What is sine and cosine. Sine, cosine, tangent, cotangent of an acute angle. Trigonometric functions

The concepts of sine, cosine, tangent and cotangent are the main categories of trigonometry - a branch of mathematics, and are inextricably linked with the definition of an angle. Possession of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. That is why trigonometric calculations often cause difficulties for schoolchildren and students. To overcome them, you should become more familiar with trigonometric functions and formulas.

Concepts in trigonometry

To understand the basic concepts of trigonometry, you must first decide what a right triangle and an angle in a circle are, and why all basic trigonometric calculations are associated with them. A triangle in which one of the angles is 90 degrees is a right triangle. Historically, this figure was often used by people in architecture, navigation, art, astronomy. Accordingly, studying and analyzing the properties of this figure, people came to the calculation of the corresponding ratios of its parameters.

The main categories associated with right triangles are the hypotenuse and the legs. The hypotenuse is the side of a triangle that is opposite the right angle. The legs, respectively, are the other two sides. The sum of the angles of any triangle is always 180 degrees.

Spherical trigonometry is a section of trigonometry that is not studied at school, but in applied sciences such as astronomy and geodesy, scientists use it. A feature of a triangle in spherical trigonometry is that it always has a sum of angles greater than 180 degrees.

Angles of a triangle

In a right triangle, the sine of an angle is the ratio of the leg opposite the desired angle to the hypotenuse of the triangle. Accordingly, the cosine is the ratio of the adjacent leg and the hypotenuse. Both of these values ​​always have a value less than one, since the hypotenuse is always longer than the leg.

The tangent of an angle is a value equal to the ratio of the opposite leg to the adjacent leg of the desired angle, or sine to cosine. The cotangent, in turn, is the ratio of the adjacent leg of the desired angle to the opposite cactet. The cotangent of an angle can also be obtained by dividing the unit by the value of the tangent.

unit circle

A unit circle in geometry is a circle whose radius is equal to one. Such a circle is constructed in the Cartesian coordinate system, with the center of the circle coinciding with the origin point, and the initial position of the radius vector is determined by the positive direction of the X axis (abscissa axis). Each point of the circle has two coordinates: XX and YY, that is, the coordinates of the abscissa and ordinate. Selecting any point on the circle in the XX plane, and dropping the perpendicular from it to the abscissa axis, we get a right triangle formed by a radius to the selected point (let us denote it by the letter C), a perpendicular drawn to the X axis (the intersection point is denoted by the letter G), and a segment the abscissa axis between the origin (the point is denoted by the letter A) and the intersection point G. The resulting triangle ACG is a right triangle inscribed in a circle, where AG is the hypotenuse, and AC and GC are the legs. The angle between the radius of the circle AC and the segment of the abscissa axis with the designation AG, we define as α (alpha). So, cos α = AG/AC. Given that AC is the radius of the unit circle, and it is equal to one, it turns out that cos α=AG. Similarly, sin α=CG.

In addition, knowing these data, you can determine the coordinate of point C on the circle, since cos α \u003d AG, and sin α \u003d CG, which means that point C has given coordinates(cos α;sin α). Knowing that the tangent is equal to the ratio of the sine to the cosine, we can determine that tg α \u003d y / x, and ctg α \u003d x / y. Considering the angles in a negative coordinate system, it can be calculated that the sine and cosine values ​​of some angles can be negative.

Calculations and basic formulas

Values ​​of trigonometric functions

Having considered the essence of trigonometric functions through the unit circle, we can derive the values ​​of these functions for some angles. The values ​​are listed in the table below.

The simplest trigonometric identities

Equations in which there is an unknown value under the sign of the trigonometric function are called trigonometric. Identities with the value sin x = α, k is any integer:

1. sin x = 0, x = πk.
2. 2. sin x \u003d 1, x \u003d π / 2 + 2πk.
3. sin x \u003d -1, x \u003d -π / 2 + 2πk.
4. sin x = a, |a| > 1, no solutions.
5. sin x = a, |a| ≦ 1, x = (-1)^k * arcsin α + πk.

