One of the areas of mathematics that students struggle with the most is trigonometry. It is not surprising: 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 use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.
Origins of trigonometry
Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.
Historically, the main object of study in this branch of mathematical science was right triangles. 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 in question 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 in art.
First stage
Initially, people talked about the relationship between angles and sides solely using 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 in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract problems. 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, and cotangent began to be used in spherical geometry, where different 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 surface marking will be “arc-shaped” in three-dimensional space.
Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.
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. It 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 the 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, which 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 equal to 180 degrees.
Definition
Finally, with a firm understanding of the geometric basis, one can turn to the definition of 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 side 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 their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.
Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.
So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.
The simplest formulas
In trigonometry you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.
The first formula 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 need to know the size of the angle rather than 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: 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 does trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, conversion rules and several basic formulas you can at any time withdraw the required more complex formulas on a piece of paper.
Formulas for double angles and addition of arguments
Two more formulas that you need to learn are related to the values of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of 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 beta angle.
Finally, note that double angle formulas can be rearranged to reduce the power 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 the figure, and the size of each side, etc.
The sine theorem states that dividing the length of each side of a triangle by the opposite angle results in the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a 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 adjacent angle - 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.
Careless mistakes
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 take a look at the most popular ones.
Firstly, you shouldn't convert fractions to decimals until you get the final result - you can leave the answer as common fraction, unless otherwise stated in the conditions. 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 your time on unnecessary mathematical operations. This is especially true for values such as the root of three or the root of two, because they are found in problems at every step. The same goes for 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 you will also demonstrate a complete lack of understanding 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 confuse them, 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 practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts that make it possible to calculate the distance to distant stars, predict the fall of a meteorite, or send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a 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're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: length three sides and the sizes of the three angles. The only difference in the tasks lies in the fact that different input data are given.
How to find sine, cosine, tangent based on known lengths legs or hypotenuse, you now know. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of a trigonometry problem is to find the roots of an ordinary equation or system of equations. And here regular school mathematics will help you.
Sine is one of the basic trigonometric functions, the use of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and calculating the sine is sometimes needed to solve various problems. In general, calculating the sine will help consolidate drawing skills and knowledge of trigonometric identities.
Games with ruler and pencil
A simple task: how to find the sine of an angle drawn on paper? To solve, you will need a regular ruler, a triangle (or compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, you first need to complete the acute angle to the shape of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. We will need to maintain an angle of exactly 90°, for which we need a clerical triangle.
Using a compass is a little more accurate, but will take more time. On one of the rays you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until the intersections of these lines are obtained. By connecting the intersection points of our circles with each other, we get a strict perpendicular to the ray of our angle; all that remains is to extend the line until it intersects with another ray.
In the resulting triangle, you need to use a ruler to measure the side opposite the corner and the long side on one of the rays. The ratio of the first dimension to the second will be the desired value of the sine acute angle.
Find the sine for an angle greater than 90°
For an obtuse angle the task is not much more difficult. We need to draw a ray from the vertex in the opposite direction using a ruler to form a straight line with one of the rays of the angle we are interested in. The resulting acute angle should be treated as described above; the sines of adjacent angles that together form a reverse angle of 180° are equal.
Calculating sine using other trigonometric functions
Also, calculating the sine is possible if the values of other trigonometric functions of the angle or at least the lengths of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.
How to find the sine with a known cosine of an angle? The first trigonometric identity, based on the Pythagorean theorem, states that the sum of the squares of the sine and cosine of the same angle is equal to one.
How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far side by the near side or dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the square sine according to the first trigonometric identity and, through simple manipulations, we reduce the equation to the calculation of the square sine through the tangent; accordingly, to calculate the sine, you will have to extract the root of the result obtained.
How to find the sine with a known cotangent of an angle? The value of the cotangent can be calculated by dividing the length of the leg closest to the angle by the length of the far one, as well as dividing the cosine by the sine, that is, the cotangent is a function inverse to the tangent relative to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α = 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with tangent, which will look like this.
How to find the sine of three sides of a triangle
There is a formula for finding the length of the unknown side of any triangle, not just a right triangle, from two known sides using the trigonometric function of the cosine of the opposite angle. She looks like this.
Average level
Right triangle. The Complete Illustrated Guide (2019)
RIGHT TRIANGLE. FIRST LEVEL.
In problems, the right angle is not at all necessary - the lower left, so you need to learn to recognize a right triangle in this form,
and in this
and in this 
What's good about a right triangle? Well... first of all, there are special beautiful names for his sides.
