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 valid:
The sum of the cosines of two angles is equal to the double 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 of
Formulas (1) and (2) can be obtained in many ways. Let us prove, for example, formula (1).
cos α cos β = 1 / 2 .
Believing in her (α + β) = NS , (α - β) = at, we come to 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, we proved the formula
Believing in her α = NS + π / 2, β = at + π / 2, we get:
But according to the casting formulas sin ( NS+ π / 2) == cos x, sin (y + π / 2) = cos y;
Hence,
Q.E.D.
We invite students to prove formula (2) on their own. Try to find at least two different ways of proof!
Exercises
1. Calculate without tables using 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 °.
v). cos 11π / 12+ cos 5π / 12.. f). sin π / 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 in at least two different ways.
4. These expressions are presented in the form of works:
a). \/ 2 + 2cos α ... v). sin x + cos y.
b). \/ 3 - 2 cos α ... G). sin x - cos y.
5 ... Simplify the expression sin 2 ( α - π / 8) - cos 2 ( α + π / 8) .
6 .Follow these expressions (No. 1156-1159):
a). 1 + sin α - cos α
b). sin α + sin (α + β) + sin β .
v). cos α + cos 2α+ cos 3α
G). 1 + sin α + cos α
7. Prove given identities
8. Prove that the cosines of the angles α and β are equal if and only if
α = ± β + 2 nπ,
where n is some integer.
Casting formulas
Casting formulas make it possible to find the values of trigonometric functions for any angles (not just acute ones). With their help, you can perform 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) Formulas for one angle:
2) Expression of some trigonometric functions in terms of others:
Comment
In these formulas, the radical sign must be preceded by a $ "+" $ or $ "-" $, depending on which quarter the corner 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, the formulas for the product of functions are useful:
Basic relationships between elements of oblique triangles
Legend:
$ a $, $ b $, $ c $ - sides of the triangle;
$ A $, $ B $, $ C $ - angles opposite to the listed sides;
$ p = \ frac (a + b + c) (2) $ - semiperimeter;
$ S $ - area;
$ R $ - the radius of the circumscribed circle;
$ r $ - radius of the inscribed circle.
Basic relationships:
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 (pa \ right) \ cdot \ left (pb \ right) \ cdot \ left (pc \ 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
There are three sides $ a $, $ b $, $ c $:
1) in a triangle, only the cosine theorem can be used to calculate angles, since only the principal 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 $ by 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 $ by the formula $ B = 180 () ^ \ circ - \ left (A + C \ right) $.
Example 3
Given two corners $ A $, $ B $ and a side $ c $:
1) find the angle $ C $ by the formula $ C = 180 () ^ \ circ - \ left (A + B \ right) $;
2) find the side $ a $ by the sine theorem $ a = \ frac (c \ cdot \ sin A) (\ sin C) $;
3) find the side $ b $ by the sine theorem $ b = \ frac (c \ cdot \ sin B) (\ sin C) $.
Example 4
Given the sides $ a $, $ b $ and the corner $ B $ opposite to the side $ b $:
1) we write the cosine theorem $ b ^ (2) = a ^ (2) + c ^ (2) -2 \ cdot a \ cdot c \ cdot \ cos B $, using the given values; from this we get 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) having solved the obtained quadratic equation, theoretically we can get one of three cases - two positive values for the $ c $ side, one positive value for the $ c $ side, the absence of positive values for the $ c $ side; accordingly, the problem will have two, one or zero solutions;
3) using a specific positive value of the side $ c $, we find the angle $ A $ by applying the cosine theorem $ \ cos A = \ frac (b ^ (2) + c ^ (2) -a ^ (2)) (2 \ cdot b \ cdot c) $ followed by the inverse trigonometric function $ A = \ arccos \ left (\ cos A \ right) $;
4) find the angle $ C $ by the formula $ C = 180 () ^ \ circ - \ left (A + B \ right) $.
The formulas for the sum and difference of sines and cosines for two angles α and β allow one to go from the sum of the indicated angles to the product of the angles α + β 2 and α - β 2. Immediately, we note that you should not confuse the formulas for the sum and difference of sines and cosines with the formulas for the sines and cosines of the sum and difference. Below we list these formulas, give their derivation, and show examples of application for specific tasks.
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 for 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, respectively, the half-sum and half-difference of the angles alpha and beta. Let's give a formulation for each formula.
Definitions of formulas for the sum and difference of sines and cosines
The sum of the sines of two angles is equal to the double product of the sine of the half-sum of these angles by the cosine of the half-difference.
Difference of sines of two angles is equal to the double product of the sine of the half-difference of these angles by 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 the 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 = 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 of sin α + sin β, 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 the sine formula of the angle differences to the second (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 Expand the brackets, present similar terms and obtain 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 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 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 values of the angles 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 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 the 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 values of the angles differ from the basic values presented in the table. Let α = 165 °, β = 75 °. Let's calculate the value of the difference between the sines of these angles.
