a rectangular triangle in which there is an acute angle x at the top of A.

To prove identities, it is necessary to use the Pythagora theorem:

(Sun) 2 + (AC) 2 \u003d (AB) 2

Now we divide on (AB) 2 of both parts of equality and remembering the definition of SIN and COS angle, we get a second identity:

(Sun) 2 / (AB) 2 + (AC) 2 / (AB) 2 \u003d 1

sIN X \u003d (BC) / (AB)

cOS X \u003d (AC) / (AB)

sIN 2 X + COS 2 x \u003d 1

To prove the third and fourth identities, we use previous proof.

To do this, both parts of the second identity are divided into COS 2 x:

sIN 2 X / COS 2 X + COS 2 X / COS 2 X \u003d 1 / COS 2 x

sIN 2 X / COS 2 X + 1 \u003d 1 / COS 2 x

Based on the first identity of TG X \u003d SIN X / COS X, we get the third:

1 + TG 2 x \u003d 1 / COS 2 x

Now we divide the second identity on SIN 2 x:

sIN 2 X / SIN 2 X + COS 2 X / SIN 2 X \u003d 1 / SIN 2 x

1+ cos 2 x / sin 2 x \u003d 1 / sin 2 x

cOS 2 X / SIN 2 X is nothing but 1 / TG \u200b\u200b2 x, so we get a fourth identity:

1 + 1 / TG \u200b\u200b2 x \u003d 1 / Sin 2 x

It is time to recall the theorem about the sum of the inner angles of the triangle, which states that the sum of the corners of the triangle \u003d 180 0. It turns out that at the top in the triangle there is an angle, the value of which is 180 0 - 90 0 - x \u003d 90 0 - x.

Recall the definitions for SIN and COS again and we get the fifth and sixth identities:

sIN X \u003d (BC) / (AB)

cOS (90 0 - X) \u003d (BC) / (AB)

cOS (90 0 - X) \u003d SIN X

Now do the following:

cOS X \u003d (AC) / (AB)

sIN (90 0 - X) \u003d (AC) / (AB)

sIN (90 0 - X) \u003d COS X

As you can see, everything is elementary here.

There are other identities that are used in solving mathematical identities, I will bring them simply in the form of reference information, because they all stem from the above.

Expressions of trigonometric functions in each other

(The choice of the sign before the root is determined by which the corner is located in the circle?)

Next, follow the formulas for addition and subtracting the corners:

Formulas of double, triple and half corners.

I note that they all stem from the previous formulas.

sin 2x \u003d 2sin x * cos x

cos 2x \u003d cos 2 x -sin 2 x \u003d 1-2sin 2 x \u003d 2cos 2 x -1

tG 2X \u003d 2TGX / (1 - TG 2 x)

cTG 2X \u003d (CTG 2 x - 1) / 2Stg x

sin3x \u003d 3sin x - 4sin 3 x

cos3x \u003d 4cos 3 x - 3cos x

tG 3X \u003d (3TGX - TG 3 x) / (1 - 3TG 2 x)

cTG 3X \u003d (CTG 3 x - 3Stg x) / (3ctg 2 x - 1)

Trigonometric expressions conversion formulas:

Trigonometric identities - These are equalities that establish a link between sine, cosine, tangent and catangent of one angle, which allows you to find any of these functions, provided that any other will be known.

tG \\ ALPHA \u003d \\ FRAC (\\ sin \\ alpha) (\\ cos \\ alpha), \\ Enspace CTG \\ Alpha \u003d \\ FRAC (\\ COS \\ Alpha) (\\ Sin \\ Alpha)

tG \\ ALPHA \\ CDOT CTG \\ Alpha \u003d 1

This identity suggests that the sum of the square of the sinus of one angle and the cosine square of one angle is equal to one, which in practice it makes it possible to calculate the sine of one angle when its cosine is known and vice versa.

When converting trigonometric expressions, this identity is very often used, which allows the unit to replace the amount of cosine and sinus squares of one angle and also produce a replacement operation in the reverse order.

Finding Tangent and Kotangence through sinus and cosine

tG \\ ALPHA \u003d \\ FRAC (\\ Sin \\ Alpha) (\\ COS \\ Alpha), \\ Enspace

These identities are formed from the definitions of sinus, cosine, tangent and catangens. After all, if you figure it out, then by definition of the ordinate y is sinus, and the x - cosine abscissa. Then the tangent will be equal to attitude \\ FRAC (Y) (X) \u003d \\ FRAC (\\ sin \\ alpha) (\\ cos \\ alpha), and attitude \\ FRAC (X) (Y) \u003d \\ FRAC (\\ COS \\ ALPHA) (\\ Sin \\ Alpha) - Will be a catangent.

We add that only for such angles \\ Alpha, in which trigonometric functions included in them make sense, identity will take place, cTG \\ Alpha \u003d \\ FRAC (\\ COS \\ Alpha) (\\ Sin \\ Alpha).