Identities with the value cos x = a, where k is any integer:

1. cos x = 0, x = π/2 + πk.
2. cos x = 1, x = 2πk.
3. cos x \u003d -1, x \u003d π + 2πk.
4. cos x = a, |a| > 1, no solutions.
5. cos x = a, |a| ≦ 1, х = ±arccos α + 2πk.

Identities with the value tg x = a, where k is any integer:

1. tg x = 0, x = π/2 + πk.
2. tg x \u003d a, x \u003d arctg α + πk.

Identities with value ctg x = a, where k is any integer:

1. ctg x = 0, x = π/2 + πk.
2. ctg x \u003d a, x \u003d arcctg α + πk.

Cast formulas

This category of constant formulas denotes methods by which you can go from trigonometric functions of the form to functions of the argument, that is, convert the sine, cosine, tangent and cotangent of an angle of any value to the corresponding indicators of the angle of the interval from 0 to 90 degrees for greater convenience of calculations.

The formulas for reducing functions for the sine of an angle look like this:

• sin(900 - α) = α;
• sin(900 + α) = cos α;
• sin(1800 - α) = sin α;
• sin(1800 + α) = -sin α;
• sin(2700 - α) = -cos α;
• sin(2700 + α) = -cos α;
• sin(3600 - α) = -sin α;
• sin(3600 + α) = sin α.

For the cosine of an angle:

• cos(900 - α) = sin α;
• cos(900 + α) = -sin α;
• cos(1800 - α) = -cos α;
• cos(1800 + α) = -cos α;
• cos(2700 - α) = -sin α;
• cos(2700 + α) = sin α;
• cos(3600 - α) = cos α;
• cos(3600 + α) = cos α.

The use of the above formulas is possible subject to two rules. First, if the angle can be represented as a value (π/2 ± a) or (3π/2 ± a), the value of the function changes:

• from sin to cos;
• from cos to sin;
• from tg to ctg;
• from ctg to tg.

The value of the function remains unchanged if the angle can be represented as (π ± a) or (2π ± a).

Secondly, the sign of the reduced function does not change: if it was initially positive, it remains so. The same is true for negative functions.

Addition Formulas

These formulas express the values ​​of the sine, cosine, tangent, and cotangent of the sum and difference of two rotation angles in terms of their trigonometric functions. Angles are usually denoted as α and β.

The formulas look like this:

1. sin(α ± β) = sin α * cos β ± cos α * sin.
2. cos(α ± β) = cos α * cos β ∓ sin α * sin.
3. tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α * tan β).
4. ctg(α ± β) = (-1 ± ctg α * ctg β) / (ctg α ± ctg β).

These formulas are valid for any angles α and β.

Double and triple angle formulas

Trigonometric formulas for double and triple angle are formulas that relate the functions of the angles 2α and 3α, respectively, to the trigonometric functions of the angle α. Derived from addition formulas:

1. sin2α = 2sinα*cosα.
2. cos2α = 1 - 2sin^2α.
3. tg2α = 2tgα / (1 - tg^2 α).
4. sin3α = 3sinα - 4sin^3α.
5. cos3α = 4cos^3α - 3cosα.
6. tg3α = (3tgα - tg^3 α) / (1-tg^2 α).

Transition from sum to product

Considering that 2sinx*cosy = sin(x+y) + sin(x-y), simplifying this formula, we obtain the identity sinα + sinβ = 2sin(α + β)/2 * cos(α − β)/2. Similarly, sinα - sinβ = 2sin(α - β)/2 * cos(α + β)/2; cosα + cosβ = 2cos(α + β)/2 * cos(α − β)/2; cosα - cosβ = 2sin(α + β)/2 * sin(α − β)/2; tgα + tgβ = sin(α + β) / cosα * cosβ; tgα - tgβ = sin(α - β) / cosα * cosβ; cosα + sinα = √2sin(π/4 ∓ α) = √2cos(π/4 ± α).

Transition from product to sum

These formulas follow from the identities for the transition of the sum to the product:

• sinα * sinβ = 1/2*;
• cosα * cosβ = 1/2*;
• sinα * cosβ = 1/2*.

