Sine and cosine formulas for the sum of arguments. Sum and difference of sines and cosines: derivation of formulas, examples. Examples of solving practical problems

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.

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Formulas for 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 \u003d 2 sin α + β 2 cos α - β 2 sin α - sin β \u003d 2 sin α - β 2 cos α + β 2

Sum and Difference Formulas for Cosines

cos α + cos β \u003d 2 cos α + β 2 cos α - β 2 cos α - cos β \u003d - 2 sin α + β 2 cos α - β 2, cos α - cos β \u003d 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 the double 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) \u003d sin α cos β + cos α sin β sin (α - β) \u003d sin α cos β - cos α sin β cos (α + β) \u003d cos α cos β - sin α sin β cos (α - β) \u003d 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, replace α and β with the expressions for these angles given above. We get

sin α + sin β \u003d 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 \u003d sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 sin α + β 2 - α - β 2 \u003d 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 \u003d 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 \u003d \u003d 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 \u003d 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 \u003d 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 \u003d \u003d 2 sin α - β 2 cos α + β 2

Derivation of the formula for the sum of cosines

cos α + cos β \u003d 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 \u003d 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 \u003d \u003d 2 cos α + β 2 cos α - β 2

Derivation of the formula for the difference of cosines

cos α - cos β \u003d 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 \u003d 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 \u003d \u003d - 2 sin α + β 2 sin α - β 2

Examples of solving practical problems

To begin with, let's check one of the formulas by substituting specific angles into it. Let α \u003d π 2, β \u003d π 6. Let's calculate the value of the sum of the sines of these angles. First, let's use the table of basic values trigonometric functionsand then apply the formula for the sum of the 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 \u200b\u200bof the angles differ from the basic values \u200b\u200bpresented in the table. Let α \u003d 165 °, β \u003d 75 °. Let's calculate the value of the difference between the sines of these angles.

Example 2. Application of the sine difference formula

α \u003d 165 °, β \u003d 75 ° sin α - sin β \u003d sin 165 ° - sin 75 ° sin 165 - sin 75 \u003d 2 sin 165 ° - sin 75 ° 2 cos 165 ° + sin 75 ° 2 \u003d \u003d 2 sin 45 ° cos 120 ° \u003d 2 2 2 - 1 2 \u003d 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 to solve trigonometric equations and when converting trigonometric expressions.

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The electronic resource is an excellent material for conducting interactive training in modern schools... It is written competently, has a clear structure and corresponds to the school plan. Thanks to detailed explanations, the topic that is presented in the video lesson will become clear to as many students in the class as possible. Teachers should remember that not all students have the same degree of perception, speed of understanding, base. Such materials will help to cope with difficulties and catch up with their peers, to correct academic performance. With the help of them, in a calm home environment, independently or together with a tutor, the student can understand a particular topic, study the theory and view examples practical application this or that formula, etc.

This video tutorial is devoted to the topic "Sine and cosine of the difference of arguments." It is assumed that students have already studied the basics of trigonometry, are familiar with the basic functions and their properties, ghost formulas and tables of trigonometric values.

Also, before proceeding to the study of this topic, it is necessary to have an understanding of the sine and cosine of the sum of arguments, to know the two basic formulas and be able to use them.

At the beginning of the video lesson, the announcer reminds the students of these two formulas. Next, the first formula is demonstrated - the sine of the difference of arguments. In addition to how the formula itself is displayed, it is shown how it is obtained from another. Thus, the student will not have to memorize a new formula without understanding, which is a common mistake. This is very important for the students in this class. You must always remember that you can add a + sign before the minus sign of everything, and the minus sign on the plus sign will eventually turn into a minus. With such a simple step, you can use the formula for the sine of the sum and get the formula for the sine of the difference of the arguments.

Similarly, the formula for the cosine of the difference is derived from the formula for the cosine of the sum of arguments.

The announcer explains everything step by step, and as a result, the general formula for the cosine of the sum and difference of the arguments and the sine is displayed, similarly.

The first example from the practical part of this video tutorial suggests finding the cosine of Pi / 12. It is proposed to represent this value in the form of some difference, in which the reduced and subtracted will be tabular values. Next, the formula for the cosine of the difference of arguments is applied. By replacing the expression, you can substitute the obtained values \u200b\u200band get the answer. The announcer reads out the answer, which is displayed at the end of the example.

The second example is an equation. On both the right and left sides, we see the cosines of the arguments' differences. The announcer resembles cast formulas that are used to replace and simplify these expressions. These formulas are written on the right side so that students can understand where these or those changes come from.

Another example, the third, is a fraction, where in both the numerator and denominator we have trigonometric expressions, namely, the difference of products.

Here, the solution also uses the formulas of the casts. Thus, students can find that skipping one topic in trigonometry makes it harder to understand the rest.

And finally, the fourth example. It is also an equation, in solving which it is necessary to use new learned and old formulas.

