Whatever real number t is taken, it can be associated with a uniquely defined number sin t. True, the matching rule is quite complex; as we saw above, it is as follows.

To find the value of sin t using the number t, you need:

1) position the number circle in the coordinate plane so that the center of the circle coincides with the origin of coordinates, and the starting point A of the circle falls at point (1; 0);

2) find a point on the circle corresponding to the number t;

3) find the ordinate of this point.

This ordinate is sin t.

In fact, we are talking about the function u = sin t, where t is any real number.

All these functions are called trigonometric functions of the numerical argument t.

There are a number of relations that connect the values ​​of various trigonometric functions; we have already obtained some of these relations:

sin 2 t+cos 2 t = 1

From the last two formulas it is easy to obtain a relationship connecting tg t and ctg t:

All of these formulas are used in cases where, knowing the value of a trigonometric function, it is necessary to calculate the values ​​of other trigonometric functions.

The terms “sine”, “cosine”, “tangent” and “cotangent” were actually familiar, however, they were still used in a slightly different interpretation: in geometry and physics they considered sine, cosine, tangent and cotangent at the head(but not

numbers, as was in the previous paragraphs).

From geometry it is known that the sine (cosine) of an acute angle is the ratio of the legs of a right triangle to its hypotenuse, and the tangent (cotangent) of an angle is the ratio of the legs of a right triangle. A different approach to the concepts of sine, cosine, tangent and cotangent was developed in the previous paragraphs. In fact, these approaches are interrelated.

Let's take an angle with degree measure b o and place it in the “numeric circle in a rectangular coordinate system” model as shown in Fig. 14

the apex of the angle is compatible with the center

circles (with the origin of the coordinate system),

and one side of the corner is compatible with

the positive ray of the x-axis. Full stop

intersection of the second side of the angle with

denote by the circle the letter M. Ordina-

Fig. 14 b o, and the abscissa of this point is the cosine of the angle b o.

To find the sine or cosine of an angle b o it is not at all necessary to do these very complex constructions every time.

It is enough to note that the arc AM makes up the same part of the length of the number circle as the angle b o makes from the corner of 360°. If the length of the arc AM is denoted by the letter t, we get:

Thus,

For example,

It is believed that 30° is a degree measure of an angle, and a radian measure of the same angle: 30° = rad. At all:

In particular, I’m glad where, in turn, we get it from.

So what is 1 radian? There are various measures of length of segments: centimeters, meters, yards, etc. There are also various measures to indicate the magnitude of angles. We consider the central angles of the unit circle. An angle of 1° is the central angle subtended by an arc that is part of a circle. An angle of 1 radian is the central angle subtended by an arc of length 1, i.e. on an arc whose length is equal to the radius of the circle. From the formula, we find that 1 rad = 57.3°.

When considering the function u = sin t (or any other trigonometric function), we can consider the independent variable t to be a numerical argument, as was the case in the previous paragraphs, but we can also consider this variable to be a measure of the angle, i.e. corner argument. Therefore, when talking about a trigonometric function, in a certain sense it makes no difference to consider it a function of a numerical or angular argument.

Lesson and presentation on the topic: "Trigonometric function of the angular argument, degree measure of angle and radians"

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What we will study:
1. Let's remember geometry.
2. Definition of the angular argument.
3. Degree measure of angle.
4. Radian measure of angle.
5. What is a radian?
6. Examples and tasks for independent solution.

Repetition of geometry

Guys, in our functions:

y= sin(t), y= cos(t), y= tg(t), y= ctg(t)

The variable t can take not only numeric values, that is, be a numeric argument, but it can also be considered as a measure of an angle - an angular argument.

Let's remember geometry!
How did we define sine, cosine, tangent, cotangent there?

Sine of an angle - the ratio of the opposite side to the hypotenuse

Cosine of the angle - the ratio of the adjacent leg to the hypotenuse

Tangent of an angle is the ratio of the opposite side to the adjacent side.

Cotangent of an angle is the ratio of the adjacent side to the opposite side.

Definition of trigonometric function of angle argument

Let's define trigonometric functions as functions of the angular argument on the number circle:
Using the number circle and coordinate system, we can always easily find the sine, cosine, tangent and cotangent of an angle:

Let's place the vertex of our angle α at the center of the circle, i.e. to the center of the coordinate axis, and position one of the sides so that it coincides with the positive direction of the abscissa axis (OA)
Then the second side intersects the number circle at point M.

Ordinate point M: sine of angle α
Abscissa point M: cosine of angle α

Note that the arc length AM is the same part of the unit circle as our angle α from 360 degrees: where t is the length of the arc AM.

Degree measure of angle

1) Guys, we got a formula for determining the degree measure of an angle through the arc length of a number circle, let's take a closer look at it:

Then we write trigonometric functions in the form:

For example:

Radian measure of angles

When calculating the degree or radian measure of an angle, remember! :
For example:

By the way! Designation rad. you can lower it!

What is a radian?

Dear friends, we are faced with a new concept - Radian. So what is it?

