Sine 2 on the number circle. Solving the simplest trigonometric equations
In this article, we will analyze in great detail the definition of the number circle, find out its main property and arrange the numbers 1,2,3, etc. How to mark other numbers on a circle (for example, \ (\ frac (π) (2), \ frac (π) (3), \ frac (7π) (4), 10π, - \ frac (29π) ( 6) \)) understands the.
Number circle is called a circle of unit radius, the points of which correspond to arranged according to the following rules:
1) The origin is at the extreme right point of the circle;
2) Counterclockwise - positive direction; clockwise - negative;
3) If we put off the distance \ (t \) on the circle in the positive direction, then we will get to the point with the value \ (t \);
4) If we put off the distance \ (t \) on the circle in the negative direction, then we will get to the point with the value \ (- t \).
Why is a circle called numeric?
Because there are numbers on it. In this, a circle is similar to a number axis - on a circle, as well as on an axis, for each number there is a certain point.
Why know what a number circle is?
The number circle is used to determine the value of sines, cosines, tangents and cotangents. Therefore, for knowledge of trigonometry and passing the exam by 60+ points, you must definitely understand what a number circle is and how to place points on it.
What in the definition do the words "... unit radius ..." mean?
This means that the radius of this circle is \ (1 \). And if we build such a circle centered at the origin, then it will intersect with the axes at the points \ (1 \) and \ (- 1 \).
It is not necessary to draw it small, you can change the "size" of the divisions along the axes, then the picture will be larger (see below).
Why is the radius exactly one? This is more convenient, because in this case, when calculating the circumference using the formula \ (l = 2πR \), we get:
The length of the number circle is \ (2π \) or approximately \ (6.28 \).
And what does "... whose points correspond to real numbers" mean?
As mentioned above, on the number circle for any real number, there will necessarily be its "place" - a point that corresponds to this number.
Why define the origin and directions on a number circle?
The main purpose of the number circle is to uniquely determine its point for each number. But how can you determine where to put a point if you don't know where to count from and where to go?
It is important not to confuse the origin on the coordinate line and on the number circle - these are two different reference systems! And also do not confuse \ (1 \) on the \ (x \) axis and \ (0 \) on the circle - these are points on different objects.
What dots correspond to numbers \ (1 \), \ (2 \), etc.?
Remember, we assumed that the radius of the number circle is \ (1 \)? This will be our unit segment (by analogy with the number axis), which we will lay on the circle.
To mark the point corresponding to the number 1 on the number circle, you need to go from 0 a distance equal to the radius in the positive direction.
To mark a point on the circle corresponding to the number \ (2 \), you need to go a distance equal to two radii from the origin, so that \ (3 \) is a distance equal to three radii, etc.
When looking at this picture, you may have 2 questions:
1. What will happen when the circle "ends" (ie we make a complete revolution)?
Answer: let's go to the second round! And when the second is over, let's go to the third, and so on. Therefore, an infinite number of numbers can be applied to a circle.
2. Where will the negative numbers be?
Answer: in the same place! They can also be placed, counting from zero the required number of radii, but now in the negative direction.
Unfortunately, it is difficult to denote integers on a number circle. This is due to the fact that the length of the numerical circle will not be equal to an integer: \ (2π \). And in the most convenient places (at the points of intersection with the axes), there will also be not whole numbers, but fractions
On the trigonometric circle, in addition to angles in degrees, we observe.
More about radians:
A radian is defined as the angular value of an arc whose length is equal to its radius. Accordingly, since the circumference is 
, then it is obvious that the radian fits into the circle, that is
1 rad ≈ 57.295779513 ° ≈ 57 ° 17′44.806 ″ ≈ 206265 ″.
Everyone knows that a radian is
So, for example, ah. That's how we learned to convert radians to angles.
Now on the contrary, let's convert degrees to radians.
Let's say we need to convert to radians. It will help us. We proceed as follows:
Since, radian, then fill in the table:
We train to find the values of the sine and cosine in a circle
Let's further clarify the following.
