The electronic resource is an excellent material for interactive learning in modern schools. It is well-written, has a clear structure and corresponds to the school plan. Thanks to detailed explanations, the topic presented in the video lesson will become clear to as many students as possible in the class. Teachers should remember that not all students have the same degree of perception, speed of understanding, base. To cope with difficulties and catch up with your peers, correct academic performance, such materials will help. With the help of them, in a calm home environment, on their own 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 learned 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, 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 the arguments. In addition to how the formula itself is derived, 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 in front of the minus sign of everything, and the minus sign on the plus sign will eventually turn into a minus. With the help of 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 the arguments.
The speaker 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 derived, similarly.
The first example from the practical part of this video tutorial suggests finding the cosine of Pi / 12. It is proposed to present this value as a certain difference, at which the reduced and the subtracted will be tabular values. Next, apply the formula for the cosine of the difference of the arguments. By replacing the expression, you can substitute the obtained values and get the answer. The announcer reads out the answer, which is displayed at the end of the example.
The second example is an equation. Both on the right and on the left sides we see the cosines of the differences of the arguments. Narrator 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 certain changes come from.
Another example, the third, is some fraction, where both in the numerator and in the denominator we have trigonometric expressions, namely, the difference between the products.
Here, too, reduction formulas are used in the solution. Thus, students can make sure that skipping one topic in trigonometry, it will be more and more difficult to understand the rest.
And finally, the fourth example. This is also an equation, in the solution of which it is necessary to use new studied and old formulas.
You can look at the examples that are given in the video tutorial in more detail and try to solve it yourself. They can be set as homework schoolchildren.
TEXT INTERPRETATION:
The topic of the lesson is “Sine and cosine of the difference of arguments”.
In the previous course, we met two trigonometric formulas sine and cosine of the sum of the arguments.
sin(x + y) = sin x cos y + cos x sin y,
cos (x + y) = cos x cos y - sin x sin y.
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 those angles and the product of the sum of those angles.
Using these formulas, we derive the formulas Sine and cosine of the difference of the arguments.
Sine of argument difference sin(x- y)
Two formulas (the sine of the sum and the sine of the difference) can be written as:
sin(xy) = 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) = cosxcosy - sinxsiny.
only for x and -y arguments. Substituting these arguments into the formula, we get cosxcos(- y) - sinxsin(- y).
sin(-y)=-siny). and 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.
So cos (x - y) = cosxcos y + sin xsin y.
the cosine of the difference of 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(xy) = cosxcos y sin xsin y.
Remember that formulas in practice can be applied both from left to right and vice versa.
Consider examples.
EXAMPLE 1. Calculate cos (cosine of pi divided by twelve).
Decision. Let's write pi divided by twelve as the difference between pi divided by three and pi divided by four: = - .
Substitute the values into the difference cosine formula: cos (x - y) = cosxcosy + sinxsiny, so cos = cos (-) = cos cos + sin sin
We know that cos = , cos = sin= , sin = . Show table of values.
Let's replace the value of sine and cosine numerical values and we get ∙ + ∙ when multiplying a fraction by a fraction, we multiply the numerators and denominators, we get
cos = cos (-) = cos cos + sin sin = ∙ + ∙ = = =.
Answer: cos =.
EXAMPLE 2. Solve cos equation(2π - 5x) = cos(- 5x) (cosine of two pi minus five x is equal to cosine of pi times two minus five x).
Decision. We apply the reduction formulas cos (2π - cos (cosine two pi minus alpha equals cosine alpha) and cos (- \u003d sin (cosine pi by two minus alpha equals sine alpha) to the left and right sides of the equation, we get cos 5x \u003d sin 5x, we give it to the form of a homogeneous equation of the first degree and we get cos 5x - sin 5x \u003d 0. This is a homogeneous equation of the first degree.We divide term by term both parts of the equation by cos 5x. We have:
cos 5x: cos 5x - sin 5x: cos 5x = 0, because cos 5x: cos 5x \u003d 1, and sin 5x: cos 5x \u003d tg 5x, then we get:
Since we already know that the equation tgt \u003d a has a solution t \u003d arctga + πn, and since we have t \u003d 5x, a \u003d 1, we get
5x \u003d arctg 1 + πn,
a arctg value 1, then tg 1= Show table
Substitute the value in the equation and solve it:
Answer: x = +.
EXAMPLE 3. Find the value of a 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 “folded” into the cosine of the sum of the arguments 75° and 65°, and in the denominator, the difference can be “folded” into the sine of the difference between the arguments 85° and 35°. Get
Answer: - 1.
EXAMPLE 4. Solve the equation: cos (-x) + sin (-x) \u003d 1 (the cosine of the difference of pi by four and x plus the sine of the difference of pi by four and x is equal to one).
Decision. We apply the formulas cosine of the difference and sine of the difference.
Show the general formula for the cosine of the difference
Then cos (-x) = cos cos x + sinsinx
Show the general formula of the sine of the difference
and sin (-x) \u003d sin cosx - cos sinx
Substitute these expressions into the equation cos(-x) + sin(-x) = 1 and get:
cos cos x + sinsin x + sin cos x - cos sin x \u003d 1,
Since cos= and sin= Show table value of sine and cosine
We get ∙ cos x + ∙ sinx + ∙ cos x - ∙ sinx \u003d 1,
the second and fourth terms are opposite, therefore they cancel each other out, it remains:
∙ cos + ∙ cos = 1,
We will decide given equation and we get that
2∙∙cos x= 1,
Since we already know that the equation cos = a has a solution t = arcosa+ 2πk, and since we have t=x, a =, we get
x \u003d arccos + 2πn,
and since the value of arccos, then cos =
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.
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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, let's use the table of basic values trigonometric functions, and then 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 ° - 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 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 when converting trigonometric expressions.
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