All formulas for cosines and sines. Basic trigonometric identities, their formulations and derivation

Sinus acute angle α of a right triangle is the ratio opposite leg to hypotenuse.
It is denoted as follows: sin α.

Cosine The acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is designated as follows: cos α.

Tangent acute angle α is the ratio opposite leg to the adjacent leg.
It is designated as follows: tg α.

Cotangent acute angle α is the ratio of the adjacent side to the opposite side.
It is designated as follows: ctg α.

The sine, cosine, tangent and cotangent of an angle depend only on the size of the angle.

Rules:

Basic trigonometric identities in a right triangle:

(α - acute angle opposite to the leg b and adjacent to the leg a . Side With – hypotenuse. β – second acute angle).

b sin α = - c	sin 2 α + cos 2 α = 1
a cos α = - c	1 1 + tan 2 α = -- cos 2 α
b tan α = - a	1 1 + cotg 2 α = -- sin 2 α
a ctg α = - b	1 1 1 + -- = -- tan 2 α sin 2 α
sin α tg α = -- cos α

As the acute angle increasessin α andtan α increase, andcos α decreases.

For any acute angle α:

sin (90° – α) = cos α

cos (90° – α) = sin α

Example-explanation:

Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.

Let's find out the sine of angle A and the cosine of angle B.

Solution .

1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of the acute angles is 90º, then angle B = 60º:

B = 90º – 30º = 60º.

2) Let's calculate sin A. We know that sine equal to the ratio opposite side to the hypotenuse. For angle A, the opposite side is side BC. So:

BC 3 1
sin A = -- = - = -
AB 6 2

3) Now let's calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC by AB - that is, perform the same actions as when calculating the sine of angle A:

BC 3 1
cos B = -- = - = -
AB 6 2

The result is:
sin A = cos B = 1/2.

sin 30º = cos 60º = 1/2.

It follows from this that in a right triangle, the sine of one acute angle is equal to the cosine of another acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° – α) = cos α
cos (90° – α) = sin α

Let's make sure of this again:

1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º – 60º) = cos 60º.
sin 30º = cos 60º.

2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° – 30º) = sin 30º.
cos 60° = sin 30º.

(For more information about trigonometry, see the Algebra section)

If the angle of the triangle is known, then you can use a special reference book and look up the sine there given angle. If the angle is not known, then you can use the theorem of sines. In a particular case, the sine of an angle in a right triangle is equal to the ratio of the opposite side to the hypotenuse.

Let's define what a sine is.

The sine of an angle (sin) in a triangle is the ratio of the opposite side to the hypotenuse.

So finding the sine of an angle is quite simple if you have the value of the leg and the hypotenuse.



To find the sine of an angle in any triangle, you need to use formulas. This figure shows the basic formulas for calculating the sine of an angle in a triangle:



Use these formulas to calculate.

If the magnitude of the angle is unknown, then this: the sine of the angle is equal to the ratio of the length of the side opposite the angle under consideration to the diameter of the circle circumscribed around the triangle. How to find this diameter? We need to find the center of the circumscribed circle. To do this, draw perpendiculars through the midpoints of any two sides of the triangle. The point of intersection of these perpendiculars is the center of the circumcircle. The distance from it to any vertex of the triangle is the radius of the circumscribed circle.

To answer this question correctly, you need to clarify the sine of the angle in which triangle you need to find. If this triangle arbitrary, then we can only do this by theorem of sines(see Alex's comprehensive answer here).

If you need to find the sine of an acute angle in rectangular triangle, then you need to use the definition of the sine of an angle (as the ratio of the opposite side to the hypotenuse). Then the answer will be: sine of angle A = BC/AV, where BC is the opposite side, AB is the hypotenuse.



Good day.

To find the sine of an angle/angles of a right triangle, you can use two methods:

• the first of them is to take a protractor and find the angle of the triangle (how many degrees), and then use the table to find the sine of this angle;
• the second method is to use the formula for finding the sine of an angle, which, as we know, is equal to the ratio of the opposite side to the hypotenuse.

You can find the sine of an angle in two ways and compare the values.

It's quite simple.

As I understand it, the problem boils down to the fact that we don’t know the angle of the triangle, and we need to find it.

In order to find the sine of an angle, and then the angle itself in an arbitrary triangle, you need to know the lengths of two sides: the side opposite the desired angle, and some other side, and also the size of the angle opposite this last side.

And then you need to apply the theorem of sines.

Let us denote the desired (unknown) angle as A, the opposite side a, the other known side b, the known angle B opposite this side.

By the law of sines: a/sin(A) = b/sin(B).