There are various measures of length, time, weight, for example: meter, kilometer, second, hour, gram, kilogram and others. So Radian is one of the measures of angle. It is worth considering central angles, that is, those located in the center of the number circle.
An angle of 1 degree is the central angle subtended by an arc equal to 1/360 of the circumference.

An angle of 1 radian is the central angle subtended by an arc equal to 1 in a unit circle, and in an arbitrary circle by an arc equal to the radius of the circle.

Examples:

Examples of conversion from a degree measure of an angle to a radian measure, and vice versa

Problems to solve independently

1. Find the radian measure of angles:
a) 55° b) 450° c) 15° d) 302°

2. Find:
a) sin(150°) b) cos(45°) c) tg(120°)

3. Find the degree measure of angles:

The video lesson “Trigonometric functions of an angular argument” provides visual material for conducting a mathematics lesson on the relevant topic. The video is designed so that the material being studied is presented as conveniently as possible for students to understand, is easy to remember, and well reveals the connection between the available information about trigonometric functions from the section on studying triangles and their definition using the unit circle. It can become an independent part of the lesson, as it fully covers this topic, supplemented with important comments during the voicing.

To clearly demonstrate the relationship between different definitions of trigonometric functions, animation effects are used. Highlighting the text with a colored font, clear, understandable constructions, and adding comments helps you quickly master and remember the material, and quickly achieve the goals of the lesson. The connections between the definitions of trigonometric functions are clearly demonstrated through animation effects and color highlighting, promoting understanding and retention of the material. The manual is aimed at increasing the effectiveness of training.

The lesson begins with the introduction of the topic. Then the definitions of sine, cosine, tangent and cotangent of an acute angle of a right triangle are recalled. The definition highlighted in the frame reminds us that sine and cosine are formed as the ratio of the leg to the hypotenuse, tangent and cotangent are formed by the ratio of the legs. Students are also reminded of recently learned material that when considering a point on the unit circle, the abscissa of the point is the cosine, and the ordinate is the sine of the number corresponding to that point. The connection between these concepts is demonstrated through construction. The screen displays a unit circle placed so that its center coincides with the origin. From the origin of coordinates, a ray is constructed that makes an angle α with the positive abscissa semi-axis. This ray intersects the unit circle at point O. From the point, perpendiculars descend to the abscissa and ordinate axis, demonstrating that the coordinates of this point determine the cosine and sine of the angle α. It is noted that the length of the arc AO from the point of intersection of the unit circle with the positive direction of the abscissa axis to point O is the same part of the entire arc as the angle α from 360°. This allows you to create the proportion α/360=t/2π, which is displayed immediately and highlighted in red for memorization. From this proportion the value t=πα/180° is derived. Taking this into account, the relationship between the definitions of sine and cosine is determined: sinα°= sint= sinπα/180, cosα°=cost=cosπα/180. For example, finding sin60° is given. Substituting the degree measure of the angle into the formula, we get sin π·60°/180°. Reducing the fraction by 60, we get sin π/3, which is equal to √3/2. It is noted that if 60° is a degree measure of an angle, then π/3 is called a radian measure of an angle. There are two possible notations for the ratio of the degree measure of an angle to the radian measure: 60°=π/3 and 60°=π/3 rad.

The concept of an angle of one degree is defined as the central angle subtended by an arc whose length 1/360 represents a part of the circumference. The following definition reveals the concept of an angle of one radian - the central angle based on an arc of length one, or equal to the radius of the circle. Definitions are marked as important and highlighted to remember.

To convert one degree measure of an angle into a radian measure and vice versa, use the formula α°=πα/180 rad. This formula is highlighted in a frame on the screen. From this formula it follows that 1° = π/180 rad. In this case, one radian corresponds to an angle of 180°/π≈57.3°. It is noted that when finding the values ​​of trigonometric functions of the independent variable t, it can be considered both a numerical argument and an angular one.

The following demonstrates examples of using the acquired knowledge in solving mathematical problems. In example 1, you need to convert the values ​​from degrees to radians 135° and 905°. On the right side of the screen there is a formula showing the relationship between degrees and radians. After substituting the value into the formula, we get (π/180)·135. After reducing this fraction by 45, we get the value 135° = 3π/4. To convert an angle of 905° to a radian measure, the same formula is used. After substituting the value into it, it turns out (π/180)·905=181π/36 rad.

In the second example, the inverse problem is solved - the degree measure of angles expressed in radians π/12, -21π/20, 2.4π is found. On the right side of the screen, we recall the studied formula for the connection between the degree and radian measure of the angle 1 rad = 180°/π. Each example is solved by substituting the radian measure into the formula. Substituting π/12, we get (180°/π)·(π/12)=15°. The values ​​of the remaining angles are found similarly -21π/20=-189° and 2.4π=432°.

The video lesson “Trigonometric functions of angular argument” is recommended for use in traditional mathematics lessons to improve learning efficiency. The material will help ensure the visibility of learning during distance learning on this topic. A detailed, understandable explanation of the topic and solutions to problems on it can help the student independently master the material.