Reduction formulas

In these identities, the square and cubic degree sine and cosine can be expressed in terms of the sine and cosine of the first power of a multiple angle:

• sin^2 α = (1 - cos2α)/2;
• cos^2α = (1 + cos2α)/2;
• sin^3 α = (3 * sinα - sin3α)/4;
• cos^3 α = (3 * cosα + cos3α)/4;
• sin^4 α = (3 - 4cos2α + cos4α)/8;
• cos^4 α = (3 + 4cos2α + cos4α)/8.

Universal substitution

The universal trigonometric substitution formulas express trigonometric functions in terms of the tangent of a half angle.

• sin x \u003d (2tgx / 2) * (1 + tg ^ 2 x / 2), while x \u003d π + 2πn;
• cos x = (1 - tg^2 x/2) / (1 + tg^2 x/2), where x = π + 2πn;
• tg x \u003d (2tgx / 2) / (1 - tg ^ 2 x / 2), where x \u003d π + 2πn;
• ctg x \u003d (1 - tg ^ 2 x / 2) / (2tgx / 2), while x \u003d π + 2πn.

Special cases

Particular cases of the simplest trigonometric equations are given below (k is any integer).

Private for sine:

Cosine quotients:

Private for tangent:

Cotangent quotients:

Theorems

Sine theorem

There are two versions of the theorem - simple and extended. Simple sine theorem: a/sin α = b/sin β = c/sin γ. In this case, a, b, c are the sides of the triangle, and α, β, γ are the opposite angles, respectively.

Extended sine theorem for an arbitrary triangle: a/sin α = b/sin β = c/sin γ = 2R. In this identity, R denotes the radius of the circle in which the given triangle is inscribed.

Cosine theorem

The identity is displayed in this way: a^2 = b^2 + c^2 - 2*b*c*cos α. In the formula, a, b, c are the sides of the triangle, and α is the angle opposite side a.

Tangent theorem

The formula expresses the relationship between the tangents of two angles, and the length of the sides opposite them. The sides are labeled a, b, c, and the corresponding opposite angles are α, β, γ. The formula of the tangent theorem: (a - b) / (a+b) = tg((α - β)/2) / tg((α + β)/2).

Cotangent theorem

Associates the radius of a circle inscribed in a triangle with the length of its sides. If a, b, c are the sides of a triangle, and A, B, C, respectively, are their opposite angles, r is the radius of the inscribed circle, and p is the half-perimeter of the triangle, the following identities hold:

• ctg A/2 = (p-a)/r;
• ctg B/2 = (p-b)/r;
• ctg C/2 = (p-c)/r.

Applications

Trigonometry is not only a theoretical science related to mathematical formulas. Its properties, theorems and rules are used in practice by various industries human activity- astronomy, aerial and maritime navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measurement work, computer graphics, cartography, oceanography, and many others.

Sine, cosine, tangent and cotangent are the basic concepts of trigonometry, with which you can mathematically express the relationship between angles and lengths of sides in a triangle, and find the desired quantities through identities, theorems and rules.

The ratio of the opposite leg to the hypotenuse is called sinus acute angle right triangle.

\sin \alpha = \frac(a)(c)

Cosine of an acute angle of a right triangle

The ratio of the nearest leg to the hypotenuse is called cosine of an acute angle right triangle.

\cos \alpha = \frac(b)(c)

Tangent of an acute angle of a right triangle

The ratio of the opposite leg to the adjacent leg is called acute angle tangent right triangle.

tg \alpha = \frac(a)(b)

Cotangent of an acute angle of a right triangle

The ratio of the adjacent leg to the opposite leg is called cotangent of an acute angle right triangle.

ctg \alpha = \frac(b)(a)

Sine of an arbitrary angle

The ordinate of the point on the unit circle to which the angle \alpha corresponds is called sine of an arbitrary angle rotation \alpha .

\sin \alpha=y

Cosine of an arbitrary angle

The abscissa of a point on the unit circle to which the angle \alpha corresponds is called cosine of an arbitrary angle rotation \alpha .

\cos \alpha=x

Tangent of an arbitrary angle

The ratio of the sine of an arbitrary rotation angle \alpha to its cosine is called tangent of an arbitrary angle rotation \alpha .

tg \alpha = y_(A)

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

Cotangent of an arbitrary angle

The ratio of the cosine of an arbitrary rotation angle \alpha to its sine is called cotangent of an arbitrary angle rotation \alpha .

ctg \alpha =x_(A)

ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

An example of finding an arbitrary angle

If \alpha is some angle AOM , where M is a point on the unit circle, then

\sin \alpha=y_(M) , \cos \alpha=x_(M) , tg \alpha=\frac(y_(M))(x_(M)), ctg \alpha=\frac(x_(M))(y_(M)).