For example: tG \\ ALPHA \u003d \\ FRAC (\\ Sin \\ Alpha) (\\ COS \\ Alpha) is just for the angles \\ alpha, which are different from \\ FRAC (\\ pi) (2) + \\ pi z, but cTG \\ Alpha \u003d \\ FRAC (\\ COS \\ Alpha) (\\ Sin \\ Alpha) - For an angle \\ Alpha, different from \\ pi z, Z - is an integer.

Dependence between Tangent and Kotangen

tG \\ ALPHA \\ CDOT CTG \\ Alpha \u003d 1

This identity is valid only for such angles \\ alpha, which are different from \\ FRAC (\\ PI) (2) Z. Otherwise or Cotangent or Tangent will not be determined.

Relying on the above items, we get that tG \\ ALPHA \u003d \\ FRAC (Y) (X), but cTG \\ Alpha \u003d \\ FRAC (X) (Y). Hence it follows that tG \\ ALPHA \\ CDOT CTG \\ Alpha \u003d \\ FRAC (Y) (X) \\ CDOT \\ FRAC (X) (Y) \u003d 1. Thus, tangent and catangenes of one angle, in which they make sense are mutually reverse numbers.

Dependencies between tangent and cosine, catangenes and sine

tG ^ (2) \\ alpha + 1 \u003d \\ FRAC (1) (\\ COS ^ (2) \\ Alpha) - The sum of the square of the tangent of the angle \\ alpha and 1 is equal to the reverse square of the cosine of this angle. This identity is true for all \\ Alpha, other than \\ FRAC (\\ pi) (2) + \\ pi z.

1 + CTG ^ (2) \\ alpha \u003d \\ FRAC (1) (\\ sin ^ (2) \\ Alpha) - Amount 1 and the square of the corner of the angle \\ alpha is equal to the reverse square of the sinus of this angle. This identity is valid for any \\ alpha, different from \\ pi z.

Examples with task solutions for the use of trigonometric identities

Example 1.

Find \\ Sin \\ Alpha and TG \\ Alpha if \\ COS \\ Alpha \u003d - \\ FRAC12 and \\ FRAC (\\ PI) (2)< \alpha < \pi ;

Show a decision

Decision

Functions \\ sin \\ alpha and \\ cos \\ alpha binds formula \\ sin ^ (2) \\ alpha + \\ cos ^ (2) \\ alpha \u003d 1. Substituting into this formula \\ COS \\ Alpha \u003d - \\ FRAC12We will get:

\\ Sin ^ (2) \\ alpha + \\ left (- \\ FRAC12 \\ RIGHT) ^ 2 \u003d 1

This equation has 2 solutions:

\\ Sin \\ Alpha \u003d \\ PM \\ SQRT (1- \\ FRAC14) \u003d \\ PM \\ FRAC (\\ SQRT 3) (2)

By condition \\ FRAC (\\ PI) (2)< \alpha < \pi . In the second quarter sinus is positive, so \\ sin \\ alpha \u003d \\ FRAC (\\ SQRT 3) (2).

In order to find TG \\ Alpha, we use the formula tG \\ ALPHA \u003d \\ FRAC (\\ Sin \\ Alpha) (\\ COS \\ Alpha)

tG \\ ALPHA \u003d \\ FRAC (\\ SQRT 3) (2): \\ FRAC12 \u003d \\ SQRT 3

Example 2.

Find \\ cos \\ alpha and ctg \\ alpha, if \\ FRAC (\\ PI) (2)< \alpha < \pi .

Show a decision

Decision

Substituting in the formula \\ sin ^ (2) \\ alpha + \\ cos ^ (2) \\ alpha \u003d 1 given by condition number \\ sin \\ alpha \u003d \\ FRAC (\\ SQRT3) (2)Receive \\ left (\\ FRAC (\\ SQRT3) (2) \\ RIGHT) ^ (2) + \\ cos ^ (2) \\ alpha \u003d 1. This equation has two solutions \\ COS \\ Alpha \u003d \\ PM \\ SQRT (1- \\ FRAC34) \u003d \\ PM \\ SQRT \\ FRAC14.

By condition \\ FRAC (\\ PI) (2)< \alpha < \pi . In the second quarter, the cosine is negative, so \\ COS \\ ALPHA \u003d - \\ SQRT \\ FRAC14 \u003d - \\ FRAC12.

In order to find CTG \\ Alpha, we use the formula cTG \\ Alpha \u003d \\ FRAC (\\ COS \\ Alpha) (\\ Sin \\ Alpha). Appropriate values \u200b\u200bare known to us.

cTG \\ Alpha \u003d - \\ FRAC12: \\ FRAC (\\ SQRT3) (2) \u003d - \\ FRAC (1) (\\ SQRT 3).

Reference data for trigonometric Sine functions (SIN X) and cosine (COS X). Geometrical definition, properties, graphs, formulas. Table of sinuses and cosines, derivatives, integrals, decompositions in the ranks, sessions, mossens. Expressions through complex variables. Communication with hyperbolic functions.

