Sine, cosine, tangent: what is it? How to find sine, cosine and tangent? Universal trigonometric substitution, derivation of formulas, examples.

I won't try to convince you not to write cheat sheets. Write! Including cheat sheets on trigonometry. Later I plan to explain why cheat sheets are needed and why cheat sheets are useful. And here is 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 “come in pairs”: cosine-cosine, sine-sine. And one more thing: cosines are “inadequate”. “Everything is not right” for them, so they change the signs: “-” to “+”, and vice versa.

Sinuses - “mix”: sine-cosine, cosine-sine.

2. Sum and difference formulas:

cosines always “come in pairs”. By adding two cosines - “koloboks”, we get a pair of cosines - “koloboks”. And by subtracting, we definitely won’t get any koloboks. We get a couple of sines. Also with a minus ahead.

Sinuses - “mix” :

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

When do we get a cosine pair? When we add cosines. That's why

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

“Mixing” is obtained both when adding and subtracting sines. What's more fun: adding or subtracting? That's right, fold. And for the formula they take addition:

In the first and third formulas, the sum is in parentheses. Rearranging the places of the terms does not change the sum. 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 amount

Cheat sheets in your pocket give you peace of mind: if you forget the formula, you can copy it. And they give you confidence: if you fail to use the cheat sheet, you can easily remember the formulas.

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).

	y = sin x	y = cos x
Scope and continuity	- ∞ < x < + ∞	- ∞ < x < + ∞
Range of values	-1 ≤ y ≤ 1	-1 ≤ y ≤ 1
Increasing
Descending
Maxima, y ​​= 1
Minima, y ​​= - 1
Zeros, y = 0
Intercept points with the ordinate axis, x = 0	y = 0	y = 1

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

The inverse functions of 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.

– there will certainly be tasks on trigonometry. Trigonometry is often disliked for the need to cram a huge number of difficult formulas, teeming with sines, cosines, tangents and cotangents. The site already once gave advice on how to remember a forgotten formula, using the example of the Euler and Peel formulas.

And in this article we will try to show that it is enough to firmly know only five simple trigonometric formulas, and have a general understanding of the rest and derive them as you go. It’s like with DNA: the molecule does not store the complete blueprints of a finished living creature. Rather, it contains instructions for assembling it from available amino acids. So in trigonometry, knowing some general principles, we will get all the necessary formulas from a small set of those that must be kept in mind.

We will rely on the following formulas:

From the formulas for sine and cosine sums, knowing about the parity of the cosine function and the oddness of the sine function, substituting -b instead of b, we obtain formulas for differences:

1. Sine of the difference: sin(a-b) = sinacos(-b)+cosasin(-b) = sinacosb-cosasinb
2. Cosine of the difference: cos(a-b) = cosacos(-b)-sinasin(-b) = cosacosb+sinasinb

Putting a = b into the same formulas, we obtain the formulas for sine and cosine of double angles:

1. Sine of double angle: sin2a = sin(a+a) = sinacosa+cosasina = 2sinacosa
2. Cosine of double angle: cos2a = cos(a+a) = cosacosa-sinasina = cos2 a-sin2 a

The formulas for other multiple angles are obtained similarly:

1. Sine of a triple angle: sin3a = sin(2a+a) = sin2acosa+cos2asina = (2sinacosa)cosa+(cos2 a-sin2 a)sina = 2sinacos2 a+sinacos2 a-sin 3 a = 3 sinacos2 a-sin 3 a = 3 sina(1-sin2 a)-sin 3 a = 3 sina-4sin 3a
2. Cosine of triple angle: cos3a = cos(2a+a) = cos2acosa-sin2asina = (cos2 a-sin2 a)cosa-(2sinacosa)sina = cos 3 a- sin2 acosa-2sin2 acosa = cos 3 a-3 sin2 acosa = cos 3 a-3(1- cos2 a)cosa = 4cos 3 a-3 cosa

Before we move on, let's look at one problem.
Given: the angle is acute.
Find its cosine if
Solution given by one student:
Because , That sina= 3,a cosa = 4.
(From math humor)

So, the definition of tangent relates this function to both sine and cosine. But you can get a formula that relates the tangent only to the cosine. To derive it, we take the main trigonometric identity: sin 2 a+cos 2 a= 1 and divide it by cos 2 a. We get:

So the solution to this problem would be:

(Since the angle is acute, when extracting the root, the + sign is taken)

The formula for the tangent of a sum is another one that is difficult to remember. Let's output it like this:

Immediately displayed and

From the cosine formula for a double angle, you can obtain the sine and cosine formulas for half angles. To do this, to the left side of the double angle cosine formula:
cos2 a = cos 2 a-sin 2 a
we add one, and to the right - a trigonometric unit, i.e. the sum of the squares of sine and cosine.
cos2a+1 = cos2 a-sin2 a+cos2 a+sin2 a
2cos 2 a = cos2 a+1
Expressing cosa through cos2 a and performing a change of variables, we get:

The sign is taken depending on the quadrant.

Similarly, subtracting one from the left side of the equality and the sum of the squares of the sine and cosine from the right, we get:
cos2a-1 = cos2 a-sin2 a-cos2 a-sin2 a
2sin 2 a = 1-cos2 a

And finally, to convert the sum of trigonometric functions into a product, we use the following technique. Let's say we need to represent the sum of sines as a product sina+sinb. Let's introduce variables x and y such that a = x+y, b+x-y. Then
sina+sinb = sin(x+y)+ sin(x-y) = sin x cos y+ cos x sin y+ sin x cos y- cos x sin y=2 sin x cos y. Let us now express x and y in terms of a and b.

