How to find the ratio of the adjacent leg to the hypotenuse. Sine, cosine, tangent and cotangent in trigonometry: definitions, examples

The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concept of angle. To understand these well at first glance, complex concepts(which cause a state of horror in many schoolchildren), and make sure that “the devil is not as scary as he is painted”, let's start from the very beginning and understand the concept of an angle.

The concept of angle: radian, degree

Let's look at the picture. The vector "turned" relative to the point by a certain amount. So the measure of this rotation relative to the initial position will be injection.

What else do you need to know about the concept of angle? Well, units of angle, of course!

Angle, both in geometry and trigonometry, can be measured in degrees and radians.

The angle at (one degree) is called the central angle in the circle, based on a circular arc equal to the part of the circle. Thus, the entire circle consists of "pieces" of circular arcs, or the angle described by the circle is equal.

That is, the figure above shows an angle that is equal, that is, this angle is based on a circular arc with the size of the circumference.

An angle in radians is a central angle in a circle, based on a circular arc, the length of which is equal to the radius of the circle. Well, did you understand? If not, then let's look at the picture.

So, the figure shows an angle equal to a radian, that is, this angle is based on a circular arc, the length of which is equal to the radius of the circle (the length is equal to the length or the radius is equal to the length of the arc). Thus, the length of the arc is calculated by the formula:

Where is the central angle in radians.

Well, knowing this, can you answer how many radians contains an angle described by a circle? Yes, for this you need to remember the formula for the circumference of a circle. There she is:

Well, now let's correlate these two formulas and get that the angle described by the circle is equal. That is, correlating the value in degrees and radians, we get that. Respectively, . As you can see, unlike "degrees", the word "radian" is omitted, since the unit of measurement is usually clear from the context.

How many radians are? That's right!

Got it? Then fasten forward:

Any difficulties? Then look answers:

Right triangle: sine, cosine, tangent, cotangent of an angle

So, with the concept of the angle figured out. But what is the sine, cosine, tangent, cotangent of an angle? Let's figure it out. For this, a right triangle will help us.

What are the sides of a right triangle called? That's right, the hypotenuse and the legs: the hypotenuse is the side that lies opposite right angle(in our example, this is the side); the legs are the two remaining sides and (those that are adjacent to the right angle), moreover, if we consider the legs with respect to the angle, then the leg is the adjacent leg, and the leg is the opposite one. So, now let's answer the question: what are the sine, cosine, tangent and cotangent of an angle?

Sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.

in our triangle.

Cosine of an angle- this is the ratio of the adjacent (close) leg to the hypotenuse.

in our triangle.

Angle tangent- this is the ratio of the opposite (far) leg to the adjacent (close).

in our triangle.

Cotangent of an angle- this is the ratio of the adjacent (close) leg to the opposite (far).

in our triangle.

These definitions are necessary remember! To make it easier to remember which leg to divide by what, you need to clearly understand that in tangent and cotangent only the legs sit, and the hypotenuse appears only in sinus and cosine. And then you can come up with a chain of associations. For example, this one:

cosine→touch→touch→adjacent;

Cotangent→touch→touch→adjacent.

First of all, it is necessary to remember that the sine, cosine, tangent and cotangent as ratios of the sides of a triangle do not depend on the lengths of these sides (at one angle). Do not believe? Then make sure by looking at the picture:

Consider, for example, the cosine of an angle. By definition, from a triangle: , but we can calculate the cosine of an angle from a triangle: . You see, the lengths of the sides are different, but the value of the cosine of one angle is the same. Thus, the values of sine, cosine, tangent and cotangent depend solely on the magnitude of the angle.

If you understand the definitions, then go ahead and fix them!

For the triangle shown in the figure below, we find.

Well, did you get it? Then try it yourself: calculate the same for the corner.

Unit (trigonometric) circle

Understanding the concepts of degrees and radians, we considered a circle with a radius equal to. Such a circle is called single. It is very useful in the study of trigonometry. Therefore, we dwell on it in a little more detail.

As you can see, this circle is built in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin, the initial position of the radius vector is fixed along the positive direction of the axis (in our example, this is the radius).