Well, well, if we are asked to calculate, say, - there is usually no confusion here - everyone starts looking at the circle first of all.
And if they ask to calculate, for example, ... Many, suddenly, start not to understand where to look for this zero ... They often look for it at the origin. Why?
1) Let's agree once and for all! What comes after or is argument = angle, and the corners are located on the circle, don't look for them on the axes!(It's just that individual points fall on both the circle and the axis ...) And the values of the sines and cosines themselves - we are looking for the axes!
2) And more! If we go from the "start" counterclock-wise(the main direction of traversing the trigonometric circle), then we postpone the positive values of the angles, the values of the angles increase when moving in this direction.
If we go from the "start" clockwise, then we postpone the negative values of the angles.
Example 1.
Find the value.
Solution:
We find it on the circle. We project a point onto the sine axis (that is, draw a perpendicular from the point to the sine axis (oh)).
We arrive at 0. Hence,.
Example 2.
Find the value.
Solution:
We find it on the circle (we go counterclockwise and more). We project a point onto the sine axis (and it already lies on the sinus axis).
We hit -1 on the sine axis.
Note that behind the point there are such points as (we could go to the point marked as, clockwise, which means a minus sign appears), and infinitely many others.
An analogy can be given:
Let's represent the trigonometric circle as treadmill stadium.
After all, you can find yourself at the "Flag" point, starting from the start counterclockwise, running, say, 300 m. Or running, say, 100 m clockwise (we count the length of the track 400 m).
And also you can be at the "Flag" point (after the "start"), having run, say, 700 m, 1100 m, 1500 m, etc. counterclockwise. You can find yourself at the "Flag" point by running 500 m or 900 m, etc. clockwise from "start".
Expand the stadium treadmill mentally into a number line. Imagine where on this line there will be, for example, the values 300, 700, 1100, 1500, etc. We will see points on the number line, equidistant from each other. Let's roll back into a circle. The dots "stick together" into one.
So it is with the trigonometric circle. There are infinitely many others hidden behind each point.
Let's say angles,,, etc. are depicted by one point. And the values of the sine, cosine in them, of course, coincide. (Did you notice that we added / subtracted or? This is the period for the sine and cosine function.)
Example 3.
Find the value.
Solution:
Let's translate for simplicity in degrees
(later when you get used to trigonometric circle, you don't need to convert radians to degrees):
We will move clockwise from the point Let's go half a circle () and more
We understand that the sine value coincides with the sine value and is equal to
Note, if we took, for example, or, etc., then we would get all the same sine value.
Example 4.
Find the value.
Solution:
However, we will not convert radians to degrees, as in the previous example.
That is, we need to go counterclockwise half a circle and another quarter of a half circle and project the resulting point onto the cosine axis (horizontal axis).
Example 5.
Find the value.
Solution:
How to postpone on the trigonometric circle?
If we pass or, yes, at least, we will still find ourselves at the point that we designated as "start". Therefore, you can immediately go to a point on the circle.
Example 6.
Find the value.
Solution:
We will find ourselves at the point (it will lead us to point zero anyway). We project the point of the circle onto the cosine axis (see trigonometric circle), we get into. That is .
Trigonometric circle - in your hands
You already understood that the main thing is to remember the meanings. trigonometric functions first quarter. In the other quarters, everything is the same, you just need to follow the signs. And I hope you will not forget the "chain-ladder" of values of trigonometric functions. 
How to find tangent and cotangent values main corners.
After that, having got acquainted with the basic values of tangent and cotangent, you can pass
On an empty circle template. Train!
Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, this can be thought of as a rectangle with one side representing lettuce and the other side representing water. The sum of these two sides will represent borscht. The diagonal and area of such a "borscht" rectangle are purely mathematical concepts and are never used in borscht recipes.
How do lettuce and water turn into borscht from a mathematical standpoint? How can the sum of two line segments turn into trigonometry? To understand this, we need linear angle functions.
You won't find anything about linear angle functions in mathematics textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.