From here: sin(A) = a * sin(B)/b;

A = arcsina * sin(B)/b.

In the case of a right triangle, the task of finding the sine of any angle comes down to just calculating the ratio of the opposite leg of the angle to the hypotenuse - the resulting value will be the sine. In an arbitrary triangle, finding the sine of an angle is more difficult, but also possible. To do this, you need to know at least something about the parameters of the triangle. For example, if three sides of a triangle are known, then the angles are found using the cosine theorem, and then, if desired, the sine of the already found angle can be easily found.

In this article we will take a comprehensive look. Basic trigonometric identities are equalities that establish the relationship between sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other.

Let us immediately list the main trigonometric identities that we will analyze in this article. Let's write them down in a table, and below we'll give the output of these formulas and provide the necessary explanations.

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Relationship between sine and cosine of one angle

Sometimes they do not talk about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind . The explanation for this fact is quite simple: the equalities are obtained from the main trigonometric identity after dividing both of its parts by and respectively, and the equalities And follow from the definitions of sine, cosine, tangent and cotangent. We'll talk about this in more detail in the following paragraphs.

That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.

Before proving the main trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.

The basic trigonometric identity is very often used when converting trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often the basic trigonometric identity is used in reverse order: unit is replaced by the sum of the squares of the sine and cosine of any angle.

Tangent and cotangent through sine and cosine

Identities connecting tangent and cotangent with sine and cosine of one angle of view and follow immediately from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, sine is the ordinate of y, cosine is the abscissa of x, tangent is the ratio of the ordinate to the abscissa, that is, , and the cotangent is the ratio of the abscissa to the ordinate, that is, .

Thanks to such obviousness of the identities and Tangent and cotangent are often defined not through the ratio of abscissa and ordinate, but through the ratio of sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.

In conclusion of this paragraph, it should be noted that the identities and take place for all angles at which the trigonometric functions included in them make sense. So the formula is valid for any , other than (otherwise the denominator will have zero, and we did not define division by zero), and the formula - for all , different from , where z is any .

Relationship between tangent and cotangent

An even more obvious trigonometric identity than the previous two is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it holds for any angles other than , otherwise either the tangent or the cotangent are not defined.

Proof of the formula very simple. By definition and from where . The proof could have been carried out a little differently. Since , That .

So, the tangent and cotangent of the same angle at which they make sense are .

Teachers believe that every student should be able to carry out calculations and know trigonometric formulas, but not every teacher explains what sine and cosine are. What is their meaning, where are they used? Why are we talking about triangles, but the textbook shows a circle? Let's try to connect all the facts together.

School subject

The study of trigonometry usually begins in grades 7-8 high school. At this time, students are explained what sine and cosine are and are asked to solve geometric problems using these functions. Later, more complex formulas and expressions appear that are required algebraically transform (double and half angle formulas, power functions), work with the trigonometric circle.

However, teachers are not always able to clearly explain the meaning of the concepts used and the applicability of the formulas. Therefore, the student often does not see the point in this subject, and the memorized information is quickly forgotten. However, it is worth once explaining to a high school student, for example, the connection between function and oscillatory motion, and logical connection will be remembered for many years, and jokes about the uselessness of the item will become a thing of the past.

Usage

For the sake of curiosity, let's look into various branches of physics. Do you want to determine the range of a projectile? Or are you calculating the friction force between an object and a certain surface? Swinging the pendulum, watching the rays passing through the glass, calculating the induction? In almost any formula they appear trigonometric concepts. So what are sine and cosine?

Definitions

The sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the same hypotenuse. There is absolutely nothing complicated here. Perhaps students are usually confused by the values ​​they see on the trigonometry table because it involves square roots. Yes, getting decimals from them is not very convenient, but who said that all numbers in mathematics must be equal?

In fact, you can find a funny hint in trigonometry problem books: most of the answers here are even and, in the worst case, contain the root of two or three. The conclusion is simple: if your answer turns out to be a “multi-story” fraction, double-check the solution for errors in calculations or reasoning. And you will most likely find them.

What to remember

Like any science, trigonometry has data that needs to be learned.

First, you should remember numeric values for sines, cosines of a right triangle 0 and 90, as well as 30, 45 and 60 degrees. These indicators are found in nine out of ten school problems. By looking at these values ​​in a textbook, you will lose a lot of time, and there will be nowhere to look at them during a test or exam.

It must be remembered that the value of both functions cannot exceed one. If anywhere in your calculations you get a value outside the 0-1 range, stop and try the problem again.

The sum of the squares of sine and cosine is equal to one. If you have already found one of the values, use this formula to find the remaining one.

Theorems

There are two basic theorems in basic trigonometry: sines and cosines.

