Determination of the sum of cubes of trigonometric functions. Universal trigonometric substitution, derivation of formulas, examples

Trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, which allows you to find any of these functions, provided that any other is known.

tg \alpha = \frac(\sin \alpha)(\cos \alpha), \enspace ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

tg \alpha \cdot ctg \alpha = 1

This identity says that the sum of the square of the sine of one angle and the square of the cosine of one angle is equal to one, which in practice makes it possible to calculate the sine of one angle when its cosine is known and vice versa.

When converting trigonometric expressions, this identity is very often used, which allows you to replace the sum of the squares of the cosine and sine of one angle with one and also perform the replacement operation in reverse order.

Finding tangent and cotangent through sine and cosine

tg \alpha = \frac(\sin \alpha)(\cos \alpha),\enspace

These identities are formed from the definitions of sine, cosine, tangent and cotangent. After all, if you look, then by definition, the ordinate of y is the sine, and the abscissa of x is the cosine. Then the tangent will be equal to the ratio \frac(y)(x)=\frac(\sin \alpha)(\cos \alpha), and the ratio \frac(x)(y)=\frac(\cos \alpha)(\sin \alpha)- will be a cotangent.

We add that only for such angles \alpha for which the trigonometric functions included in them make sense, the identities will take place, ctg \alpha=\frac(\cos \alpha)(\sin \alpha).

For example: tg \alpha = \frac(\sin \alpha)(\cos \alpha) is valid for \alpha angles that are different from \frac(\pi)(2)+\pi z, but ctg \alpha=\frac(\cos \alpha)(\sin \alpha)- for an angle \alpha other than \pi z , z is an integer.

Relationship between tangent and cotangent

tg \alpha \cdot ctg \alpha=1

This identity is valid only for angles \alpha that are different from \frac(\pi)(2) z. Otherwise, either cotangent or tangent will not be determined.

Based on the points above, we get that tg \alpha = \frac(y)(x), but ctg\alpha=\frac(x)(y). Hence it follows that tg \alpha \cdot ctg \alpha = \frac(y)(x) \cdot \frac(x)(y)=1. Thus, the tangent and cotangent of one angle at which they make sense are mutually reciprocal numbers.

Relationships between tangent and cosine, cotangent and sine

tg^(2) \alpha + 1=\frac(1)(\cos^(2) \alpha)- the sum of the square of the tangent of the angle \alpha and 1 is equal to the inverse square of the cosine of this angle. This identity is valid for all \alpha other than \frac(\pi)(2)+ \pi z.

1+ctg^(2) \alpha=\frac(1)(\sin^(2)\alpha)- the sum of 1 and the square of the cotangent of the angle \alpha , equals the inverse square of the sine of the given angle. This identity is valid for any \alpha other than \pi z .

Examples with solutions to problems using trigonometric identities

Example 1

Find \sin \alpha and tg \alpha if \cos \alpha=-\frac12 And \frac(\pi)(2)< \alpha < \pi ;

Show Solution

Solution

The functions \sin \alpha and \cos \alpha are linked by the formula \sin^(2)\alpha + \cos^(2) \alpha = 1. Substituting into this formula \cos \alpha = -\frac12, we get:

\sin^(2)\alpha + \left (-\frac12 \right)^2 = 1

This equation has 2 solutions:

\sin \alpha = \pm \sqrt(1-\frac14) = \pm \frac(\sqrt 3)(2)

By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter, the sine is positive, so \sin \alpha = \frac(\sqrt 3)(2).

To find tg \alpha , we use the formula tg \alpha = \frac(\sin \alpha)(\cos \alpha)

tg \alpha = \frac(\sqrt 3)(2) : \frac12 = \sqrt 3

Example 2

Find \cos \alpha and ctg \alpha if and \frac(\pi)(2)< \alpha < \pi .

Show Solution

Solution

Substituting into the formula \sin^(2)\alpha + \cos^(2) \alpha = 1 conditional number \sin \alpha=\frac(\sqrt3)(2), we get \left (\frac(\sqrt3)(2)\right)^(2) + \cos^(2) \alpha = 1. This equation has two solutions \cos \alpha = \pm \sqrt(1-\frac34)=\pm\sqrt\frac14.

By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter, the cosine is negative, so \cos \alpha = -\sqrt\frac14=-\frac12.

In order to find ctg \alpha , we use the formula ctg \alpha = \frac(\cos \alpha)(\sin \alpha). We know the corresponding values.

ctg \alpha = -\frac12: \frac(\sqrt3)(2) = -\frac(1)(\sqrt 3).

I will not convince you not to write cheat sheets. Write! Including cheat sheets on trigonometry. Later I plan to explain why cheat sheets are needed and how cheat sheets are useful. And here - 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 "go in pairs": cosine-cosine, sine-sine. And one more thing: cosines are “inadequate”. They “everything is wrong”, so they change the signs: “-” to “+”, and vice versa.

Sinuses - "mix": sine-cosine, cosine-sine.

