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Cast formulas

The reduction formulas make it possible to find the values ​​of trigonometric functions for any angles (and not just acute ones). With their help, you can make transformations that simplify the form of trigonometric expressions.

Picture 1.

In addition to the reduction formulas, the following basic formulas are used in solving problems.

1) One angle formulas:

2) Expression of some trigonometric functions in terms of others:

Comment

In these formulas, the sign of the radical must be preceded by the sign $"+"$ or $"-"$, depending on which quarter the angle is in.

Sum and difference of sines, sum and difference of cosines

Formulas for the sum and difference of functions:

In addition to the formulas for the sum and difference of functions, when solving problems, formulas for the product of functions are useful:

Basic relations between the elements of oblique triangles

Designations:

$a$, $b$, $c$ - triangle sides;

$A$, $B$, $C$ - angles opposite to the listed sides;

$p=\frac(a+b+c)(2) $ - half-perimeter;

$S$ - area;

$R$ - radius of the circumscribed circle;

$r$ - radius of the inscribed circle.

Basic ratios:

1) $\frac(a)(\sin A) =\frac(b)(\sin B) =\frac(c)(\sin C) =2\cdot R$ - sine theorem;

2) $a^(2) =b^(2) +c^(2) -2\cdot b\cdot c\cdot \cos A$ - cosine theorem;

3) $\frac(a+b)(a-b) =\frac(tg\frac(A+B)(2) )(tg\frac(A-B)(2) ) $ - tangent theorem;

4) $S=\frac(1)(2) \cdot a\cdot b\cdot \sin C=\sqrt(p\cdot \left(p-a\right)\cdot \left(p-b\right)\cdot \ left(p-c\right)) =r\cdot p=\frac(a\cdot b\cdot c)(4\cdot R) $ - area formulas.

Solving oblique triangles

The solution of oblique triangles involves the definition of all its elements: sides and corners.

Example 1

Given three sides $a$, $b$, $c$:

1) in a triangle, only the cosine theorem can be used to calculate angles, since only the main value of the arccosine is within $0\le \arccos x\le +\pi $ corresponding to the triangle;

3) find the angle $B$ by applying the cosine theorem $\cos B=\frac(a^(2) +c^(2) -b^(2) )(2\cdot a\cdot c) $, and then inverse trigonometric function $B=\arccos \left(\cos B\right)$;

Example 2

Given two sides $a$, $b$ and an angle $C$ between them:

1) find the side $c$ using the cosine theorem $c^(2) =a^(2) +b^(2) -2\cdot a\cdot b\cdot \cos C$;

2) find the angle $A$ by applying the cosine theorem $\cos A=\frac(b^(2) +c^(2) -a^(2) )(2\cdot b\cdot c) $, and then inverse trigonometric function $A=\arccos \left(\cos A\right)$;

3) find the angle $B$ using the formula $B=180()^\circ -\left(A+C\right)$.

Example 3

Given two angles $A$, $B$ and a side $c$:

1) find the angle $C$ using the formula $C=180()^\circ -\left(A+B\right)$;

2) find the side $a$ using the sine theorem $a=\frac(c\cdot \sin A)(\sin C) $;

3) find the side $b$ using the sine theorem $b=\frac(c\cdot \sin B)(\sin C) $.

Example 4

Given sides $a$, $b$ and angle $B$ opposite side $b$:

1) write the cosine theorem $b^(2) =a^(2) +c^(2) -2\cdot a\cdot c\cdot \cos B$ using the given values; hence we obtain the quadratic equation $c^(2) -\left(2\cdot a\cdot \cos B\right)\cdot c+\left(a^(2) -b^(2) \right)=0$ with respect to sides $c$;

2) by solving the resulting quadratic equation, theoretically we can get one of three cases - two positive values ​​for side $c$, one positive value for side $c$, no positive values ​​for side $c$; accordingly, the problem will have two, one or zero solutions;

3) using a specific positive value of the side $c$, find the angle $A$ by applying the cosine theorem $\cos A=\frac(b^(2) +c^(2) -a^(2) )(2\cdot b\cdot c) $ and then the inverse trigonometric function $A=\arccos \left(\cos A\right)$;

4) find the angle $C$ using the formula $C=180()^\circ -\left(A+B\right)$.

). These formulas allow you to go from the sum or difference of the sines and cosines of the angles and to the product of the sines and/or cosines of the angles and . In this article, we will first list these formulas, then show their derivation, and finally consider some examples of their application.

Page navigation.

List of formulas

Let's write down the formulas for the sum and difference of sines and cosines. As you understand, there are four of them: two for sines and two for cosines.

We now give their formulations. When formulating the formulas for the sum and difference of sines and cosines, the angle is called the half-sum of the angles and, and the angle is called the half-difference. So,

It is worth noting that the formulas for the sum and difference of sines and cosines are valid for any angles and.