Geometric definition of sinus and cosine

| BD | - Arc length of circle with center at point A..
α - angle, expressed in radians.

Definition
Sinus (sin α) - It is a trigonometric function depending on the angle α between the hypothenooma and a rigid triangle cathet, equal to the ratio of the length of the opposite category | BC | To the length of hypotenuse | AC |.

Cosine (COS α) - It is a trigonometric function, depending on the angle α between the hypothenooma and the cathe of the rectangular triangle, equal to the ratio of the length of the adjacent category | AB | To the length of hypotenuse | AC |.

Accepted designations

;
;
.

Sinus function graph, y \u003d sin x

Schedule Function Kosinus, Y \u003d COS X

Properties of sinus and cosine

Periodicity

Functions y \u003d sIN X. and y \u003d. cOS X. Periodic with a period 2 π..

Parity

The sinus function is odd. The cosine function is even.

Scope of definition and values, extremes, increasing, decrease

The functions of sine and cosine are continuous on their definition area, that is, for all x (see proof of continuity). Their basic properties are presented in table (n - whole).

	y \u003d. sIN X.	y \u003d. cOS X.
Definition and continuity area	- ∞ < x < + ∞	- ∞ < x < + ∞
Region of values	-1 ≤ y ≤ 1	-1 ≤ y ≤ 1
Ascending
Disarmament
Maxima, y \u200b\u200b\u003d 1
Minima, y \u200b\u200b\u003d - 1
Zeros, y \u003d 0
Point of intersection with the ordinate axis, x \u003d 0	y \u003d. 0	y \u003d. 1

Basic formulas

Sinus and cosine squares

Formulas of sinus and cosine from the amount and difference

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;

Formulas works of sinuses and cosine

Formulas of the sum and difference

Sinus expression through cosine

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;
;
.

Cosine expression through sinus

;
;
;
.

Expression through tangent

; .

When we have:
; .

With:
; .

Sinus and Cosine Table, Tangents and Kotangers

This table shows the values \u200b\u200bof sinuses and cosines at some values \u200b\u200bof the argument.

Expressions through complex variables

;

Formula Euler

{ -∞ < x < +∞ }

Sean, Kosakhans

Reverse functions

Inverse functions to sinus and cosine are arcsinus and arquosine, respectively.

Arksinus, Arcsin.

Arkkosinus, Arccos.

References:
I.N. Bronstein, K.A. Semendyaev, a reference book on mathematics for engineers and students of the attendants, "Lan", 2009.

In this article, we will comprehensively consider. The main trigonometric identities are equivals that establish the relationship between sine, cosine, tangent and catangent of one angle, and allow you to find any of these trigonometric functions through a well-known other.

Immediately list the basic trigonometric identities that we will analyze in this article. We write them to the table, and below we will give the output of these formulas and give the necessary explanations.

Navigating page.

Communication between sine and cosine of one corner

Sometimes they say not about the basic trigonometric identities listed in the table above, but about one single the main trigonometric identity View . Explanation of this fact is quite simple: equality is obtained from the main trigonometric identity after dividing both parts of it on and, accordingly, and equality and Follow the definitions of sinus, cosine, tangent and catangens. We will talk about this in the following paragraphs.

That is, it is particular interest to the equality that the name of the main trigonometric identity was given.

Before proving the main trigonometric identity, we will give it the wording: the sum of the squares of the sine and the cosine of one angle is identically equal to one. Now we prove it.

The main trigonometric identity is very often used when transformation of trigonometric expressions. It allows the sum of the squares of the sine and the cosine of one angle to replace the unit. No less often the main trigonometric identity is used in the reverse order: the unit is replaced by the sum of the sinus squares and the cosine of any corner.

Tangent and Kotangenes through sinus and cosine

Identities bonding tangent and catangenes with sine and cosine of one angle of type and Immediately follow the definitions of sinus, cosine, tangent and catangent. Indeed, by definition sinus there is an order y, cosine is the abscissa X, Tangent is the ratio of the ordinate to the abscissa, that is, , and Kothangence is the abscissa ratio to ordinate, that is, .

Due to the evidence of identities and Often the definitions of Tangent and Kotangenes give not through the ratio of the abscissa and ordinate, but through the ratio of sinus and cosine. So a tangent of the angle is called the ratio of the sinus to the cosine of this angle, and Kotangent is the attitude of the cosine to sinus.

In conclusion of this item, it should be noted that identities and They take place for all such angles in which trigonometric functions in them make sense. So the formula is valid for any other than (otherwise in the denominator will be zero, and we did not define the division to zero), and the formula - For all other than the Z - any.

Communication between Tangent and Kotangen

An even more apparent trigonometric identity than two previous ones is a identity that connects the Tangent and Cotangent of one angle of type . It is clear that it takes place for any angles other than, otherwise, either Tangent, or Cotangenes is not defined.

Proof of formula very simple. By definition and where . It was possible to spend proof and a little different. As I. T. .

So, Tangent and Kotnence of the same angle, in which they make sense.