The examples given in the video tutorial can be considered in more detail and try to solve it yourself. They can be set as homework schoolchildren.

TEXT CODE:

Topic of the lesson "Sine and cosine of the difference of arguments."

In the previous course, we got acquainted with two trigonometric formulas sine and cosine of the sum of arguments.

sin (x + y) \u003d sin x cos y + cos x sin y,

cos (x + y) \u003d cos x cos y - sin x sin y.

the sine of the sum of two angles is equal to the sum between the product of the sine of the first angle and the cosine of the second angle and the product of the cosine of the first angle and the sine of the second angle;

the cosine of the sum of two angles is equal to the difference between the product of the cosines of these angles and the product of the sum of these angles.

Using these formulas, we derive the formulas Sine and cosine of the difference of arguments.

Sine of Argument Difference sin (x- y)

Two formulas (sine of sum and sine of difference) can be written as:

sin (x y) \u003d sin x cos ycos x sin y.

Similarly, we derive the formula for the cosine of the difference:

We rewrite the cosine of the difference of the arguments as a sum and apply the already known formula for the cosine of the sum: cos (x + y) \u003d cosxcosy - sinxsiny.

only for arguments x and -y. Substituting these arguments into the formula, we get cosxcos (- y) - sinxsin (- y).

sin (- y) \u003d - siny). and we get the final expression cosxcosy + sinxsiny.

cos (x - y) \u003d cos (x + (- y)) \u003d cos xcos (- y) - sin x sin (- y) \u003d cosx cos y + sin xsin y.

Hence, cos (x - y) \u003d cosxcos y + sin xsin y.

the cosine of the difference between two angles is equal to the sum between the product of the cosines of these angles and the product of the sines of these angles.

Combining two formulas (the cosine of the sum and the cosine of the difference) into one, we write

cos (x y) \u003d cosxcos y sin xsin y.

Remember that formulas in practice can be applied both from left to right and vice versa.

Let's consider some examples.

EXAMPLE 1. Calculate cos (cosine pi divided by twelve).

Decision. Let's write pi divided by twelve as the difference between pi and three and pi divided by four: \u003d -.

Substitute the values \u200b\u200binto the difference cosine formula: cos (x - y) \u003d cosxcosy + sinxsiny, thus cos \u003d cos (-) \u003d cos cos + sin sin

We know that cos \u003d, cos \u003d sin \u003d, sin \u003d. Show table of values.

Replace the value of the sine and cosine with numerical values \u200b\u200band get ∙ + ∙ when multiplying a fraction by a fraction, we multiply the numerators and denominators, we get

cos \u003d cos (-) \u003d cos cos + sin sin \u003d ∙ + ∙ \u003d \u003d \u003d.

Answer: cos \u003d.

EXAMPLE 2. Solve the equation cos (2π - 5x) \u003d cos (- 5x) (the cosine of two pi minus five x is equal to the cosine of pi by two minus five x).

Decision. To the left and right sides of the equation, apply the reduction formulas cos (2π - cos (cosine two pi minus alpha equal to cosine alpha) and cos (- \u003d sin (cosine pi by two minus alpha is equal to sine alpha), we get cos 5x \u003d sin 5x, bring it to the form of a homogeneous equation of the first degree and get cos 5x - sin 5x \u003d 0. This is a homogeneous equation of the first degree .Divide both sides of the equation term by cos 5x.

cos 5x: cos 5x - sin 5x: cos 5x \u003d 0, because cos 5x: cos 5x \u003d 1, and sin 5x: cos 5x \u003d tg 5x, we get:

Since we already know that the equation tgt \u003d a has a solution t \u003d arctana + πn, and since we have t \u003d 5x, a \u003d 1, we get

5x \u003d arctan 1 + πn,

and the arctan value is 1, then tg 1 \u003d Show table

plug the value into the equation and solve it:

Answer: x \u003d +.

EXAMPLE 3. Find the value of the fraction. (in the numerator, the difference between the product of the cosines of seventy-five degrees and sixty-five degrees and the product of the sines of seventy-five degrees and sixty-five degrees, and in the denominator, the difference between the product of the sine of eighty-five degrees and the cosine of thirty-five degrees and the product of the cosine of eighty-five degrees and the sine of thirty-five degrees) ...

Decision. In the numerator of this fraction, the difference can be "rolled up" into the cosine of the sum of the arguments 75 ° and 65 °, and in the denominator, the difference can be "rolled up" into the sine of the difference between the arguments 85 ° and 35 °. We get

Answer: - 1.

EXAMPLE 4. Solve the equation: cos (-x) + sin (-x) \u003d 1 (the cosine of the difference pi by four and x plus the sine of the difference pi by four and x is equal to one).

Decision. Let's apply the formulas cosine of difference and sine of difference.

Show general formula for cosine of difference

Then cos (-x) \u003d cos cos x + sinsinx