Derivation of formulas

To derive formulas for the sum and difference of sines, you can use addition formulas, in particular, the formulas
sum sine,
sine difference,
cosine of the sum and
cosine of the difference .

We also need the representation of angles in the form and . This representation is valid, since and for any angles and .

Now let's analyze in detail derivation of the formula for the sum of the sines of two angles kind.

First, we replace the sum by , and on , and we get . Now to apply the formula for the sine of the sum, and to - the formula of the sine of the difference:

After reducing like terms, we get . As a result, we have a formula for the sum of sines of the form.

To derive the rest of the formulas, you just need to do similar steps. We present the derivation of the formulas for the difference of sines, as well as the sum and difference of cosines:

For the difference of cosines, we have given formulas of two types or . They are equivalent because , which follows from the properties of the sines of opposite angles.

So, we have analyzed the proof of all the formulas for the sum and difference of sines and cosines.

Examples of using

Let's analyze a few examples of using the formulas for the sum of sines and cosines, as well as the difference between sines and cosines.

For example, let's check the validity of the formula for the sum of sines of the form , taking and . To do this, we calculate the values ​​of the left and right parts of the formula for these angles. Since and (if necessary, see the table of the main values ​​​​of sines and cosines), then. For and we have and , then . Thus, the values ​​of the left and right parts of the formula for the sum of sines for and coincide, which confirms the validity of this formula.

In some cases, the use of the formulas for the sum and difference of sines and cosines allows you to calculate the values ​​of trigonometric expressions when the angles are different from the main angles ( ). Let us give an example solution that confirms this idea.

Example.

Calculate the exact value of the difference between the sines of 165 and 75 degrees.

Decision.

We do not know the exact values ​​of the sines of 165 and 75 degrees, so we cannot directly calculate the value of the given difference. But the sine difference formula allows us to answer the question of the problem. Indeed, the half-sum of the angles of 165 and 75 degrees is 120, and the half-difference is 45, and the exact values ​​of the sine of 45 degrees and the cosine of 120 degrees are known.

Thus, we have

Answer:

.

Undoubtedly, the main value of the formulas for the sum and difference of sines and cosines lies in the fact that they allow you to go from the sum and difference to the product of trigonometric functions (for this reason, these formulas are often called formulas for the transition from the sum to the product of trigonometric functions). And this, in turn, can be useful, for example, when transformation of trigonometric expressions or when solving trigonometric equations. But these topics require a separate discussion.

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.

The formulas for the sum and difference of sines and cosines for two angles α and β allow you to go from the sum of the indicated angles to the product of the angles α + β 2 and α - β 2 . We note right away 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 derivation and show examples of application for specific problems.

Formulas for the sum and difference of sines and cosines

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

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, respectively, the half-sum and half-difference of the angles alpha and beta. We give a formulation for each formula.

Definitions of sum and difference formulas for sines and cosines

The sum of the 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.

The sum of the 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.

Derivation of 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. We present 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

We also represent the angles themselves as the sum of half-sums and half-differences.

α \u003d α + β 2 + α - β 2 \u003d α 2 + β 2 + α 2 - β 2 β \u003d α + β 2 - α - β 2 \u003d 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. 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 the sine formula of the angle differences to the second one (see the 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

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 for deriving the rest of the 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 cosine difference formula

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

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

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

α \u003d π 2, β \u003d π 6 sin π 2 + sin π 6 \u003d 1 + 1 2 \u003d 3 2 sin π 2 + sin π 6 \u003d 2 sin π 2 + π 6 2 cos π 2 - π 6 2 \u003d 2 sin π 3 cos π 6 \u003d 2 3 2 3 2 \u003d 3 2

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

Example 2. Applying the sine difference formula

α = 165 ° , β = 75 ° sin α - sin β = sin 165 ° - sin 75 ° sin 165 - sin 75 = 2 sin 165 ° - 75 ° 2 cos 165 ° + 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 go from the sum or difference to the product of trigonometric functions. Often these formulas are called formulas for the transition from sum to product. The formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and in converting trigonometric expressions.

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Lesson topic. Sum and difference of sines. Sum and difference of cosines.

(A lesson in learning new knowledge.)

Lesson goals.

Didactic:

derive formulas for the sum of sines and the sum of cosines and facilitate their assimilation in the course of solving problems;

continue the formation of skills in the application of trigonometric formulas;

control the degree of assimilation of the material on the topic.

Developing:

to promote the development of the skill of independent application of knowledge;

develop skills of self-control and mutual control;

to continue work on the development of logical thinking and oral mathematical speech in the search for a solution to the problem.

Educational:

to teach the ability to communicate and listen to others;

cultivate attentiveness and observation;

stimulate motivation and interest in the study of trigonometry.

Equipment: presentation, interactive board, formulas.

During the classes:

Organizing time. - 2 minutes.

Updating of basic knowledge. Repetition. – 12 min.

