In trigonometry important concept is angle of rotation. Below we will consistently give an idea of the turn, and introduce all related concepts. Let's start with general idea about a turn, let's say a full turn. Next, we turn to the concept of the angle of rotation and consider its main characteristics, such as the direction and amount of rotation. Finally, let's define the rotation of a figure around a point. We will supply the entire theory in the text with explanatory examples and graphic illustrations.
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What is called the rotation of a point around a point?
We note right away that along with the phrase “turn around a point”, we will also use the phrases “turn around a point” and “turn around a point”, which mean the same thing.
Let's introduce the concept of turning a point around a point.
First, let's define the center of rotation.
Definition.
The point about which the rotation is made is called pivot point.
Now let's say what happens as a result of point rotation.
As a result of the rotation of some point A about the center of rotation O, a point A 1 is obtained (which, in the case of a certain amount, can coincide with A ), and the point A 1 lies on a circle centered at the point O of radius OA . In other words, when turning about the point O, the point A goes to the point A 1 lying on a circle centered at the point O of radius OA .
It is believed that the point O, when turning around itself, goes into itself. That is, as a result of rotation around the center of rotation O, the point O goes into itself.
It is also worth noting that the rotation of point A around point O should be considered as a movement as a result of the movement of point A along a circle centered at point O with radius OA.
For clarity, we give illustrations of the rotation of point A around point O, in the figures below, we will show the movement of point A to point A 1 with the help of an arrow.
Full turn
It is possible to perform such a rotation of the point A relative to the center of rotation O, that the point A, having passed all the points of the circle, will be in the same place. In this case, they say that the point A has made around the point O.
Let's give a graphic illustration of a full turn.
If you do not stop at one turn, but continue to move the point around the circle, then you can perform two, three, and so on full turns. The drawing below on the right shows how two full turnover, and on the left - three turns.
The concept of the angle of rotation
From the concept of point rotation introduced in the first paragraph, it is clear that there are an infinite number of options for turning point A around point O. Indeed, any point of the circle centered at the point O of radius OA can be considered as a point A 1 obtained as a result of the rotation of the point A . Therefore, in order to distinguish one rotation from another, we introduce concept of angle of rotation.
One of the characteristics of the angle of rotation is turning direction. The direction of rotation is used to judge whether the point is rotated clockwise or counterclockwise.
Another characteristic of the angle of rotation is its magnitude. Rotation angles are measured in the same units as : degrees and radians are the most common. It is worth noting here that the angle of rotation can be expressed in degrees by any real number from minus infinity to plus infinity, in contrast to the angle in geometry, the value of which in degrees is positive and does not exceed 180 .
To indicate the angle of rotation are usually used lower case Greek alphabet: etc. To designate a large number rotation angles often use one letter with subscripts, for example, 
.
Now let's talk about the characteristics of the angle of rotation in more detail and in order.
Turn direction
Let the points A and A 1 be marked on the circle centered at the point O. You can get to point A 1 from point A by turning around the center O either clockwise or counterclockwise. It is logical to consider these turns different.
Let us illustrate the turns in the positive and negative direction. The drawing below shows rotation in the positive direction on the left and negative rotation on the right.
Rotation angle value, angle of arbitrary value
The angle of rotation of a point other than the center of rotation is completely determined by specifying its magnitude, on the other hand, by the magnitude of the angle of rotation, one can judge how this rotation was carried out.
As we mentioned above, the angle of rotation in degrees is expressed as a number from −∞ to +∞. The plus sign corresponds to clockwise rotation, and the minus sign corresponds to counterclockwise rotation.
Now it remains to establish a correspondence between the value of the angle of rotation and what rotation it corresponds to.
Let's start with a rotation angle of zero degrees. This angle of rotation corresponds to the displacement of point A towards itself. In other words, when rotating 0 degrees around point O, point A remains in place.
We turn to the rotation of point A around point O, in which the rotation occurs within half a turn. We will assume that point A goes to point A 1 . In this case, the absolute value of the angle AOA 1 in degrees does not exceed 180 . If the rotation took place in a positive direction, then the value of the angle of rotation is considered equal to angle AOA 1 , and if the rotation occurred in a negative direction, then its value is considered equal to the value of the angle AOA 1 with a minus sign. For example, here is a figure showing rotation angles of 30, 180, and −150 degrees.
Rotation angles greater than 180 degrees and less than −180 degrees are determined based on the following rather obvious properties of successive turns: several successive rotations of the point A around the center O are equivalent to one rotation, the value of which is equal to the sum of the values of these rotations.
Let's give an example illustrating this property. Let's rotate point A relative to point O by 45 degrees, and then rotate this point by 60 degrees, after which we will rotate this point by −35 degrees. Let's designate intermediate points at these turns as A 1 , A 2 and A 3 . We could get to the same point A 3 by making one turn of point A through an angle of 45+60+(−35)=70 degrees.
So, we will represent the rotation angles greater than 180 degrees as several successive rotations by angles, the sum of the values of which gives the value of the initial rotation angle. For example, a rotation angle of 279 degrees corresponds to consecutive rotations of 180 and 99 degrees, or 90 , 90 , 90 and 9 degrees, or 180 , 180 and -81 degrees, or 279 consecutive 1 degree rotations.
Rotation angles smaller than −180 degrees are defined similarly. For example, a rotation angle of −520 degrees can be interpreted as successive rotations of a point by −180 , −180 and −160 degrees.
Summarize. We have defined the angle of rotation, the value of which in degrees is expressed by some real number from the interval from −∞ to +∞. In trigonometry, we will work with rotation angles, although the word "rotation" is often omitted, and just "angle" is said. Thus, in trigonometry we will work with angles of arbitrary magnitude, by which we mean the angles of rotation.