Attention to the drawing!
Remember and don't confuse: there are two legs, and there is only one hypotenuse(one and only, unique and longest)!
Well, we’ve discussed the names, now the most important thing: the Pythagorean Theorem.
Pythagorean theorem.
This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought a lot of benefit to those who know it. And the best thing about it is that it is simple.
So, Pythagorean theorem:
Do you remember the joke: “Pythagorean pants are equal on all sides!”?
Let's draw these same Pythagorean pants and look at them. 
Doesn't it look like some kind of shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:
"Sum areas of squares, built on the legs, is equal to square area, built on the hypotenuse."
Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came out.
In this picture, the sum of the areas of the small squares is equal to the area of the large square. And so that children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with this joke about Pythagorean pants.
Why are we now formulating the Pythagorean theorem?
Did Pythagoras suffer and talk about squares?
You see, in ancient times there was no... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:
It should be easy now:
| Square of the hypotenuse equal to the sum squares of legs. |
Well, the most important theorem about right triangles has been discussed. If you are interested in how it is proven, read the following levels of theory, and now let's move on... to dark forest... trigonometry! To the terrible words sine, cosine, tangent and cotangent.
Sine, cosine, tangent, cotangent in a right triangle.
In fact, everything is not so scary at all. Of course, the “real” definition of sine, cosine, tangent and cotangent should be looked at in the article. But I really don’t want to, do I? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:
Why is everything just about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!
1.
Actually it sounds like this:
What about the angle? Is there a leg that is opposite the corner, that is, an opposite (for an angle) leg? Of course have! This is a leg!
What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and
Now, pay attention! Look what we got:
See how cool it is:
Now let's move on to tangent and cotangent.
How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?
See how the numerator and denominator have swapped places?
And now the corners again and made an exchange:
Summary
Let's briefly write down everything we've learned.
 |
Pythagorean theorem: |
The main theorem about right triangles is the Pythagorean theorem.
Pythagorean theorem
By the way, do you remember well what legs and hypotenuse are? If not very good, then look at the picture - refresh your knowledge
It is quite possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true? How can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.
See how cleverly we divided its sides into lengths and!
Now let's connect the marked dots
Here we, however, noted something else, but you yourself look at the drawing and think why this is so.
What is the area of the larger square? Right, . What about a smaller area? Certainly, . The total area of the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses. What happened? Two rectangles. This means that the area of the “cuts” is equal.
Let's put it all together now.
Let's transform:
So we visited Pythagoras - we proved his theorem in an ancient way.
Right triangle and trigonometry
For a right triangle, the following relations hold:
The sine of an acute angle is equal to the ratio of the opposite side to the hypotenuse
The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.
The tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side.
The cotangent of an acute angle is equal to the ratio of the adjacent side to the opposite side.
And once again all this in the form of a tablet:
It is very comfortable!
Signs of equality of right triangles
I. On two sides
II. By leg and hypotenuse
III. By hypotenuse and acute angle
IV. Along the leg and acute angle
a) 
b) 
Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:
THEN TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.
Need to in both triangles the leg was adjacent, or in both it was opposite.
Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Look at the topic “and pay attention to the fact that for equality of “ordinary” triangles, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides. But for the equality of right triangles, only two corresponding elements are enough. Great, right?
The situation is approximately the same with the signs of similarity of right triangles.
Signs of similarity of right triangles
I. Along an acute angle
II. On two sides
III. By leg and hypotenuse
Median in a right triangle
Why is this so?
Instead of a right triangle, consider a whole rectangle.
Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?
And what follows from this?
So it turned out that
- - median:
Remember this fact! Helps a lot!
What’s even more surprising is that the opposite is also true.
What good can be obtained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture
Look carefully. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But there is only one point in the triangle, the distances from which from all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCLE. So what happened?
So let's start with this “besides...”.
Let's look at and.
But similar triangles have all equal angles!
The same can be said about and
Now let's draw it together:
What benefit can be derived from this “triple” similarity?
Well, for example - two formulas for the height of a right triangle.
Let us write down the relations of the corresponding parties:
To find the height, we solve the proportion and get the first formula "Height in a right triangle":
So, let's apply the similarity: .
What will happen now?
Again we solve the proportion and get the second formula:
You need to remember both of these formulas very well and use the one that is more convenient. Let's write them down again
Pythagorean theorem:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: .
Signs of equality of right triangles:
- on two sides:
- by leg and hypotenuse: or
- along the leg and adjacent acute angle: or
- along the leg and the opposite acute angle: or
- by hypotenuse and acute angle: or.