Example 2. Application of the formula for the difference of sines
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. These formulas are often called sum-to-product transition formulas. The formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and when converting trigonometric expressions.
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Lesson topic. Sum and difference of sinuses. Sum and difference of cosines.
(A lesson in assimilating new knowledge.)
Lesson objectives.
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 use of trigonometric formulas;
control the degree of assimilation of the material on the topic.
Developing:
contribute to the development of the skill of independent application of knowledge;
develop skills of self-control and mutual control;
continue to work on the development of logical thinking and oral mathematical speech in the search for a solution to the problem posed.
Educational:
to teach the ability to communicate and listen to others;
educate attentiveness and observation;
stimulate motivation and interest in learning trigonometry.
Equipment: presentation, interactive whiteboard, formulas.
During the classes:
Organizing time. - 2 minutes.
Updating basic knowledge. Repetition. - 12 minutes
Goal setting. - 1 minute.
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 minute.
Homework. - 1 minute.
1. Organizing time.(slide 1)
- Hello! Trigonometry is one of the most interesting areas of mathematics, but for some reason most students find 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 trigonometry problems, 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 one who walks, and the math - the thinking one." Let's think!
2. Updating basic knowledge. Repetition.
1) mathematical dictation with mutual check(slides 2-5)
First task. Using the learned formulas calculate:
Option 1
Option 2
sin 390 0
cos 420 0
1 - cos 2 30 0
1 - sin 2 60 0
сos 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) problematic task(slide 6) - student's report.
Simplify expression using trigonometric formulas:
Is it possible to solve this problem differently? (Yes, with new formulas.)
3. Goal setting(slide 7)
Lesson topic:
Sum and difference of sinuses. 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 apply them in 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)
Memorization rules!
In the proof of what other trigonometric formulas were addition formulas used?
5. Application of the acquired knowledge.(slides 10-11)
With new formulas:
1) Calculate: (at the blackboard) - What will be the answer? (number)
Dictation with a teacher
6. Analysis of achievements and correction of activities.(slide 13)
Differentiated self-test with self-test
Calculate:
7. Reflection.(slide 14)
Are you satisfied with your work in the lesson?
How would you rate yourself for the entire lesson?
What was the most interesting moment in the lesson?
Where did you have to concentrate the most?
8. Homework: learn formulas, individual tasks on cards.
). These formulas allow from the sum or difference of sines and cosines of angles and go to the product of sines and / or cosines of angles and. In this article, we will first list these formulas, then show their derivation, and in conclusion, consider several 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 can imagine, there are four of them: two for sines and two for cosines.
Now let us 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 should be noted 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
sine sum,
sine difference,
the cosine of the sum and
the cosine of the difference.
We also need the representation of angles in the form 
and 
... This representation is valid, as well as for any angles and.
Now we will analyze in detail derivation of the formula for the sum of the sines of two angles species.
First, in total, we replace with and on , and we get. Now to 
we apply the sine formula of the sum, and to 
- the formula for the sine of the difference: 
After reducing such terms, we obtain 
... As a result, we have a formula for the sum of sines of the form.
To display the rest of the formulas, you just need to do the same. Here is 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 since 
, which follows from the properties of the sines of opposite angles.
So, we have analyzed the proof of all formulas for the sum and difference of sines and cosines.
Examples of using
Let's look at several 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, let's calculate the values of the left and right sides of the formula for these angles. Since and (if necessary, see the table of basic values of sines and cosines), then. For and we have 
and 
, then . Thus, the values of the left and right sides of the formula for the sum of sines for and coincide, which confirms the validity of this formula.
In some cases, the use of 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 fundamental angles ( 
). Let's give a solution to an example that confirms this idea.
Example.
Calculate the exact value of the difference between the sines of 165 and 75 degrees.
Solution.
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 formula for the difference of sines allows us to answer the question of the problem. Indeed, the half-sum of the angles 165 and 75 degrees is 120, and the half-difference is 45, and the exact values of the sine 45 degrees and cosine 120 degrees are known.
Thus, we have 
Answer:
.
Undoubtedly, the main value of the formulas for the sum and difference of sines and cosines is 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 the formulas for the transition from the sum to the product of trigonometric functions). And this, in turn, can be useful, for example, when converting trigonometric expressions or at solving trigonometric equations... But these topics require a separate discussion.
Bibliography.
- Algebra: Textbook. for 9 cl. wednesday school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M .: Education, 1990.- 272 p .: ill.- ISBN 5-09-002727-7
- Bashmakov M.I. Algebra and the beginning of analysis: Textbook. for 10-11 cl. wednesday shk. - 3rd ed. - M .: Education, 1993 .-- 351 p .: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of the analysis: Textbook. for 10-11 cl. general education. institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M .: Education, 2004. - 384 p .: ill. - ISBN 5-09-013651-3.
- Gusev V.A., Mordkovich A.G. Mathematics (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.