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Addition formula ctg. Group II. Addition formulas

I won't try to convince you not to write cheat sheets. Write! Including cheat sheets on trigonometry. Later I plan to explain why cheat sheets are needed and why cheat sheets are useful. And here is information on how not to learn, but to remember some trigonometric formulas. So - trigonometry without a cheat sheet! We use associations for memorization.

1. Addition formulas:

Cosines always “come in pairs”: cosine-cosine, sine-sine. And one more thing: cosines are “inadequate”. “Everything is wrong” for them, so they change the signs: “-” to “+”, and vice versa.

Sinuses - “mix”: sine-cosine, cosine-sine.

2. Sum and difference formulas:

cosines always “come in pairs”. By adding two cosines - “koloboks”, we get a pair of cosines - “koloboks”. And by subtracting, we definitely won’t get any koloboks. We get a couple of sines. Also with a minus ahead.

Sinuses - “mix” :

3. Formulas for converting a product into a sum and difference.

When do we get a cosine pair? When we add cosines. That's why

When do we get a couple of sines? When subtracting cosines. From here:

“Mixing” is obtained both when adding and subtracting sines. What's more fun: adding or subtracting? That's right, fold. And for the formula they take addition:

In the first and third formulas, the sum is in parentheses. Rearranging the places of the terms does not change the sum. 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 amount

Cheat sheets in your pocket give you peace of mind: if you forget the formula, you can copy it off. And they give you confidence: if you fail to use the cheat sheet, you can easily remember the formulas.

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

Definition 1

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

To begin with, we will give a complete list of addition formulas, then we will prove them and analyze several illustrative examples.

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Basic addition formulas in trigonometry

There are eight basic formulas: sine of the sum and sine of the difference of two angles, cosines of the sum and difference, 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 and the cosine of the second;

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

Add up the resulting values.

The 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 should not be added, but subtracted from each other. Thus, we calculate the products of the sine of the first angle and the cosine of the second and the cosine of the first angle and 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 of the difference: calculate the products of sines and cosines of these 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, the numerator of which is the sum of the tangents of the required angles, and the denominator is a unit from which the product of the tangents of the desired angles is subtracted. Everything is clear from its graphical notation: t g (α + β) = t g α + t g β 1 - t g α · t g β

6. Tangent of the difference. We calculate the values of the difference and product of the tangents of these angles and proceed 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 amount. To calculate using this formula, we will need the product and the sum of the cotangents of these angles, 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 the difference . The formula is similar to the previous one, but the numerator and denominator are minus, 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 similar in pairs. Using the signs ± (plus-minus) and ∓ (minus-plus), we can group them for ease of recording:

sin (α ± β) = sin α · cos β ± cos α · sin β cos (α ± β) = cos α · cos β ∓ sin α · sin β t g (α ± β) = t g α ± t g β 1 ∓ t g α · t g β c t g (α ± β) = - 1 ± c t g α · c t g β c t g α ± c t g 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 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. The rest of the evidence can then be easily deduced from it.

Let's clarify the basic concepts. We will need a unit circle. It will work out if we take a certain point A and rotate the angles α and β around the center (point O). 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 full 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)

Result: 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).

Let us recall the definitions of sine and cosine: sine is a function of the angle, equal to the ratio of the leg of the opposite angle to the hypotenuse, cosine is the sine of the complementary angle. Therefore, the points A 1 And A 2 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 is 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 unit 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 derive the equality:

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

Thus, the difference cosine formula is proven.

Now we will prove the following formula - the cosine of the sum. This is easier because we can use the previous calculations. Let's 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 cosine sum formula. The last line uses the property of sine and cosine of opposite angles.

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

of the form sin (α + β) = cos (π 2 (α + β)). So
sin (α + β) = cos (π 2 (α + β)) = cos ((π 2 - α) - β) = = 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 difference sine formula:

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 addition formulas for tangent and cotangent. Let's remember 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 the following formula: 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 tangent of the difference 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

Formulas for cotangent are proved in a similar way:
c t g (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 + c t g α · c t g β c t g α + c t g 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

Addition formulas are used to express through the sines and cosines of 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, points Ma, M-b, M(a+b) are obtained by rotating point Mo by 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)), M-b (cos(-b); sin(-b)), M(a+b) (cos(a+ b); sin(a+b)). AngleMoOM(a+b) = angleM-bOMa, therefore the triangles MoOM(a+b) and M-bOMa are equal, and they are isosceles. This means that the bases MoM(a-b) and M-bMa are 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).

Let's give similar ones and reduce them 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 reduction formulas and replacing b with -b. There are also addition formulas for tangents and cotangents, but they will not be valid for all arguments.