The first states that the ratio of each side of a triangle to the sine of the opposite angle is the same. The second is that the square of any side can be obtained by adding the squares of the two remaining sides and subtracting their double product multiplied by the cosine of the angle lying between them.

Thus, if we substitute the value of an angle of 90 degrees into the cosine theorem, we get... the Pythagorean theorem. Now, if you need to calculate the area of ​​a figure that is not a right triangle, you don’t have to worry anymore - the two theorems discussed will significantly simplify the solution of the problem.

Goals and objectives

Learning trigonometry will become much easier when you realize one simple fact: all the actions you perform are aimed at achieving just one goal. Any parameters of a triangle can be found if you know the bare minimum of information about it - this could be the value of one angle and the length of two sides or, for example, three sides.

To determine the sine, cosine, tangent of any angle, these data are sufficient, and with their help you can easily calculate the area of ​​the figure. Almost always, the answer requires one of the mentioned values, and they can be found using the same formulas.

Inconsistencies in learning trigonometry

One of the confusing questions that students prefer to avoid is discovering the connections between different concepts in trigonometry. It would seem that triangles are used to study the sines and cosines of angles, but for some reason the symbols are often found in the figure with a circle. In addition, there is a completely incomprehensible wave-like graph called a sine wave, which has no external resemblance neither with a circle nor with triangles.

Moreover, angles are measured either in degrees or in radians, and the number Pi, written simply as 3.14 (without units), for some reason appears in the formulas, corresponding to 180 degrees. How is all this connected?

Units of measurement

Why is Pi exactly 3.14? Do you remember what this meaning is? This is the number of radii that fit in an arc on half a circle. If the diameter of the circle is 2 centimeters, the circumference will be 3.14 * 2, or 6.28.

Second point: you may have noticed the similarity between the words “radian” and “radius”. The fact is that one radian is numerically equal to the angle taken from the center of the circle to an arc one radius long.

Now we will combine the acquired knowledge and understand why “Pi in half” is written on the top of the coordinate axis in trigonometry, and “Pi” is written on the left. This is an angular value measured in radians, because a semicircle is 180 degrees, or 3.14 radians. And where there are degrees, there are sines and cosines. It is easy to draw a triangle from the desired point, setting aside segments to the center and to the coordinate axis.

Let's look into the future

Trigonometry, studied in school, deals with a rectilinear coordinate system, where, no matter how strange it may sound, a straight line is a straight line.

But there is more complex ways working with space: the sum of the angles of the triangle here will be more than 180 degrees, and the straight line in our view will look like a real arc.

Let's move from words to action! Take an apple. Make three cuts with a knife so that when viewed from above you get a triangle. Take out the resulting piece of apple and look at the “ribs” where the peel ends. They are not straight at all. The fruit in your hands can be conventionally called round, but now imagine how complex the formulas must be with which you can find the area of ​​the cut piece. But some specialists solve such problems every day.

Trigonometric functions in life

Have you noticed that the most short route does an airplane from point A to point B on the surface of our planet have a pronounced arc shape? The reason is simple: the Earth is spherical, which means you can’t calculate much using triangles - you have to use more complex formulas.

You cannot do without the sine/cosine of an acute angle in any questions related to space. It’s interesting that a whole lot of factors come together here: trigonometric functions are required when calculating the motion of planets along circles, ellipses and various trajectories more complex shapes; the process of launching rockets, satellites, shuttles, undocking research vehicles; observing distant stars and studying galaxies that humans will not be able to reach in the foreseeable future.

In general, the field of activity for a person who knows trigonometry is very wide and, apparently, will only expand over time.

Conclusion

Today we learned, or at least repeated, what sine and cosine are. These are concepts that you don’t need to be afraid of - just want them and you will understand their meaning. Remember that trigonometry is not a goal, but only a tool that can be used to satisfy real human needs: build houses, ensure traffic safety, even explore the vastness of the universe.

Indeed, science itself may seem boring, but as soon as you find in it a way to achieve your own goals and self-realization, the learning process will become interesting, and your personal motivation will increase.

As homework Try to find ways to apply trigonometric functions in an area of ​​activity that interests you personally. Imagine, use your imagination, and then you will probably find that new knowledge will be useful to you in the future. And besides, mathematics is useful for general development thinking.

Instructions

Use the arcsine function to calculate the value of an angle in degrees if you know the value of the angle. If corner denoted by the letter α, in general view the solution can be written as follows: α = arcsin(sin(α)).