For example, if \angle AOM = -\frac(\pi)(4), then: the ordinate of the point M is -\frac(\sqrt(2))(2), the abscissa is \frac(\sqrt(2))(2) and that's why

\sin \left (-\frac(\pi)(4) \right)=-\frac(\sqrt(2))(2);

\cos \left (\frac(\pi)(4) \right)=\frac(\sqrt(2))(2);

tg;

ctg \left (-\frac(\pi)(4) \right)=-1.

Table of values ​​of sines of cosines of tangents of cotangents

The values ​​of the main frequently encountered angles are given in the table:

Trigonometry is a branch of mathematics that studies trigonometric functions and their use in geometry. The development of trigonometry began at the time ancient Greece. During the Middle Ages, scientists from the Middle East and India made an important contribution to the development of this science.

This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the main trigonometric functions: sine, cosine, tangent and cotangent. Their meaning in the context of geometry is explained and illustrated.

Yandex.RTB R-A-339285-1

Initially, the definitions of trigonometric functions, whose argument is an angle, were expressed through the ratio of the sides of a right triangle.

Definitions of trigonometric functions

The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.

The cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.

The tangent of the angle (t g α) is the ratio of the opposite leg to the adjacent one.

The cotangent of the angle (c t g α) is the ratio of the adjacent leg to the opposite one.

These definitions are given for an acute angle of a right triangle!

Let's give an illustration.

In triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.

The definitions of sine, cosine, tangent, and cotangent make it possible to calculate the values ​​of these functions from the known lengths of the sides of a triangle.

Important to remember!

The range of sine and cosine values: from -1 to 1. In other words, sine and cosine take values ​​from -1 to 1. The range of tangent and cotangent values ​​is the entire number line, that is, these functions can take any value.

The definitions given above refer to acute angles. In trigonometry, the concept of the angle of rotation is introduced, the value of which, unlike an acute angle, is not limited by frames from 0 to 90 degrees. The angle of rotation in degrees or radians is expressed by any real number from - ∞ to + ∞.

In this context, one can define the sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Imagine a unit circle centered at the origin of the Cartesian coordinate system.

The starting point A with coordinates (1 , 0) rotates around the center of the unit circle by some angle α and goes to point A 1 . The definition is given through the coordinates of the point A 1 (x, y).

Sine (sin) of the rotation angle

The sine of the rotation angle α is the ordinate of the point A 1 (x, y). sinα = y

Cosine (cos) of the angle of rotation

The cosine of the angle of rotation α is the abscissa of the point A 1 (x, y). cos α = x

Tangent (tg) of rotation angle

The tangent of the angle of rotation α is the ratio of the ordinate of the point A 1 (x, y) to its abscissa. t g α = y x

Cotangent (ctg) of rotation angle

The cotangent of the angle of rotation α is the ratio of the abscissa of the point A 1 (x, y) to its ordinate. c t g α = x y

Sine and cosine are defined for any angle of rotation. This is logical, because the abscissa and ordinate of the point after the rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is not defined when the point after the rotation goes to the point with zero abscissa (0 , 1) and (0 , - 1). In such cases, the expression for the tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with the cotangent. The difference is that the cotangent is not defined in cases where the ordinate of the point vanishes.

Important to remember!

Sine and cosine are defined for any angles α.

The tangent is defined for all angles except α = 90° + 180° k , k ∈ Z (α = π 2 + π k , k ∈ Z)

The cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)

When deciding practical examples don't say "sine of the angle of rotation α". The words "angle of rotation" are simply omitted, implying that from the context it is already clear what is at stake.

Numbers

What about the definition of the sine, cosine, tangent and cotangent of a number, and not the angle of rotation?

Sine, cosine, tangent, cotangent of a number

Sine, cosine, tangent and cotangent of a number t a number is called, which is respectively equal to the sine, cosine, tangent and cotangent in t radian.

