Cos sums. Basic trigonometric identities, their formulations and derivation
We continue our conversation about the most used formulas in trigonometry. The most important of them are the addition formulas.
Definition 1
Addition formulas allow you to express the functions of the difference or the sum of two angles using the trigonometric functions of these angles.
To begin with, we will present full list addition formulas, then we will prove them and analyze a few illustrative examples.
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Basic addition formulas in trigonometry
There are eight basic formulas: the sine of the sum and the sine of the difference of two angles, the cosines of the sum and difference, the 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 by the cosine of the second;
Multiply the cosine of the first angle by the sine of the first;
Add up the resulting values.
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 must not be added, but subtracted from each other. Thus, we calculate the products of the sine of the first angle by the cosine of the second and the cosine of the first angle by 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 difference: we calculate the products of the sines and cosines of the given 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, in the numerator of which is the sum of the tangents of the desired angles, and in the denominator is the unit from which the product of the tangents of the desired angles is subtracted. Everything is clear from her graphic notation: t g (α + β) = t g α + t g β 1 - t g α t g β
6. Tangent of difference. We calculate the values of the difference and the product of the tangents of these angles and deal 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 sum. For calculations using this formula, we need the product and the sum of the cotangents of these angles, with 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 difference . The formula is similar to the previous one, but in the numerator and denominator - minus, and 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 pairwise similar. Using the signs ± (plus-minus) and ∓ (minus-plus), we can group them for ease of notation:
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 the 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. From it, you can then easily deduce the rest of the evidence.
Let us clarify the basic concepts. We need a unit circle. It will turn out if we take a certain point A and rotate around the center (point O) the angles α and β. 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 complete 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)
Bottom line: 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) .
Recall the definitions of sine and cosine: sine is a function of an angle equal to the ratio of the leg of the opposite angle to the hypotenuse, cosine is the sine of the additional angle. Therefore, the points A 1 and A2 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's 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 single 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 deduce the equality:
cos (α - β) = cos α cos β + sin α sin β
Thus, the formula for the cosine of the difference is proved.
Now we will prove the following formula - the cosine of the sum. This is easier because we can use the previous calculations. 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 formula for the cosine of the sum. The last line uses the property of the sine and cosine of opposite angles.
The formula for the sine of the sum can be derived from the formula for the cosine of the difference. Let's take the reduction formula for this:
of the form sin (α + β) = cos (π 2 (α + β)) . So
sin (α + β) \u003d cos (π 2 (α + β)) \u003d cos ((π 2 - α) - β) \u003d \u003d 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 formula for the sine of the difference:
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 the addition formulas for the tangent and cotangent. Let us recall 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 a formula of the following form: 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 difference tangent 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
The formulas for the 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
- 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 everything 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:
- Sine of difference: sin(a-b) = sinacos(-b)+cosasin(-b) = sinacosb-cosasinb
- 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:
- Sine of a double angle: sin2a = sin(a+a) = sinacosa+cosasina = 2sinacosa
- Cosine of a double angle: cos2a = cos(a+a) = cosacosa-sinasina = cos2a-sin2a
The formulas for other multiple angles are obtained similarly:
- 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
- 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 through 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
- 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.
One of the branches of mathematics with which schoolchildren cope with the greatest difficulties is trigonometry. No wonder: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to apply trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to deduce complex logical chains.
Origins of trigonometry
Acquaintance with this science should begin with the definition of the sine, cosine and tangent of the angle, but first you need to figure out what trigonometry does in general.
Historically, right triangles have been the main object of study in this section of mathematical science. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values of all parameters of the figure under consideration using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy, and even art.
First stage
Initially, people talked about the relationship of angles and sides solely on the example right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in Everyday life this branch of mathematics.
The study of trigonometry at school today begins with right triangles, after which the acquired knowledge is used by students in physics and solving abstract trigonometric equations, work with which begins in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where other rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence, at least because earth's surface, and the surface of any other planet is convex, which means that any marking of the surface will be "arc-shaped" in three-dimensional space.
Take the globe and thread. Attach the thread to any two points on the globe so that it is taut. Pay attention - it has acquired the shape of an arc. It is with such forms that spherical geometry, which is used in geodesy, astronomy, and other theoretical and applied fields, deals.
Right triangle
Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.
The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. She is the longest. We remember that, according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.
For example, if two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.
The two remaining sides that form a right angle are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is 180 degrees.
Definition
Finally, with a solid understanding of the geometric base, we can turn to the definition of the sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.
Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means that their ratio will always be less than one. Thus, if you get a sine or cosine with a value greater than 1 in the answer to the problem, look for an error in calculations or reasoning. This answer is clearly wrong.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. The same result will give the division of the sine by the cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, after which we divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same ratio as in the definition of tangent.
The cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing the unit by the tangent.
So, we have considered the definitions of what sine, cosine, tangent and cotangent are, and we can deal with formulas.
The simplest formulas
In trigonometry, one cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? And this is exactly what is required when solving problems.
The first formula that you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you want to know the value of the angle, not the side.
Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: after all, this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, the conversion rules and a few basic formulas you can at any time yourself display the required more complex formulas on a sheet of paper.
Double angle formulas and addition of arguments
Two more formulas that you need to learn are related to the values \u200b\u200bof the sine and cosine for the sum and difference of the angles. They are shown in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second, the pairwise product of the sine and cosine is added.
There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself by taking the alpha angle equal to the angle beta.
Finally, note that the double angle formulas can be converted to lower the degree of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of \u200b\u200bthe figure, and the size of each side, etc.
The sine theorem states that as a result of dividing the length of each of the sides of the triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all points of the given triangle.
The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the angle adjacent to them - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.
Mistakes due to inattention
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's get acquainted with the most popular of them.
Firstly, you should not convert ordinary fractions to decimals until the final result is obtained - you can leave the answer in the form common fraction unless the condition states otherwise. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author's idea, should be reduced. In this case, you will waste time on unnecessary mathematical operations. This is especially true for values such as the root of three or two, because they occur in tasks at every step. The same applies to rounding "ugly" numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but also demonstrate a complete misunderstanding of the subject. This is worse than a careless mistake.
Thirdly, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to mix them up, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to start studying trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on the surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
Finally
So you are sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole essence of trigonometry boils down to the fact that unknown parameters must be calculated from the known parameters of the triangle. There are six parameters in total: the lengths of three sides and the magnitudes of three angles. The whole difference in the tasks lies in the fact that different input data are given.
How to find the sine, cosine, tangent based on the known lengths of the legs or the hypotenuse, you now know. Since these terms mean nothing more than ratio, and ratio is a fraction, main goal finding the roots of an ordinary equation or a system of equations becomes a trigonometric problem. And here you will be helped by ordinary school mathematics.
We begin our study of trigonometry with a right triangle. Let's define what sine and cosine are, as well as tangent and cotangent acute angle. These are the basics of trigonometry.
Recall that right angle is an angle equal to 90 degrees. In other words, half of the unfolded corner.
Sharp corner- less than 90 degrees.
Obtuse angle- greater than 90 degrees. In relation to such an angle, "blunt" is not an insult, but a mathematical term :-)
Let's draw a right triangle. A right angle is usually denoted . Note that the side opposite the corner is denoted by the same letter, only small. So, the side lying opposite the angle A is denoted.
An angle is denoted by the corresponding Greek letter.
Hypotenuse A right triangle is the side opposite the right angle.
Legs- sides opposite sharp corners.
The leg opposite the corner is called opposite(relative to angle). The other leg, which lies on one side of the corner, is called adjacent.
Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:
Tangent acute angle in a right triangle - the ratio of the opposite leg to the adjacent:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent leg to the opposite (or, equivalently, the ratio of cosine to sine):
Pay attention to the basic ratios for sine, cosine, tangent and cotangent, which are given below. They will be useful to us in solving problems.
Let's prove some of them.
Okay, we have given definitions and written formulas. But why do we need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is.
We know the relationship between parties right triangle. This is the Pythagorean theorem: .
It turns out that knowing two angles in a triangle, you can find the third one. Knowing two sides in a right triangle, you can find the third. So, for angles - their ratio, for sides - their own. But what to do if in a right triangle one angle (except for a right one) and one side are known, but you need to find other sides?
This is what people faced in the past, making maps of the area and the starry sky. After all, it is not always possible to directly measure all the sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric functions of the angle- give the ratio between parties and corners triangle. Knowing the angle, you can find all of it trigonometric functions according to special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of sine, cosine, tangent and cotangent values for "good" angles from to.
Notice the two red dashes in the table. For the corresponding values of the angles, the tangent and cotangent do not exist.
Let's analyze several problems in trigonometry from the Bank of FIPI tasks.
1. In a triangle, the angle is , . Find .
The problem is solved in four seconds.
Because the , .
2. In a triangle, the angle is , , . Find .
Let's find by the Pythagorean theorem.
Problem solved.
Often in problems there are triangles with angles and or with angles and . Memorize the basic ratios for them by heart!
For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.
A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.
We considered problems for solving right triangles - that is, for finding unknown sides or angles. But that's not all! AT USE options in mathematics, there are many problems where the sine, cosine, tangent or cotangent of the outer angle of the triangle appears. More on this in the next article.
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