Each point of the circle corresponds to two numbers: the coordinate along the axis and the coordinate along the axis. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, remember about the considered right-angled triangle. In the figure above, you can see two whole right triangles. Consider a triangle. It is rectangular because it is perpendicular to the axis.

What is equal to from a triangle? That's right. In addition, we know that is the radius of the unit circle, and therefore, . Substitute this value into our cosine formula. Here's what happens:

And what is equal to from a triangle? Well, of course, ! Substitute the value of the radius into this formula and get:

So, can you tell me what are the coordinates of a point that belongs to the circle? Well, no way? And if you realize that and are just numbers? What coordinate does it correspond to? Well, of course, the coordinate! What coordinate does it correspond to? That's right, coordinate! Thus, the point.

And what then are equal and? That's right, let's use the appropriate definitions of tangent and cotangent and get that, a.

What if the angle is larger? Here, for example, as in this picture:

What has changed in this example? Let's figure it out. To do this, we again turn to a right-angled triangle. Consider a right triangle: an angle (as adjacent to an angle). What is the value of the sine, cosine, tangent and cotangent of an angle? That's right, we adhere to the corresponding definitions of trigonometric functions:

Well, as you can see, the value of the sine of the angle still corresponds to the coordinate; the value of the cosine of the angle - the coordinate; and the values of tangent and cotangent to the corresponding ratios. Thus, these relations are applicable to any rotations of the radius vector.

It has already been mentioned that the initial position of the radius vector is along the positive direction of the axis. So far we have rotated this vector counterclockwise, but what happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain size, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles, and when rotating clockwise - negative.

So, we know that a whole revolution of the radius vector around the circle is or. Is it possible to rotate the radius vector by or by? Well, of course you can! In the first case, therefore, the radius vector will make one complete revolution and stop at position or.

In the second case, that is, the radius vector will make three complete revolutions and stop at position or.

Thus, from the above examples, we can conclude that angles that differ by or (where is any integer) correspond to the same position of the radius vector.

The figure below shows an angle. The same image corresponds to the corner, and so on. This list can be continued indefinitely. All these angles can be written with the general formula or (where is any integer)

Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values \u200b\u200bare equal to:

Here's a unit circle to help you:

Any difficulties? Then let's figure it out. So we know that:

From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner at corresponds to a point with coordinates, therefore:

Does not exist;

Further, adhering to the same logic, we find out that the corners in correspond to points with coordinates, respectively. Knowing this, it is easy to determine the values of trigonometric functions at the corresponding points. Try it yourself first, then check the answers.

Answers:

Does not exist

Thus, we can make the following table:

There is no need to remember all these values. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values of trigonometric functions:

But the values \u200b\u200bof the trigonometric functions of the angles in and, given in the table below, must be remembered:

Do not be afraid, now we will show one of the examples rather simple memorization of the corresponding values:

To use this method, it is vital to remember the values of the sine for all three measures of the angle (), as well as the value of the tangent of the angle in. Knowing these values, it is quite easy to restore the entire table - the cosine values are transferred in accordance with the arrows, that is:

Knowing this, you can restore the values for. The numerator " " will match and the denominator " " will match. Cotangent values are transferred in accordance with the arrows shown in the figure. If you understand this and remember the diagram with arrows, then it will be enough to remember the entire value from the table.

Coordinates of a point on a circle

Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation?

Well, of course you can! Let's bring out general formula for finding the coordinates of a point.

Here, for example, we have such a circle:

We are given that the point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the point by degrees.

As can be seen from the figure, the coordinate of the point corresponds to the length of the segment. The length of the segment corresponds to the coordinate of the center of the circle, that is, it is equal to. The length of a segment can be expressed using the definition of cosine:

Then we have that for the point the coordinate.

By the same logic, we find the value of the y coordinate for the point. In this way,

So in general view point coordinates are determined by the formulas:

Circle center coordinates,

circle radius,

Angle of rotation of the radius vector.

As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are zero, and the radius is equal to one:

Well, let's try these formulas for a taste, practicing finding points on a circle?

