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 teach, but 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. That's why

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.

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 trigonometric functions, the property of symmetry, as well as 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. corner

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 formulas double angle.

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

Reduction Formulas

Trigonometric formulas for decreasing degrees designed to facilitate the transition from natural degrees trigonometric functions to sines and cosines to 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

main destination sum and difference formulas for trigonometric functions is to pass to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equations, since 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.

Universal trigonometric substitution

We complete the review of the basic formulas of trigonometry with formulas expressing trigonometric functions in terms of the tangent of a half angle. This replacement is called universal trigonometric substitution. Its convenience lies in the fact that all trigonometric functions are expressed in terms of the tangent of a half angle rationally without roots.

Bibliography.

• Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
• 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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The addition formulas are used to express through the sines and cosines of the angles a and b, the values ​​of the functions cos (a + b), cos (a-b), sin (a + b), sin (a-b).

Addition formulas for sines and cosines

Theorem: For any a and b, the following equality is true cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b).

Let's prove this theorem. Consider the following figure:

On it, the points Ma, M-b, M(a+b) are obtained by rotating the point Mo through angles a, -b, and a+b, respectively. From the definitions of sine and cosine, the coordinates of these points will be the following: Ma(cos(a); sin(a)), Mb (cos(-b); sin(-b)), M(a+b) (cos(a+ b);sin(a+b)). Angle MoOM (a + b) \u003d angle M-bOM, therefore the triangles MoOM (a + b) and M-bOM are equal, and they are isosceles. So, the bases MoM (a-b) and M-bMa are also equal. Therefore, (MoM(a-b))^2 = (M-bMa)^2. Using the formula for the distance between two points, we get:

(1 - cos(a+b))^2 + (sin(a+b))^2 = (cos(-b) - cos(a))^2 + (sin(-b) - sin(a) )^2.

sin(-a) = -sin(a) and cos(-a) = cos(a). Let's transform our equality, taking into account these formulas and the square of the sum and difference, then:

1 -2*cos(a+b) + (cos(a+b))^2 +(sin(a+b))^2 = (cos(b))^2 - 2*cos(b)*cos (a) + (cos(a)^2 +(sin(b))^2 +2*sin(b)*sin(a) + (sin(a))^2.

Now we apply the basic trigonometric identity:

2-2*cos(a+b) = 2 - 2*cos(a)*cos(b) + 2*sin(a)*sin(b).

We give similar ones and reduce by -2:

cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b). Q.E.D.

The following formulas are also valid:

• cos(a-b) = cos(a)*cos(b) + sin(a)*sin(b);
• sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b);
• sin(a-b) = sin(a)*cos(b) - cos(a)*sin(b).

These formulas can be obtained from the one proved above, using the reduction formulas and replacing b with -b. For tangents and cotangents, there are also addition formulas, but they will not be valid for any arguments.

Formulas for adding tangents and cotangents

For any angles a,b except a=pi/2+pi*k, b=pi/2 +pi*n and a+b =pi/2 +pi*m, for any integers k,n,m the following formula will be valid:

tg(a+b) = (tg(a) + tg(b))/(1-tg(a)*tg(b)).

For any angles a,b except a=pi/2+pi*k, b=pi/2 +pi*n and a-b =pi/2 +pi*m, for any integers k,n,m the following formula will hold:

tg(a-b) = (tg(a)-tg(b))/(1+tg(a)*tg(b)).

For any angles a,b except a=pi*k, b=pi*n, a+b = pi*m and for any integers k,n,m the following formula will hold true:

ctg(a+b) = (ctg(a)*ctg(b) -1)/(ctg(b)+ctg(a)).

We continue our conversation about the most used formulas in trigonometry. The most important of them are the addition formulas.

Definition 1

Addition formulas allow you to express the functions of the difference or sum of two angles using the trigonometric functions of these angles.

To begin with, we will present full list addition formulas, then we will prove them and analyze a few illustrative examples.

Yandex.RTB R-A-339285-1

Basic addition formulas in trigonometry

There are eight basic formulas: the sine of the sum and the sine of the difference of two angles, the cosines of the sum and difference, the tangents and cotangents of the sum and difference, respectively. Below are their standard formulations and calculations.

1. The sine of the sum of two angles can be obtained as follows:

We calculate the product of the sine of the first angle by the cosine of the second;

Multiply the cosine of the first angle by the sine of the first;

Add up the resulting values.

Graphical writing of the formula looks like this: sin (α + β) = sin α cos β + cos α sin β

2. The sine of the difference is calculated in almost the same way, only the resulting products must not be added, but subtracted from each other. Thus, we calculate the products of the sine of the first angle by the cosine of the second and the cosine of the first angle by the sine of the second and find their difference. The formula is written like this: sin (α - β) = sin α cos β + sin α sin β

3. Cosine of the sum. For it, we find the products of the cosine of the first angle by the cosine of the second and the sine of the first angle by the sine of the second, respectively, and find their difference: cos (α + β) = cos α cos β - sin α sin β

4. Cosine difference: we calculate the products of the sines and cosines of the given angles, as before, and add them. Formula: cos (α - β) = cos α cos β + sin α sin β

5. Tangent of the sum. This formula is expressed as a fraction, in the numerator of which is the sum of the tangents of the desired angles, and in the denominator is the unit from which the product of the tangents of the desired angles is subtracted. Everything is clear from her graphic notation: t g (α + β) = t g α + t g β 1 - t g α t g β

6. Tangent of difference. We calculate the values ​​of the difference and the product of the tangents of these angles and deal with them in a similar way. In the denominator, we add to one, and not vice versa: t g (α - β) = t g α - t g β 1 + t g α t g β

7. Cotangent of the sum. For calculations using this formula, we need the product and the sum of the cotangents of these angles, with which we proceed as follows: c t g (α + β) = - 1 + c t g α c t g β c t g α + c t g β

8. Cotangent of difference . The formula is similar to the previous one, but in the numerator and denominator - minus, and not plus c t g (α - β) = - 1 - c t g α c t g β c t g α - c t g β.

