Formula for the sine of the sum of two angles. Examples with solutions to problems using trigonometric identities

In this article we will take a comprehensive look. Basic trigonometric identities represent equalities that establish a connection between sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other.

Let us immediately list the main trigonometric identities that we will analyze in this article. Let's write them down in a table, and below we'll give the output of these formulas and provide the necessary explanations.

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Relationship between sine and cosine of one angle

Sometimes they do not talk about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind . The explanation for this fact is quite simple: the equalities are obtained from the main trigonometric identity after dividing both of its parts by and, respectively, and the equalities And follow from the definitions of sine, cosine, tangent and cotangent. We'll talk about this in more detail in the following paragraphs.

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

Before proving the main trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.

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

Tangent and cotangent through sine and cosine

Identities connecting tangent and cotangent with sine and cosine of one angle of view and follow immediately from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, sine is the ordinate of y, cosine is the abscissa of x, tangent is the ratio of the ordinate to the abscissa, that is, , and the cotangent is the ratio of the abscissa to the ordinate, that is, .

Thanks to such obviousness of the identities and Tangent and cotangent are often defined not through the ratio of abscissa and ordinate, but through the ratio of sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.

In conclusion of this paragraph, it should be noted that the identities and take place for all angles at which the trigonometric functions included in them make sense. So the formula is valid for any , other than (otherwise the denominator will have zero, and we did not define division by zero), and the formula - for all , different from , where z is any .

Relationship between tangent and cotangent

An even more obvious trigonometric identity than the previous two is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it holds for any angles other than , otherwise either the tangent or the cotangent are not defined.

Proof of the formula very simple. By definition and from where . The proof could have been carried out a little differently. Since , That .

So, the tangent and cotangent of the same angle at which they make sense are .

Reference information on the trigonometric functions sine (sin x) and cosine (cos x). Geometric definition, properties, graphs, formulas. Table of sines and cosines, derivatives, integrals, series expansions, secant, cosecant. Expressions through complex variables. Connection with hyperbolic functions.

Geometric definition of sine and cosine

|BD|- length of the arc of a circle with center at a point A.
α - angle expressed in radians.

Definition
Sine (sin α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the hypotenuse |AC|.

Cosine (cos α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the hypotenuse |AC|.

Accepted notations

;
;
.

;
;
.

Graph of the sine function, y = sin x

Graph of the cosine function, y = cos x

Properties of sine and cosine

Periodicity

Functions y = sin x and y = cos x periodic with period 2π.

Parity

The sine function is odd. The cosine function is even.

Domain of definition and values, extrema, increase, decrease

The sine and cosine functions are continuous in their domain of definition, that is, for all x (see proof of continuity). Their main properties are presented in the table (n - integer).

Basic formulas

Sum of squares of sine and cosine

Formulas for sine and cosine from sum and difference

;
;

Formulas for the product of sines and cosines

Sum and difference formulas

Expressing sine through cosine

;
;
;
.

Expressing cosine through sine

;
;
;
.

Expression through tangent

; .

When , we have:
; .

At :
; .

Table of sines and cosines, tangents and cotangents

This table shows the values ​​of sines and cosines for certain values ​​of the argument.

Expressions through complex variables

;

Euler's formula

{ -∞ < x < +∞ }

Secant, cosecant

Inverse functions

Inverse functions to sine and cosine are arcsine and arccosine, respectively.

Arcsine, arcsin

Arccosine, arccos

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

At the very beginning of this article, we examined the concept of trigonometric functions. Their main purpose is to study the basics of trigonometry and study periodic processes. And it was not in vain that we drew the trigonometric circle, because in most cases trigonometric functions are defined as the ratio of the sides of a triangle or its certain segments in a unit circle. I also mentioned the undeniably enormous importance 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 provisions to real, and sometimes to complex numbers.

Trigonometry formulas There are several types. Let's look at them in order.

1. Ratios of trigonometric functions of the same angle

Here we come to consider such a concept as basic trigonometric identities.

A trigonometric identity is an equality that consists of trigonometric relations and which holds for all values ​​of the angles that are included in it.

Let's look at the most important trigonometric identities and their proofs:

The first identity follows from the very definition of tangent.