1. Find the coordinates of a point on a unit circle obtained by turning a point on.

2. Find the coordinates of a point on a unit circle obtained by rotating a point on.

3. Find the coordinates of a point on a unit circle obtained by turning a point on.

4. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.

5. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.

Having trouble finding the coordinates of a point on a circle?

Solve these five examples (or understand the solution well) and you will learn how to find them!

It can be seen that. And we know what corresponds to a full turn of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:

2. The circle is unit with a center at a point, which means that we can use simplified formulas:

It can be seen that. We know what corresponds to two complete rotations of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:

Sine and cosine are table values. We remember their values and get:

Thus, the desired point has coordinates.

3. The circle is unit with a center at a point, which means that we can use simplified formulas:

It can be seen that. Let's depict the considered example in the figure:

The radius makes angles with the axis equal to and. Knowing that the tabular values of the cosine and sine are equal, and having determined that the cosine here takes a negative value, and the sine is positive, we have:

Similar examples are analyzed in more detail when studying the formulas for reducing trigonometric functions in the topic.

Thus, the desired point has coordinates.

Angle of rotation of the radius vector (by condition)

To determine the corresponding signs of sine and cosine, we construct a unit circle and an angle:

As you can see, the value, that is, is positive, and the value, that is, is negative. Knowing the tabular values of the corresponding trigonometric functions, we obtain that:

Let's substitute the obtained values into our formula and find the coordinates:

Thus, the desired point has coordinates.

5. To solve this problem, we use formulas in general form, where

The coordinates of the center of the circle (in our example,

Circle radius (by condition)

Angle of rotation of the radius vector (by condition).

Substitute all the values into the formula and get:

and - table values. We remember and substitute them into the formula:

Thus, the desired point has coordinates.

SUMMARY AND BASIC FORMULA

The sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.

The cosine of an angle is the ratio of the adjacent (close) leg to the hypotenuse.

The tangent of an angle is the ratio of the opposite (far) leg to the adjacent (close).

The cotangent of an angle is the ratio of the adjacent (close) leg to the opposite (far).

In this article, we will show how definitions of sine, cosine, tangent and cotangent of angle and number in trigonometry. Here we will talk about notation, give examples of records, give graphic illustrations. In conclusion, we draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.

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Definition of sine, cosine, tangent and cotangent

Let's follow how the concept of sine, cosine, tangent and cotangent is formed in the school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent is given acute angle in a right triangle. And later trigonometry is studied, which refers to the sine, cosine, tangent and cotangent of the angle of rotation and the number. We give all these definitions, give examples and give the necessary comments.

Acute angle in a right triangle

From the course of geometry, the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle are known. They are given as the ratio of the sides of a right triangle. We present their formulations.

Definition.

Sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse.

Definition.

Cosine of an acute angle in a right triangle- it's an attitude adjacent leg to the hypotenuse.

Definition.

Tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.

Definition.

Cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite leg.

The notation of sine, cosine, tangent and cotangent is also introduced there - sin, cos, tg and ctg, respectively.

For example, if ABC is a right triangle with a right angle C, then the sine of the acute angle A is equal to the ratio of the opposite leg BC to the hypotenuse AB, that is, sin∠A=BC/AB.

These definitions allow you to calculate the values of the sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from the known values of the sine, cosine, tangent, cotangent and the length of one of the sides, find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is 3 and the hypotenuse AB is 7 , then we could calculate the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7 .

Angle of rotation

In trigonometry, they begin to look at the angle more widely - they introduce the concept of angle of rotation. The angle of rotation, unlike an acute angle, is not limited to frames from 0 to 90 degrees, the angle of rotation in degrees (and in radians) can be expressed by any real number from −∞ to +∞.

In this light, the definitions of the sine, cosine, tangent and cotangent are no longer an acute angle, but an angle of arbitrary magnitude - the angle of rotation. They are given through the x and y coordinates of the point A 1 , into which the so-called initial point A(1, 0) passes after it rotates through an angle α around the point O - the beginning of a rectangular Cartesian coordinate system and the center of the unit circle.