If you have the opportunity to use a computer, the easiest way to do practical calculations is to use the built-in operating system. In the last two versions of Windows OS, you can launch it like this: press the Win key, type “ka” and press Enter. In earlier releases of this OS, look for the “Calculator” link in the “Standard” subsection of the “All programs” section of the system’s main menu.

After launching the application, switch it to a mode that allows you to work with trigonometric functions. This can be done by selecting the “Engineering” line in the “View” section of the calculator menu or by pressing Alt + 2.

Enter the sine value. By default, the calculator interface does not have a button for calculating the arcsine. To be able to use this function, you need to invert the default button values ​​- click on the Inv key in the program window. In more earlier versions this button is replaced by a checkbox with the same designation - check it.

You can also use various services in calculations, of which there are more than enough on the Internet. For example, go to http://planetcalc.com/326/, scroll down a little and enter the sine value in the Input field. To start the calculation procedure, there is a button labeled Calculate - click on it. You will find the calculation result in the first row of the table under this button. In addition to the arc sine, it displays both the magnitudes and the arc tangent of the entered value.

The inverse of sine is called a trigonometric function arcsine. It can take values ​​within half of Pi, both positive and negative when measured in radians. When measured in degrees, these values ​​will respectively be in the range from -90° to +90°.

Instructions

Some “round” values ​​do not need to be calculated; they are easier to remember. For example: - if the function argument is zero, then the arcsine of it is also zero; - of 1/2 is equal to 30° or 1/6 Pi, if measured; - arcsine of -1/2 is -30° or -1/ 6 from the number Pi in; - the arcsine of 1 is equal to 90° or 1/2 of the number Pi in radians; - the arcsine of -1 is equal to -90° or -1/2 of the number Pi in radians;

To measure the values ​​of this function from other arguments, the easiest way is to use a standard Windows calculator, if you have one at hand. To start, open the main menu on the “Start” button (or by pressing the WIN key), go to the “All Programs” section, and then to the “Accessories” subsection and click “Calculator”.

Switch the calculator interface to the operating mode that allows you to calculate trigonometric functions. To do this, open the “View” section in its menu and select “Engineering” or “Scientific” (depending on the operating system used).

Enter the value of the argument from which the arctangent should be calculated. This can be done by clicking the calculator interface buttons with the mouse, or by pressing the keys on , or by copying the value (CTRL + C) and then pasting it (CTRL + V) into the calculator input field.

Select the units of measurement in which you need to obtain the result of the function calculation. Below the input field there are three options, from which you need to select (by clicking it with the mouse) one - , radians or rads.

Check the checkbox that inverts the functions indicated on the calculator interface buttons. Next to it is a short inscription Inv.

Click the sin button. The calculator will invert the function associated with it, perform the calculation and present you with the result in the specified units.

Video on the topic

On the right triangle, as the simplest of polygons, various scientists honed their knowledge in the field of trigonometry back in the days when no one even called this area of ​​mathematics by that word. Therefore, indicate the author who revealed patterns in the ratios of side lengths and angle values ​​in this plane geometric figure, is not possible today. Such relationships are called trigonometric functions and are divided into several groups, the main of which are conventionally considered to be “direct” functions. This group includes only two functions and one of them is sine.

Instructions

By definition, in a right triangle, one of the angles is equal to 90°, and due to the fact that the sum of its angles in Euclidean geometry must be equal to 180°, the other two angles are (i.e. 90°). The patterns of relationships between precisely these angles and side lengths describe trigonometric functions.

A function called the sine of an acute angle determines the ratio between the lengths of two sides of a right triangle, one of which lies opposite this acute angle, and the other is adjacent to it and lies opposite right angle. Since the side lying opposite the right angle in such a triangle is called the hypotenuse, and the other two are called the legs, the sine function can be formulated as the ratio between the lengths of the leg and the hypotenuse.

In addition to this simplest definition of this trigonometric function, there are more complex ones: through a circle in Cartesian coordinates, through series, through differential and functional equations. This function is continuous, that is, its arguments (“domain”) can be any number - from infinitely negative to infinitely positive. And the maximum values ​​of this function are limited to the range from -1 to +1 - this is the “range of its values”. Minimum value the sine takes place at an angle of 270°, which corresponds to 3/Pi, and the maximum is obtained at 90° (½ of Pi). The function values ​​become zero at 0°, 180°, 360°, etc. From all this it follows that sine is a periodic function and its period is equal to 360° or twice the number Pi.

For practical calculations of the values ​​of this function from a given argument, you can use the vast majority of them (including the software calculator built into operating system your computer) has an appropriate option.

Video on the topic

Sinus And cosine- these are direct trigonometric functions for which there are several definitions - through a circle in a Cartesian coordinate system, through solutions of a differential equation, through acute angles in a right triangle. Each of these definitions allows us to derive the relationship between these two functions. Below is perhaps the simplest way to express cosine through the sine - through their definitions for the acute angles of a right triangle.