Since a = x+y, b = x-y, then . That's why

You can withdraw immediately

1. Formula for partitioning products of sine and cosine V amount: sinacosb = 0.5(sin(a+b)+sin(a-b))

We recommend that you practice and derive formulas on your own for converting the difference of sines and the sum and difference of cosines into the product, as well as for dividing the products of sines and cosines into the sum. Having completed these exercises, you will thoroughly master the skill of deriving trigonometric formulas and will not get lost even in the most difficult test, olympiad or testing.

Formulas for the sum and difference of sines and cosines for two angles α and β allow us to move from the sum of these angles to the product of angles α + β 2 and α - β 2. Let us immediately note that you should not confuse the formulas for the sum and difference of sines and cosines with the formulas for sines and cosines of the sum and difference. Below we list these formulas, give their derivations and show examples of application for specific problems.

Yandex.RTB R-A-339285-1

Formulas for the sum and difference of sines and cosines

Let's write down what the sum and difference formulas look like for sines and cosines

Sum and difference formulas for sines

sin α + sin β = 2 sin α + β 2 cos α - β 2 sin α - sin β = 2 sin α - β 2 cos α + β 2

Sum and difference formulas for cosines

cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β = - 2 sin α + β 2 cos α - β 2 , cos α - cos β = 2 sin α + β 2 · β - α 2

These formulas are valid for any angles α and β. The angles α + β 2 and α - β 2 are called the half-sum and half-difference of the angles alpha and beta, respectively. Let us give the formulation for each formula.

Definitions of formulas for sums and differences of sines and cosines

Sum of sines of two angles is equal to twice the product of the sine of the half-sum of these angles and the cosine of the half-difference.

Difference of sines of two angles is equal to twice the product of the sine of the half-difference of these angles and the cosine of the half-sum.

Sum of cosines of two angles is equal to twice the product of the cosine of the half-sum and the cosine of the half-difference of these angles.

Difference of cosines of two angles is equal to twice the product of the sine of the half-sum and the cosine of the half-difference of these angles, taken with a negative sign.

Deriving formulas for the sum and difference of sines and cosines

To derive formulas for the sum and difference of the sine and cosine of two angles, addition formulas are used. Let's list them below

sin (α + β) = sin α · cos β + cos α · sin β sin (α - β) = sin α · cos β - cos α · sin β cos (α + β) = cos α · cos β - sin α sin β cos (α - β) = cos α cos β + sin α sin β

Let’s also imagine the angles themselves as a sum of half-sums and half-differences.

α = α + β 2 + α - β 2 = α 2 + β 2 + α 2 - β 2 β = α + β 2 - α - β 2 = α 2 + β 2 - α 2 + β 2

We proceed directly to the derivation of the sum and difference formulas for sin and cos.

Derivation of the formula for the sum of sines

In the sum sin α + sin β, we replace α and β with the expressions for these angles given above. We get

sin α + sin β = sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2

Now we apply the addition formula to the first expression, and to the second - the formula for the sine of angle differences (see formulas above)

sin α + β 2 + α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 Open the brackets, add similar terms and get the required formula

sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α + β 2 cos α - β 2

The steps to derive the remaining formulas are similar.

Derivation of the formula for the difference of sines

sin α - sin β = sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 - sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α - β 2 cos α + β 2

Derivation of the formula for the sum of cosines

cos α + cos β = cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 + cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = 2 cos α + β 2 cos α - β 2

Derivation of the formula for the difference of cosines

cos α - cos β = cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 - cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = - 2 sin α + β 2 sin α - β 2

Examples of solving practical problems

First, let's check one of the formulas by substituting specific angle values ​​into it. Let α = π 2, β = π 6. Let us calculate the value of the sum of the sines of these angles. First, we will use the table of basic values ​​of trigonometric functions, and then we will apply the formula for the sum of sines.

Example 1. Checking the formula for the sum of sines of two angles

α = π 2, β = π 6 sin π 2 + sin π 6 = 1 + 1 2 = 3 2 sin π 2 + sin π 6 = 2 sin π 2 + π 6 2 cos π 2 - π 6 2 = 2 sin π 3 cos π 6 = 2 3 2 3 2 = 3 2

Let us now consider the case when the angle values ​​differ from the basic values ​​presented in the table. Let α = 165°, β = 75°. Let's calculate the difference between the sines of these angles.

Example 2. Application of the difference of sines formula

α = 165 °, β = 75 ° sin α - sin β = sin 165 ° - sin 75 ° sin 165 - sin 75 = 2 sin 165 ° - sin 75 ° 2 cos 165 ° + sin 75 ° 2 = = 2 sin 45° cos 120° = 2 2 2 - 1 2 = 2 2

Using the formulas for the sum and difference of sines and cosines, you can move from the sum or difference to the product of trigonometric functions. Often these formulas are called formulas for moving from a sum to a product. The formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and in converting trigonometric expressions.

If you notice an error in the text, please highlight it and press Ctrl+Enter

In this article we will take a comprehensive look. Basic trigonometric identities are equalities that establish a connection between the sine, cosine, tangent and cotangent of one angle, and allow one 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.

Page navigation.

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 converting 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 the 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 .