Definition.

Sine of rotation angleα is the ordinate of the point A 1 , that is, sinα=y .

Definition.

cosine of the angle of rotationα is called the abscissa of the point A 1 , that is, cosα=x .

Definition.

Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tgα=y/x .

Definition.

The cotangent of the angle of rotationα is the ratio of the abscissa of the point A 1 to its ordinate, that is, ctgα=x/y .

The sine and cosine are defined for any angle α, since we can always determine the abscissa and ordinate of the point, which is obtained by rotating the starting point by the angle α. And tangent and cotangent are not defined for any angle. The tangent is not defined for such angles α at which the initial point goes to a point with zero abscissa (0, 1) or (0, −1) , and this takes place at angles 90°+180° k , k∈Z (π /2+π k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for such angles α at which the starting point goes to a point with zero ordinate (1, 0) or (−1, 0) , and this is the case for angles 180° k , k ∈Z (π k rad).

So, the sine and cosine are defined for any rotation angles, the tangent is defined for all angles except 90°+180° k , k∈Z (π/2+π k rad), and the cotangent is for all angles except 180° ·k , k∈Z (π·k rad).

The notations already known to us appear in the definitions sin, cos, tg and ctg, they are also used to denote the sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the notation tan and cotcorresponding to tangent and cotangent). So the sine of the rotation angle of 30 degrees can be written as sin30°, the records tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α . Recall that when writing the radian measure of an angle, the notation "rad" is often omitted. For example, the cosine of a rotation angle of three pi rads is usually denoted cos3 π .

In conclusion of this paragraph, it is worth noting that in talking about the sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase "sine of the angle of rotation alpha", they usually use the phrase "sine of the angle of alpha" or even shorter - "sine of alpha". The same applies to cosine, and tangent, and cotangent.

Let's also say that the definitions of the sine, cosine, tangent, and cotangent of an acute angle in a right triangle are consistent with the definitions just given for the sine, cosine, tangent, and cotangent of a rotation angle ranging from 0 to 90 degrees. We will substantiate this.

Numbers

Definition.

Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the angle of rotation in t radians, respectively.

For example, the cosine of the number 8 π is, by definition, the number cosine angle of 8 π rad. And the cosine of the angle in 8 π rad is equal to one, therefore, the cosine of the number 8 π is equal to 1.

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is assigned a point of the unit circle centered at the origin of the rectangular coordinate system, and the sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's dwell on this in more detail.

Let us show how the correspondence between real numbers and points of the circle is established:

the number 0 is assigned the starting point A(1, 0) ;
positive number t corresponds to the point of the unit circle, which we will get to if we move along the circle from the starting point in the counterclockwise direction and go through a path of length t;
negative number t corresponds to the point of the unit circle, which we will get to if we move around the circle from the starting point in a clockwise direction and go through a path of length |t| .

Now let's move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point of the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).

Definition.

The sine of a number t is the ordinate of the unit circle point corresponding to the number t , that is, sint=y .

Definition.

The cosine of a number t is called the abscissa of the point of the unit circle corresponding to the number t , that is, cost=x .

Definition.

Tangent of a number t is the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of the number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost .

Definition.

Cotangent of a number t is the ratio of the abscissa to the ordinate of the point of the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is as follows: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t : ctgt=cost/sint .

Here we note that the definitions just given agree with the definition given at the beginning of this subsection. Indeed, the point of the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point through an angle of t radians.

It is also worth clarifying this point. Let's say we have a sin3 entry. How to understand whether the sine of the number 3 or the sine of the rotation angle of 3 radians is in question? This is usually clear from the context, otherwise it probably doesn't matter.

Trigonometric functions of angular and numerical argument

According to the definitions given in the previous paragraph, each rotation angle α corresponds to a well-defined value sin α , as well as the value cos α . In addition, all rotation angles other than 90°+180° k , k∈Z (π/2+π k rad) correspond to the values tgα , and other than 180° k , k∈Z (π k rad ) are the values of ctgα . Therefore sinα, cosα, tgα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.