You probably noticed that these formulas are pairwise similar. Using the signs ± (plus-minus) and ∓ (minus-plus), we can group them for ease of notation:

sin (α ± β) = sin α cos β ± cos α sin β cos (α ± β) = cos α cos β ∓ sin α sin β tg (α ± β) = tg α ± tg β 1 ∓ tg α tg β ctg (α ± β) = - 1 ± ctg α ctg β ctg α ± ctg β

Accordingly, we have one recording formula for the sum and difference of each value, just in one case we pay attention to the upper sign, in the other - to the lower one.

Definition 2

We can take any angles α and β , and the addition formulas for cosine and sine will work for them. If we can correctly determine the values ​​of the tangents and cotangents of these angles, then the addition formulas for the tangent and cotangent will also be valid for them.

Like most concepts in algebra, addition formulas can be proven. The first formula we will prove is the difference cosine formula. From it, you can then easily deduce the rest of the evidence.

Let us clarify the basic concepts. We'll need unit circle. It will turn out if we take a certain point A and rotate around the center (point O) the angles α and β. Then the angle between the vectors O A 1 → and O A → 2 will be equal to (α - β) + 2 π z or 2 π - (α - β) + 2 π z (z is any integer). The resulting vectors form an angle that is equal to α - β or 2 π - (α - β) , or it may differ from these values ​​by an integer number of complete revolutions. Take a look at the picture:

We used the reduction formulas and got the following results:

cos ((α - β) + 2 π z) = cos (α - β) cos (2 π - (α - β) + 2 π z) = cos (α - β)

Bottom line: the cosine of the angle between the vectors O A 1 → and O A 2 → is equal to the cosine of the angle α - β, therefore, cos (O A 1 → O A 2 →) = cos (α - β) .

Recall the definitions of sine and cosine: sine is a function of an angle, equal to the ratio the leg of the opposite angle to the hypotenuse, the cosine is the sine of the complementary angle. Therefore, the points A 1 And A2 have coordinates (cos α , sin α) and (cos β , sin β) .

We get the following:

O A 1 → = (cos α , sin α) and O A 2 → = (cos β , sin β)

If it's not clear, look at the coordinates of the points located at the beginning and end of the vectors.

The lengths of the vectors are equal to 1, because we have a single circle.

Let us now analyze the scalar product of the vectors O A 1 → and O A 2 → . In coordinates it looks like this:

(O A 1 → , O A 2) → = cos α cos β + sin α sin β

From this we can deduce the equality:

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

Thus, the formula for the cosine of the difference is proved.

Now we will prove the following formula is the cosine of the sum. This is easier because we can use the previous calculations. Take the representation α + β = α - (- β) . We have:

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

This is the proof of the formula for the cosine of the sum. The last line uses the property of the sine and cosine of opposite angles.

The formula for the sine of the sum can be derived from the formula for the cosine of the difference. Let's take the reduction formula for this:

of the form sin (α + β) = cos (π 2 (α + β)) . So
sin (α + β) \u003d cos (π 2 (α + β)) \u003d cos ((π 2 - α) - β) \u003d \u003d cos (π 2 - α) cos β + sin (π 2 - α) sin β = = sin α cos β + cos α sin β

And here is the proof of the formula for the sine of the difference:

sin (α - β) = sin (α + (- β)) = sin α cos (- β) + cos α sin (- β) = = sin α cos β - cos α sin β
Note the use of the sine and cosine properties of opposite angles in the last calculation.

Next, we need proofs of the addition formulas for the tangent and cotangent. Let us recall the basic definitions (tangent is the ratio of sine to cosine, and cotangent is vice versa) and take the formulas already derived in advance. We made it:

t g (α + β) = sin (α + β) cos (α + β) = sin α cos β + cos α sin β cos α cos β - sin α sin β

We have a complex fraction. Next, we need to divide its numerator and denominator by cos α cos β , given that cos α ≠ 0 and cos β ≠ 0 , we get:
sin α cos β + cos α sin β cos α cos β cos α cos β - sin α sin β cos α cos β = sin α cos β cos α cos β + cos α sin β cos α cos β cos α cos β cos α cos β - sin α sin β cos α cos β

Now we reduce the fractions and get a formula of the following form: sin α cos α + sin β cos β 1 - sin α cos α s i n β cos β = t g α + t g β 1 - t g α t g β.
We got t g (α + β) = t g α + t g β 1 - t g α · t g β . This is the proof of the tangent addition formula.

The next formula that we will prove is the difference tangent formula. Everything is clearly shown in the calculations:

t g (α - β) = t g (α + (- β)) = t g α + t g (- β) 1 - t g α t g (- β) = t g α - t g β 1 + t g α t g β

The formulas for the cotangent are proved in a similar way:
ctg (α + β) = cos (α + β) sin (α + β) = cos α cos β - sin α sin β sin α cos β + cos α sin β = = cos α cos β - sin α sin β sin α sin β sin α cos β + cos α sin β sin α sin β = cos α cos β sin α sin β - 1 sin α cos β sin α sin β + cos α sin β sin α sin β = = - 1 + ctg α ctg β ctg α + ctg β
Further:
c t g (α - β) = c t g   (α + (- β)) = - 1 + c t g α c t g (- β) c t g α + c t g (- β) = - 1 - c t g α c t g β c t g α - c t g β