For example, the sine of 10 π equal to the sine rotation angle of 10 π rad.

There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's consider it in more detail.

Any real number t a point on the unit circle is put in correspondence with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are defined in terms of the coordinates of this point.

The starting point on the circle is point A with coordinates (1 , 0).

positive number t

Negative number t corresponds to the point to which the starting point will move if it moves counterclockwise around the circle and passes the path t .

Now that the connection between the number and the point on the circle has been established, we proceed to the definition of sine, cosine, tangent and cotangent.

Sine (sin) of the number t

Sine of a number t- ordinate of the point of the unit circle corresponding to the number t. sin t = y

Cosine (cos) of t

Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x

Tangent (tg) of t

Tangent of a number t- the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t. t g t = y x = sin t cos t

The latter definitions are consistent with and do not contradict the definition given at the beginning of this section. Point on a circle corresponding to a number t, coincides with the point to which the starting point passes after turning through the angle t radian.

Trigonometric functions of angular and numerical argument

Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° · k , k ∈ Z (α = π 2 + π · k , k ∈ Z) corresponds to a certain value of the tangent. The cotangent, as mentioned above, is defined for all α, except for α = 180 ° k , k ∈ Z (α = π k , k ∈ Z).

We can say that sin α , cos α , t g α , c t g α are functions of the angle alpha, or functions of the angular argument.

Similarly, one can speak of sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a specific value of the sine or cosine of a number t. All numbers other than π 2 + π · k , k ∈ Z, correspond to the value of the tangent. The cotangent is similarly defined for all numbers except π · k , k ∈ Z.

Basic functions of trigonometry

Sine, cosine, tangent and cotangent are the basic trigonometric functions.

It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.

Let's return to the data at the very beginning of the definitions and the angle alpha, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are in full agreement with the geometric definitions given by the ratios of the sides of a right triangle. Let's show it.

Take a unit circle centered on a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw from the resulting point A 1 (x, y) perpendicular to the x-axis. In the resulting right triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of the point A 1 (x, y) . The length of the leg opposite the corner is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.

In accordance with the definition from geometry, the sine of the angle α is equal to the ratio of the opposite leg to the hypotenuse.

sin α \u003d A 1 H O A 1 \u003d y 1 \u003d y

This means that the definition of the sine of an acute angle in a right triangle through the aspect ratio is equivalent to the definition of the sine of the angle of rotation α, with alpha lying in the range from 0 to 90 degrees.

Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Initially, sine and cosine arose due to the need to calculate quantities in right triangles. It was noticed that if the value of 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 leg to the hypotenuse, and the cosine is the ratio of the adjacent leg to the hypotenuse.

Theorems of cosines and sines

But cosines and sines can be used not only in right triangles. To find the value of an obtuse or acute angle, the side of any triangle, it is enough to apply the cosine and sine theorem.

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 these sides by the cosine of the angle between them."

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

Derivatives

A derivative is a mathematical tool that shows how quickly a function changes with respect 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 ​​\u200b\u200bof the derivatives of trigonometric functions: sine and cosine. The derivative of the sine is the cosine, and the derivative of the cosine is the sine, but with a minus sign.

Application in mathematics

Especially often, sines and cosines are used in solving right triangles and problems related to 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 complex shapes and objects into "simple" triangles. Engineers and, often dealing with calculations of aspect ratios and degree measures, spent a lot of time and effort calculating cosines and sines of non-table angles.

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

Radian - the angular value of the arc, along the length equal to the radius or 57.295779513 ° degrees.

Degree (in geometry) - 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°.

The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concept of angle. In order to understand well these, 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 radius equal to length arcs). 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. Here 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 legs: the hypotenuse is the side that lies opposite the right angle (in our example, this is the side); the legs are the two remaining sides and (those adjacent to right angle), moreover, if we consider the legs relative 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, it will turn out the same angle a certain amount, 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

Does not exist

Does not exist

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. Thus,

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!

1.

It can be seen that. But we know what corresponds to a full turn 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 tabular 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 table values ​​of the cosine and sine are equal, and having determined that the cosine here takes negative meaning, and the sine is positive, we have:

More similar examples understand when studying formulas for reducing trigonometric functions in the topic.

Thus, the desired point has coordinates.

4.

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).