TEXT DECODING:

"Trigonometric functions of the angular argument."

We already know from geometry that the sine (cosine) of an acute angle of a right triangle is the ratio of the leg to the hypotenuse, and the tangent (cotangent) is the ratio of the legs. And in algebra we call the abscissa of a point on the unit circle a cosine, and the ordinate of this point a sine. Let's make sure that all this is closely interconnected.

Let's place an angle with degree measure α° (alpha degrees), as shown in Figure 1: the vertex of the angle is compatible with the center of the unit circle (with the origin of the coordinate system), and one side of the angle is compatible with the positive ray of the abscissa axis. The second side of the angle intersects the circle at point O. The ordinate of point O is the sine of the angle alpha, and the abscissa of this point is the cosine of alpha.

Note that the arc AO is the same part of the length of the unit circle as the angle alpha is from the angle of three hundred and sixty degrees. Let us denote the length of the arc AO by t(te), then we will compose the proportion =

(alpha is to trusts sixty as te is to two pi). From here we find te: t = = (te is equal to pi alpha divided by one hundred and eighty).

Thus, to find the sine or cosine of the angle alpha degrees, you can use the formula:

sin α° = sint = sin (sine alpha degrees is equal to sine te and equal to the sine of the partial pi alpha to one hundred and eighty),

cosα° = cost = cos (the cosine of alpha degrees is equal to the cosine of te and is equal to the cosine of the partial pi alpha to one hundred and eighty).

For example, sin 60° = sin = sin = (the sine of sixty degrees is equal to the sine of pi by three, according to the table of basic values ​​of sines, it is equal to the root of three by two).

It is believed that 60° is a degree measure of an angle, and (pi by three) is a radian measure of the same angle, that is, 60° = glad(Sixty degrees is equal to pi times three radians). For brevity, we agreed on the designation glad omit, that is, the following entry is acceptable: 60°= (show abbreviations radian measure = rad.)

An angle of one degree is a central angle that subtends an arc that is (one three hundred and sixtieth) part of the arc. An angle of one radian is the central angle that rests on an arc of length one, that is, an arc whose length is equal to the radius of the circle (we consider the central angles of a unit circle to show an angle in pi radians on a circle).

Let us remember the important formula for converting degrees to radians:

α° = glad. (alpha equals pi alpha divided by one hundred eighty, radians) Specifically, 1° = glad(one degree is equal to pi divided by one hundred and eighty, radians).

From this we can find that one radian is equal to the ratio of one hundred eighty degrees to pi and is approximately equal to fifty-seven point three degrees: 1 glad= ≈ 57.3°.

From the above: when we talk about any trigonometric function, for example about the function s = sint (es is equal to sine te), the independent variable t(te) can be considered both a numerical argument and an angular argument.

Let's look at examples.

EXAMPLE 1. Convert from degrees to radians: a) 135°; b) 905°.

Solution. Let's use the formula for converting degrees to radians:

a) 135° = 1° ∙ 135 = glad ∙ 135 = glad

(one hundred thirty-five degrees is equal to pi times one hundred eighty radians multiplied by one hundred thirty-five, and after reduction equals three pi times four radians)

b) Similarly, using the formula for converting a degree measure into a radian measure, we obtain

905° = glad ∙ 905 = glad.

(nine hundred five degrees equals one hundred eighty-one pi times thirty-six radians).

EXAMPLE 2. Express in degrees: a) ; b) - ; c) 2.4π

(pi over twelve; minus twenty-one pi over twenty; two point four pi).

Solution. a) Let's express pi by twelve in degrees, use the formula for converting the radian measure of an angle to a degree in 1 glad=, we get

glad = 1 glad∙ = ∙ = 15° (pi times twelve radians is equal to the product of one radian and pi times twelve. Substituting one hundred and eighty for pi instead of one radian and reducing, we get fifteen degrees)

Similar to b) - = 1 glad∙ (-) = ∙ (-)= - 189° (minus twenty-one pi times twenty equals minus one hundred eighty-nine degrees),

c) 2.4π = 1 glad∙ 2.4π = ∙ 2.4π = 432° (two point four pi equals four hundred thirty-two degrees).

Trigonometric functions of numeric argument we sorted it out. We took point A on the circle and looked for the sines and cosines of the resulting angle β.

We designated the point as A, but in algebra it is often designated as t and all formulas/functions with it are given. We will also not deviate from the canons. Those. t - this will be a certain number, therefore numeric function(for example, sint)

It is logical that since we have a circle with a radius of one, then

Trigonometric functions of angle argument We also successfully analyzed it - according to the canons, we will write for such functions: sin α°, meaning by α° any angle with the number of degrees we need.

The ray of this angle will give us the second point on the circle (OA - point A) and the corresponding points C and B for the numerical argument function, if we need it: sin t = sin α°

Lines of sines, cosines, tangents and cotangents

Never forget that Y axis is the line of sines, X axis is line of cosines! The points obtained from the circle are marked on these axes.

A the lines of tangents and cotangents are parallel to them and pass through the points (1; 0) and (0; 1) respectively.