Similarly, we can talk about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a well-defined value of sint , as well as cost . In addition, all numbers other than π/2+π·k , k∈Z correspond to the values tgt , and the numbers π·k , k∈Z correspond to the values ctgt .

The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.

It is usually clear from the context that we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can consider the independent variable as both a measure of the angle (the angle argument) and a numeric argument.

However, the school mainly studies numeric functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking about functions, it is reasonable to consider trigonometric functions as functions of numerical arguments.

Connection of definitions from geometry and trigonometry

If we consider the angle of rotation α from 0 to 90 degrees, then the data in the context of trigonometry of the definition of the sine, cosine, tangent and cotangent of the angle of rotation are fully consistent with the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's substantiate this.

Draw a unit circle in the rectangular Cartesian coordinate system Oxy. Note the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get the point A 1 (x, y) . Let's drop the perpendicular A 1 H from the point A 1 to the Ox axis.

It is easy to see that in a right triangle the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of the point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of the point A 1 , that is, |A 1 H|=y , and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y . And by definition from trigonometry, the sine of the angle of rotation α is equal to the ordinate of the point A 1, that is, sinα=y. This shows that the definition of the sine of an acute angle in a right triangle is equivalent to the definition of the sine of the angle of rotation α for α from 0 to 90 degrees.

Similarly, it can be shown that the definitions of the cosine, tangent, and cotangent of an acute angle α are consistent with the definitions of the cosine, tangent, and cotangent of the angle of rotation α.

Bibliography.

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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 exclusively on the example of 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 knowledge gained is used by students 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, 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 the 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 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 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 independently derive 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 case, 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, taking the angle of alpha equal to the angle of 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 task, 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: lengths three parties and the dimensions of the 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.

In life, we often have to deal with math problems: at school, at university, and then helping your child with homework. People of certain professions will encounter mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article, we will analyze one of them: finding the leg of a right triangle.

What is a right triangle

First, let's remember what a right triangle is. Right triangle- it geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles of this figure is 90 degrees. The sides that form a right angle are called the legs, and the side that lies opposite the right angle is called the hypotenuse.

Finding the leg of a right triangle

There are several ways to find out the length of the leg. I would like to consider them in more detail.

Pythagorean theorem to find the leg of a right triangle

If we know the hypotenuse and the leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs. We transform the formula and get: a²=c²-b².

Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c²=a²+b² → a²=c²-b². Next, we decide: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).

Trigonometric relations to find the leg of a right triangle

It is also possible to find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. To solve the problems, the table below will help us. Let's consider these options.

Find the leg of a right triangle using the sine

The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d a / c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.

Example. The hypotenuse is 10 cm and angle A is 30 degrees. According to the table, we calculate the sine of angle A, it is equal to 1/2. Then, using the transformed formula, we solve: a=sin∠A*c; a=1/2*10; a=5 (cm).

Find the leg of a right triangle using cosine

The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos \u003d b / c, where b is the leg adjacent to the given angle, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.

Example. Angle A is 60 degrees, the hypotenuse is 10 cm. According to the table, we calculate the cosine of angle A, it is equal to 1/2. Next, we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).

Find the leg of a right triangle using the tangent

The tangent of an angle (tg) is the ratio of the opposite leg to the adjacent one. Formula: tg \u003d a / b, where a is the leg opposite to the corner, and b is adjacent. Let's transform the formula and get: a=tg*b.

Example. Angle A is 45 degrees, the hypotenuse is 10 cm. According to the table, we calculate the tangent of angle A, it is equal to Solve: a=tg∠A*b; a=1*10; a=10 (cm).

Find the leg of a right triangle using the cotangent

The cotangent of an angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg \u003d b / a, where b is the leg adjacent to the corner, and is opposite. In other words, the cotangent is the "inverted tangent". We get: b=ctg*a.

Example. Angle A is 30 degrees, the opposite leg is 5 cm. According to the table, the tangent of angle A is √3. Calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).

So, now you know how to find the leg in a right triangle. As you can see, it is not so difficult, the main thing is to remember the formulas.

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