To conclude this paragraph, we note that a complete rotation in the positive direction corresponds to a rotation angle of 360 degrees (or 2 π radians), and in the negative direction, a rotation angle of −360 degrees (or −2 π rad). In this case, it is convenient to represent large angles of rotation as a number of complete revolutions and one more rotation through an angle of −180 to 180 degrees. For example, let's take a rotation angle of 1,340 degrees. It is easy to represent 1 340 as 360 4+(−100) . That is, the initial angle of rotation corresponds to 4 full turns in the positive direction and the subsequent rotation by −100 degrees. Another example: a rotation angle of −745 degrees can be interpreted as two turns counterclockwise and then a rotation of −25 degrees, since −745=(−360) 2+(−25) .
Rotate a shape around a point by an angle
The concept of turning a point is easily extended to rotate any shape around a point by an angle (we are talking about such a rotation that both the point about which the rotation is carried out and the figure that is being rotated lie in the same plane).
Under the rotation of the figure, we mean the rotation of all points of the figure around a given point by a given angle.
As an example, let's illustrate the following action: let's rotate the segment AB by an angle relative to the point O, this segment, when rotated, will go into the segment A 1 B 1 .
Bibliography.
- Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: ill.- isbn 5-09-002727-7
- Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Subject: Observation of the phenomena of interference and diffraction of light.
Goal of the work: experimentally study the phenomenon of interference and diffraction.
Equipment:
- glasses with a solution of soap;
- wire ring with a handle;
- nylon fabric;
- CD;
- incandescent lamp;
- calipers;
- two glass plates;
- blade;
- tweezers;
- nylon fabric.
Theoretical part
Interference is a phenomenon characteristic of waves of any nature: mechanical, electromagnetic. Wave interference is the addition of two (or several) waves in space, in which at its different points an amplification or weakening of the resulting wave is obtained. To form a stable interference pattern, coherent (matched) wave sources are needed. Coherent waves are waves that have the same frequency and constant phase difference.
Maximum Conditions Δd = ±kλ, minimum conditions, Δd = ± (2k + 1)λ/2 where k =0; ± 1; ±2; ± 3;...(the difference in the path of the waves is equal to an even number of half-waves
An interference pattern is a regular alternation of areas of increased and decreased light intensity. Light interference is the spatial redistribution of the energy of light radiation when two or more light waves are superimposed. Consequently, in the phenomena of interference and diffraction of light, the law of conservation of energy is observed. In the area of interference, light energy is only redistributed without being converted into other types of energy. The increase in energy at some points of the interference pattern relative to the total light energy is compensated by its decrease at other points (total light energy is the light energy of two light beams from independent sources).
Light stripes correspond to energy maxima, dark stripes correspond to energy minima.
Diffraction is the phenomenon of wave deviation from rectilinear propagation when passing through small holes and rounding small obstacles by the wave. Condition for the manifestation of diffraction: d< λ, Where d- the size of the obstacle, λ - wavelength. The dimensions of the obstacles (holes) must be smaller than or commensurate with the wavelength. The existence of this phenomenon (diffraction) limits the scope of the laws of geometric optics and is the reason for the limiting resolution of optical instruments. Diffraction grating - an optical device, which is a periodic structure of a large number regularly spaced elements on which light is diffracted. Strokes with a profile defined and constant for a given diffraction grating are repeated at regular intervals d(lattice period). The ability of a diffraction grating to decompose a beam of light incident on it into wavelengths is its main property. There are reflective and transparent diffraction gratings. IN modern appliances mainly reflective diffraction gratings are used. Condition for observing the diffraction maximum: d sin(φ) = ± kλ
Instructions for work
1. Dip the wire frame in the soap solution. Observe and draw the interference pattern in the soap film. When the film is illuminated with white light (from a window or a lamp), coloring occurs light stripes: top - Blue colour, below - in red. Use a glass tube to blow a soap bubble. Watch him. When illuminated with white light, the formation of colored interference rings is observed. As the film thickness decreases, the rings expand and move down.
Answer the questions:
- Why bubble are iridescent?
- What shape are the rainbow stripes?
- Why does the color of the bubble change all the time?
2. Thoroughly wipe the glass plates, put them together and squeeze with your fingers. Due to the non-ideal shape of the contacting surfaces, the thinnest air voids are formed between the plates, giving bright iridescent annular or closed irregularly shaped stripes. When the force compressing the plates changes, the location and shape of the bands change both in reflected and transmitted light. Draw the pictures you see.
Answer the questions:
- Why are bright iridescent annular or irregularly shaped stripes observed in separate places of contact between the plates?
- Why does the shape and location of the obtained interference fringes change with a change in pressure?
3. Lay a CD horizontally at eye level. What are you observing? Explain the observed phenomena. Describe the interference pattern.
4. Look through nylon fabric on the filament of a burning lamp. By turning the fabric around the axis, achieve a clear diffraction pattern in the form of two diffraction bands crossed at right angles. Sketch the observed diffraction cross.
5. Observe two diffraction patterns when examining the filament of a burning lamp through a slit formed by the jaws of a caliper (with a slit width of 0.05 mm and 0.8 mm). Describe the change in the nature of the interference pattern when the caliper is smoothly rotated around the vertical axis (with a slit width of 0.8 mm). Repeat this experiment with two blades, pressing them against each other. Describe the nature of the interference pattern
Record your findings. Indicate in which of your experiments the phenomenon of interference was observed? diffraction?