Signs of similarity of right triangles:
- one acute corner: or
- from the proportionality of two legs:
- from the proportionality of the leg and hypotenuse: or.
Sine, cosine, tangent, cotangent in a right triangle
- The sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse:
- The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
- The tangent of an acute angle of a right triangle is the ratio of the opposite side to the adjacent side:
- The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .
Height of a right triangle: or.
In a right triangle, the median drawn from the vertex right angle, is equal to half the hypotenuse: .
Area of a right triangle:
- via legs:
Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.
Geometric definition
|BD| - length of the arc of a circle with center at point A.
α is the angle expressed in radians.
Tangent ( tan α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| .
Cotangent ( ctg α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .
Tangent
Where n- whole.
In Western literature, tangent is denoted as follows:
.
;
;
.
Graph of the tangent function, y = tan x
Cotangent
Where n- whole.
In Western literature, cotangent is denoted as follows:
.
The following notations are also accepted:
;
;
.
Graph of the cotangent function, y = ctg x
Properties of tangent and cotangent
Periodicity
Functions y = tg x and y = ctg x are periodic with period π.
Parity
The tangent and cotangent functions are odd.
Areas of definition and values, increasing, decreasing
The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table ( n- whole).
| y= tg x | y= ctg x | |
| Scope and continuity | ||
| Range of values | -∞ < y < +∞ | -∞ < y < +∞ |
| Increasing | - | |
| Descending | - | |
| Extremes | - | - |
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y= 0 | - |
Formulas
Expressions using sine and cosine
;
;
;
;
;
Formulas for tangent and cotangent from sum and difference
The remaining formulas are easy to obtain, for example
Product of tangents
Formula for the sum and difference of tangents
This table presents the values of tangents and cotangents for certain values of the argument. 
Expressions using complex numbers
Expressions through hyperbolic functions
;
;
Derivatives
; .
.
Derivative of the nth order with respect to the variable x of the function:
.
Deriving formulas for tangent > > > ; for cotangent > > >
Integrals
Series expansions
To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in power series for functions sin x And cos x and divide these polynomials by each other, . This produces the following formulas.
At .
at .
Where Bn- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to Laplace's formula: 
Inverse functions
Inverse functions to tangent and cotangent are arctangent and arccotangent, respectively.
Arctangent, arctg
, Where n- whole.
Arccotangent, arcctg
, Where n- whole.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.
We will begin our study of trigonometry with the right triangle. Let's define what sine and cosine are, as well as tangent and cotangent of an acute angle. This is the basics of trigonometry.
Let us recall that right angle is an angle equal to 90 degrees. In other words, half a turned angle.
Sharp corner- less than 90 degrees.
Obtuse angle- greater than 90 degrees. When applied to such an angle, “obtuse” is not an insult, but a mathematical term :-)
Let's draw a right triangle. A right angle is usually denoted by . Please note that the side opposite the corner is indicated by the same letter, only small. Thus, the side opposite angle A is designated .
The angle is denoted by the corresponding Greek letter.
Hypotenuse of a right triangle is the side opposite the right angle.
Legs- sides lying opposite acute angles.
The leg lying opposite the angle is called opposite(relative to angle). The other leg, which lies on one of the sides of the angle, is called adjacent.
Sinus The acute angle in a right triangle is the ratio of the opposite side 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 side to the adjacent:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of the angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent side to the opposite (or, which is the same, the ratio of cosine to sine):
Note the basic relationships for sine, cosine, tangent, and cotangent below. They will be useful to us when solving problems.
Let's prove some of them.
Okay, we have given definitions and written down formulas. But why do we still need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is equal to.
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. Knowing the two sides of a right triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what should you do if in a right triangle you know one angle (except the right angle) and one side, but you need to find the other sides?
This is what people in the past encountered when making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric angle functions- give relationships between parties And corners triangle. Knowing the angle, you can find all its trigonometric functions using 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 the values of sine, cosine, tangent and cotangent for “good” angles from to.
Please note the two red dashes in the table. At appropriate angle values, tangent and cotangent do not exist.
Let's look at several trigonometry problems from the FIPI Task Bank.
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 it using the Pythagorean theorem.
The problem is solved.
Often in problems there are triangles with angles and or with angles and. Remember 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 looked at problems solving right triangles - that is, finding unknown sides or angles. But that's not all! IN Unified State Exam options in mathematics there are many problems where the sine, cosine, tangent or cotangent of the external angle of a triangle appears. More on this in the next article.