2. Sum and difference formulas:

cosines always "go in pairs". Having added two cosines - "buns", we get a pair of cosines - "koloboks". And subtracting, we definitely won’t get koloboks. We get a couple of sines. Still with a minus ahead.

Sinuses - "mix" :

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

When do we get a pair of cosines? When adding the cosines. That's why

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

"Mixing" is obtained both by adding and subtracting sines. Which is more fun: adding or subtracting? That's right, fold. And for the formula take addition:

In the first and third formulas in brackets - the amount. From the rearrangement of the places of the terms, the sum does not change. 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 sum

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

- surely there will be tasks in trigonometry. Trigonometry is often disliked for having to cram a huge amount of difficult formulas teeming with sines, cosines, tangents and cotangents. The site already once gave advice on how to remember a forgotten formula, using the example of the Euler and Peel formulas.

And in this article we will try to show that it is enough to firmly know only five of the simplest trigonometric formulas, and to have about the rest general idea and take them out as you go. It's like with DNA: they are not stored in a molecule complete drawings finished living being. It contains, rather, instructions for assembling it from the available amino acids. So it is in trigonometry, knowing some general principles, we will get all the necessary formulas from a small set of those that must be kept in mind.

We will rely on the following formulas:

From the formulas for the sine and cosine of the sums, knowing that the cosine function is even and that the sine function is odd, substituting -b for b, we obtain formulas for the differences:

1. Sine of difference: sin(a-b) = sinacos(-b)+cosasin(-b) = sinacosb-cosasinb
2. cosine difference: cos(a-b) = cosacos(-b)-sinasin(-b) = cosacosb+sinasinb

Putting a \u003d b into the same formulas, we obtain the formulas for the sine and cosine of double angles:

1. Sine of a double angle: sin2a = sin(a+a) = sinacosa+cosasina = 2sinacosa
2. Cosine of a double angle: cos2a = cos(a+a) = cosacosa-sinasina = cos2a-sin2a

The formulas for other multiple angles are obtained similarly:

1. Sine of a triple angle: sin3a = sin(2a+a) = sin2acosa+cos2asina = (2sinacosa)cosa+(cos2a-sin2a)sina = 2sinacos2a+sinacos2a-sin 3 a = 3 sinacos2a-sin 3 a = 3 sina(1-sin2a)-sin 3 a = 3 sina-4sin 3a
2. Cosine of a triple angle: cos3a = cos(2a+a) = cos2acosa-sin2asina = (cos2a-sin2a)cosa-(2sinacosa)sina = cos 3a- sin2acosa-2sin2acosa = cos 3a-3 sin2acosa = cos 3 a-3(1- cos2a)cosa = 4cos 3a-3 cosa

Before moving on, let's consider one problem.
Given: the angle is acute.
Find its cosine if
Solution given by one student:
Because , then sina= 3,a cosa = 4.
(From mathematical humor)

So, the definition of tangent connects this function with both sine and cosine. But you can get a formula that gives the connection of the tangent only with the cosine. To derive it, we take the basic trigonometric identity: sin 2 a+cos 2 a= 1 and divide it by cos 2 a. We get:

So the solution to this problem would be:

(Because the angle is acute, the + sign is taken when extracting the root)

The formula for the tangent of the sum is another one that is hard to remember. Let's output it like this:

immediately output and

From the cosine formula for a double angle, you can get the sine and cosine formulas for a half angle. To do this, to the left side of the double angle cosine formula:
cos2 a = cos 2 a-sin 2 a
we add a unit, and to the right - a trigonometric unit, i.e. sum of squares of sine and cosine.
cos2a+1 = cos2a-sin2a+cos2a+sin2a
2cos 2 a = cos2 a+1
expressing cosa across cos2 a and performing a change of variables, we get:

The sign is taken depending on the quadrant.

Similarly, subtracting one from the left side of the equality, and the sum of the squares of the sine and cosine from the right side, we get:
cos2a-1 = cos2a-sin2a-cos2a-sin2a
2sin 2 a = 1-cos2 a

And finally, to convert the sum of trigonometric functions into a product, we use the following trick. Suppose we need to represent the sum of sines as a product sina+sinb. Let's introduce variables x and y such that a = x+y, b+x-y. Then
sina+sinb = sin(x+y)+ sin(x-y) = sin x cos y+ cos x sin y+ sin x cos y- cos x sin y=2 sin x cos y. Let us now express x and y in terms of a and b.

Since a = x+y, b = x-y, then . That's why

You can withdraw immediately

1. Partition formula products of sine and cosine in amount: sinacosb = 0.5(sin(a+b)+sin(a-b))

We recommend that you practice and derive formulas for converting the product of the difference of sines and the sum and difference of cosines into a product, as well as for splitting the products of sines and cosines into a sum. Having done these exercises, you will thoroughly master the skill of deriving trigonometric formulas and will not get lost even in the most difficult control, olympiad or testing.

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 their application to 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, 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 ° - 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 in converting trigonometric expressions.

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