Goal setting. - 1 min.

Perception and comprehension of new knowledge. - 3 min.

Application of acquired knowledge. - 20 minutes.

Analysis of achievements and correction of activities. - 5 minutes.

Reflection. - 1 min.

Homework. - 1 min.

1. Organizing time.(slide 1)

- Hello! Trigonometry is one of the most interesting branches of mathematics, but for some reason most students consider it the most difficult. This can most likely be explained by the fact that there are more formulas in this section than in any other. To successfully solve problems in trigonometry, you need to be confident in numerous formulas. Many formulas have already been studied, but it turns out, not all. Therefore, the motto of this lesson will be the saying of Pythagoras "The road will be mastered by the walking one, and mathematics by the thinking one." Let's think!

2. Actualization of basic knowledge. Repetition.

1) mathematical dictation with mutual verification(slides 2-5)

First task. Using learned formulas calculate:

1 option

Option 2

sin 390 0

cos 420 0

1 – cos 2 30 0

1 - sin 2 60 0

cos 120 0 ∙cos 30 0 + sin 120 0 ∙sin 30 0

sin 30 0 ∙cos 150 0 + cos 30 0 ∙sin 150 0

Answers: ; 1 ; -; ; - ; - 1 ; 1 ; ; ; 0; ; 3 . - mutual verification.

Evaluation criteria: (works are handed over to the teacher)

"4" - 10 - 11

2) a task of a problematic nature(slide 6) - student's report.

Simplify the expression using trigonometric formulas:

Can this problem be solved in another way? (Yes, with new formulas.)

3. Goal setting(slide 7)

Lesson topic:
Sum and difference of sines. Sum and difference of cosines. - writing in a notebook

Lesson Objectives:

derive formulas for the sum and difference of sines, the sum and difference of cosines;

be able to put them into practice.

4. Perception and comprehension of new knowledge. ( slide 8-9)

We derive the formula for the sum of sines: - teacher

The remaining formulas are proved similarly: (formulas for converting a sum into a product)

Memory Rules!

In the proof of what other trigonometric formulas, addition formulas were used?

5. Application of acquired knowledge.(slides 10-11)

With new formulas:

1) Calculate: (at the blackboard) - What will be the answer? (number)

Under dictation with a teacher

6. Analysis of achievements and correction of activities.(slide 13)

Differentiated independent work with self-checking

Calculate:

7. Reflection.(slide 14)

Are you satisfied with your work in class?

What grade would you give yourself for the whole lesson?

What was the most interesting part of the lesson?

Where did you have to concentrate the most?

8. Homework: learn formulas, individual tasks on cards.

Converting the sum (difference) of the cosines of two angles into a product

For the sum and difference of the cosines of two angles, the following formulas are true:

The sum of the 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.

The difference between the cosines of two angles is equal to minus twice the product of the sine of the half-sum and the sine of the half-difference of these angles.

Examples

Formulas (1) and (2) can be obtained in many ways. Let us prove, for example, formula (1).

cos α cos β = 1 / 2 .

Putting in her (α + β) = X , (α - β) = at, we arrive at formula (1). This method is similar to the one with which the formula for the sum of the sines of two angles was obtained in the previous paragraph.

2nd way. In the previous section, the formula was proved

Putting in her α = X + π / 2, β = at + π / 2, we get:

But according to the reduction formulas sin( X+ π / 2) == cos x, sin (y + π / 2) = cos y;

Consequently,

Q.E.D.

We offer students to prove formula (2) on their own. Try to find at least two different proofs!

Exercises

1. Calculate without tables using the formulas for the sum and difference of the cosines of two angles:

a). cos 105° + cos 75°. G). cos 11π / 12- cos 5π / 12..

b). cos 105° - cos 75°. e). cos 15° -sin 15°.

in). cos 11π / 12+ cos 5π / 12.. e). sin p / 12+ cos 11π / 12.

2 . Simplify these expressions:

a). cos ( π / 3 + α ) + cos( π / 3 - α ).

b). cos( π / 3 + α ) - cos( π / 3 - α ).

3. Each of the identities

sin α + cos α = \/ 2 sin( α + π / 4)

sin α - cos α = \/ 2 sin( α - π / 4)

prove at least two different ways.

4. Present these expressions in the form of products:

a). \/ 2 + 2cos α . in). sin x + cos y.

b). \/ 3 - 2cos α . G). sin x - cos y.

5 . Simplify the expression sin 2 ( α - π / 8) - cos 2 ( α + π / 8) .

6 .Factorize these expressions (No. 1156-1159):

a). 1+ sin α - cos α

b). sin α + sin (α + β) + sin β .

in). cos α + cos 2α+ cos 3α

G). 1+ sin α + cos α

7. Prove given identities

8. Prove that the cosines of angles α and β equal if and only if

α = ± β + 2npi,

where n is some integer.

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