Take a right triangle that has an acute angle x at vertex A.



To prove the identities, you need to use the Pythagorean theorem:

(BC) 2 + (AC) 2 = (AB) 2

Now we divide both sides of the equality by (AB) 2 and recalling the definitions of sin and cos angle, we obtain the second identity:

(BC) 2 /(AB) 2 + (AC) 2 /(AB) 2 = 1

sin x = (BC)/(AB)

cos x = (AC)/(AB)

sin 2 x + cos 2 x = 1

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

To do this, divide both sides of the second identity by cos 2 x:

sin 2 x/ cos 2 x + cos 2 x/ cos 2 x = 1/ cos 2 x

sin 2 x/ cos 2 x + 1 = 1/ cos 2 x

Based on the first identity tg x = sin x /cos x we ​​obtain the third:

1 + tan 2 x = 1/cos 2 x

Now let's divide the second identity by sin 2 x:

sin 2 x/ sin 2 x + cos 2 x/ sin 2 x = 1/ sin 2 x

1+ cos 2 x/ sin 2 x = 1/ sin 2 x

cos 2 x/ sin 2 x is nothing more than 1/tg 2 x, so we get the fourth identity:

1 + 1/tg 2 x = 1/sin 2 x

It's time to remember the theorem about the sum of interior angles of a triangle, which states that the sum of the angles of a triangle = 180 0. It turns out that at vertex B of the triangle there is an angle whose value is 180 0 – 90 0 – x = 90 0 – x.



Let us again recall the definitions for sin and cos and obtain the fifth and sixth identities:

sin x = (BC)/(AB)

cos(90 0 – x) = (BC)/(AB)

cos(90 0 – x) = sin x

Now let's do the following:

cos x = (AC)/(AB)

sin(90 0 – x) = (AC)/(AB)

sin(90 0 – x) = cos x

As you can see, everything is elementary here.

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



2. Expressing trigonometric functions through each other

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







3. The following are the formulas for adding and subtracting angles:



4. Formulas for double, triple and half angles.

   I note that they all stem from the previous formulas.

sin 2x =2sin x*cos x

cos 2x =cos 2 x -sin 2 x =1-2sin 2 x =2cos 2 x -1

tg 2x = 2tgx/(1 - tg 2 x)

сtg 2x = (сtg 2 x - 1) /2сtg x

sin3x =3sin x - 4sin 3 x

cos3х =4cos 3 x - 3cos x

tg 3x = (3tgx – tg 3 x) /(1 - 3tg 2 x)

сtg 3x = (сtg 3 x – 3сtg x) /(3сtg 2 x - 1)









5. Formulas for converting trigonometric expressions:

Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.

Geometric definition

|BD| - length of the arc of a circle with center at point A.
α is the angle expressed in radians.

Tangent ( tan α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| .

Cotangent ( ctg α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .

Tangent

Where n- whole.

In Western literature, tangent is denoted as follows:
.
;
;
.

Graph of the tangent function, y = tan x

Cotangent

Where n- whole.

In Western literature, cotangent is denoted as follows:
.
The following notations are also accepted:
;
;
.

Graph of the cotangent function, y = ctg x

Properties of tangent and cotangent

Periodicity

Functions y = tg x and y = ctg x are periodic with period π.

Parity

The tangent and cotangent functions are odd.

Areas of definition and values, increasing, decreasing

The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table ( n- whole).

Formulas

Expressions using sine and cosine

; ;
; ;
;

Formulas for tangent and cotangent from sum and difference

The remaining formulas are easy to obtain, for example

Product of tangents

Formula for the sum and difference of tangents

This table presents the values ​​of tangents and cotangents for certain values ​​of the argument.

Expressions using complex numbers

Expressions through hyperbolic functions

;
;

Derivatives

; .

.
Derivative of the nth order with respect to the variable x of the function:
.
Deriving formulas for tangent > > > ; for cotangent > > >

Integrals

Series expansions

To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in power series for functions sin x And cos x and divide these polynomials by each other, . In this case it turns out following formulas.

At .

at .
Where Bn- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to Laplace's formula:

Inverse functions

The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively.

Arctangent, arctg

, Where n- whole.

Arccotangent, arcctg

, Where n- whole.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.

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