Instructions

Express the sine of an acute angle of a right triangle in terms of the lengths of the sides of this figure. According to the definition, the sine of an angle (α) must be the ratio of the length of the side (a) lying opposite it - the leg - to the length of the side (c) opposite the right angle - the hypotenuse: sin(α) = a/c.

Find a similar formula for cosine but the same angle. By definition, this value should be expressed as the ratio of the length of the side (b) adjacent to this angle (the second leg) to the length of the side (c) lying opposite the right angle: cos(a) = a/c.

Rewrite the equality following from the Pythagorean theorem so that it involves the relationships between the legs and the hypotenuse derived in the previous two steps. To do this, first divide both of the original theorem (a² + b² = c²) by the square of the hypotenuse (a²/c² + b²/c² = 1), and then rewrite the resulting equality in this form: (a/c)² + (b/c )² = 1.

In the resulting expression, replace the ratio of the lengths of the legs and the hypotenuse with trigonometric functions, based on the formulas of the first and second steps: sin²(a) + cos²(a) = 1. Express cosine from the resulting equality: cos(a) = √(1 - sin²(a)). With this, the problem can be solved in general form.

If, in addition to the general one, you need to get a numerical result, use, for example, a calculator built into the operating room Windows system. A link to launch it in the “Standard” subsection of the “All programs” section of the OS menu. This link is formulated succinctly - “Calculator”. To be able to calculate trigonometric functions with this program, enable its “engineering” interface - press the key combination Alt + 2.

Enter the value of the sine of the angle in the conditions and click on the interface button marked x² - this will square the original value. Then type *-1 on the keyboard, press Enter, enter +1 and press Enter again - in this way you will subtract the square of the sine from one. Click on the radical key to extract the square and get the final result.

The study of triangles has been carried out by mathematicians for several millennia. The science of triangles - trigonometry - uses special quantities: sine and cosine.

Right triangle

Sine and cosine originally arose from the need to calculate quantities in right triangles. It was noticed that if the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how much these sides change in length, always remains the same.

This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the side adjacent to the hypotenuse.

Theorems of cosines and sines

But cosines and sines can be used for more than just right triangles. To find the value of an obtuse or acute angle or side of any triangle, it is enough to apply the theorem of cosines and sines.

The cosine theorem is quite simple: “The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them.”

There are two interpretations of the sine theorem: small and extended. According to the minor: “In a triangle, the angles are proportional to the opposite sides.” This theorem is often expanded due to the property of the circumscribed circle of a triangle: “In a triangle, the angles are proportional to the opposite sides, and their ratio is equal to the diameter of the circumscribed circle.”

Derivatives

The derivative is a mathematical tool that shows how quickly a function changes relative to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.

When solving problems, you need to know the tabular values ​​of the derivatives of trigonometric functions: sine and cosine. The derivative of a sine is a cosine, and a cosine is a sine, but with a minus sign.

Application in mathematics

Sines and cosines are especially often used when solving right triangles and tasks associated with them.

The convenience of sines and cosines is also reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking down complex shapes and objects into “simple” triangles. Engineers who often deal with calculations of aspect ratios and degree measures spent a lot of time and effort calculating the cosines and sines of non-tabular angles.

Then Bradis tables came to the rescue, containing thousands of values ​​of sines, cosines, tangents and cotangents of different angles. IN Soviet era some teachers forced their students to memorize pages of Bradis tables.

Radian is the angular value of an arc whose length is equal to the radius or 57.295779513° degrees.

A degree (in geometry) is 1/360th of a circle or 1/90th of a right angle.

π = 3.141592653589793238462… (approximate value of Pi).

Cosine table for angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°.

Angle x (in degrees)	0°	30°	45°	60°	90°	120°	135°	150°	180°	210°	225°	240°	270°	300°	315°	330°	360°
Angle x (in radians)	0	π/6	π/4	π/3	π/2	2 x π/3	3 x π/4	5 x π/6	π	7 x π/6	5 x π/4	4 x π/3	3 x π/2	5 x π/3	7 x π/4	11 x π/6	2 x π
cos x	1	√3/2 (0,8660)	√2/2 (0,7071)	1/2 (0,5)	0	-1/2 (-0,5)	-√2/2 (-0,7071)	-√3/2 (-0,8660)	-1	-√3/2 (-0,8660)	-√2/2 (-0,7071)	-1/2 (-0,5)	0	1/2 (0,5)	√2/2 (0,7071)	√3/2 (0,8660)	1