Linear angle functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.
Can linear angle functions be dispensed with? You can, because mathematicians still do without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. Everything. We do not know other tasks and are not able to solve them. What to do if we only know the result of addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angle functions. Then we ourselves choose what one term can be, and the linear angle functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. V Everyday life we can do just fine without decomposing the sum; subtraction is enough for us. But with scientific research the laws of nature, the decomposition of the sum into terms can be very useful.
Another law of addition that mathematicians don't like to talk about (another trick of theirs) requires that the terms have the same units of measurement. For salad, water and borscht, these can be units of measure for weight, volume, value, or units of measure.
The figure shows two levels of difference for math. The first level is the differences in the field of numbers, which are indicated a, b, c... This is what mathematicians do. The second level is the differences in the area of units, which are shown in square brackets and indicated by the letter U... This is what physicists do. We can understand the third level - differences in the area of the described objects. Different objects can have the same number of the same units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same designation of units of measurement of different objects, we can say exactly which mathematical value describes a particular object and how it changes over time or in connection with our actions. By letter W I will designate water, with the letter S I will designate the salad and the letter B- Borsch. This is what the linear angular functions for borsch would look like.
If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we are doing it is not clear what, it is not clear why, and we very poorly understand how this relates to reality, because of the three levels of difference, mathematics operates only one. It would be more correct to learn how to switch from one measurement unit to another.
And bunnies, and ducks, and animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a childish version of the problem. Let's take a look at a similar problem for adults. What happens when you add bunnies and money? There are two possible solutions here.
First option... We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.
Second option... You can add the number of bunnies to the number of banknotes we have. We will receive the number of movable property in pieces.
As you can see, the same addition law produces different results. It all depends on what exactly we want to know.
But back to our borscht. Now we can see what will happen when different meanings angle of linear angular functions.
The angle is zero. We have salad, but no water. We cannot cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. Zero borscht can be at zero salad (right angle).
For me personally, this is the main mathematical proof of the fact that. Zero does not change the number when added. This is because the addition itself is impossible if there is only one term and there is no second term. You can relate to this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "for the knock-out point zero" and other nonsense. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses all meaning: how can we consider a number that is not a number. It's like asking what color an invisible color should be. Adding zero to a number is like painting with paint that doesn't exist. We waved with a dry brush and told everyone that "we have painted". But I digress a little.
The angle is greater than zero, but less than forty-five degrees. We have a lot of salad, but not enough water. As a result, we get a thick borscht.
The angle is forty-five degrees. We have equal amounts of water and salad. This is the perfect borscht (yes, the cooks will forgive me, it's just math).
The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You get liquid borscht.
Right angle. We have water. From the salad, only memories remain, as we continue to measure the angle from the line that once stood for the salad. We cannot cook borscht. The amount of borscht is zero. In that case, hold on and drink the water while you have it)))
Here. Something like this. I can tell other stories here that will be more than appropriate here.
Two friends had their shares in the common business. After killing one of them, everything went to the other.
The emergence of mathematics on our planet.
All of these stories are told in the language of mathematics using linear angle functions. Some other time I'll show you the real place of these functions in the structure of mathematics. In the meantime, let's return to the trigonometry of the borscht and consider the projections.
Saturday, 26 October 2019
Wednesday, 7 August 2019
Concluding the conversation about, there is an infinite number to consider. The result is that the concept of "infinity" acts on mathematicians like a boa constrictor on a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:
The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take as an example an infinite set of natural numbers, then the considered examples can be presented in the following form:
For a visual proof of their correctness, mathematicians have come up with many different methods. Personally, I look at all these methods as dancing shamans with tambourines. Essentially, they all boil down to the fact that either some of the rooms are not occupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of the century. Of course, the time factor can be stupidly ignored, but it will already be from the category "the law is not written for fools." It all depends on what we are doing: adjusting reality to fit mathematical theories or vice versa.
What is an "endless hotel"? An endless hotel is a hotel that always has any number of vacant places, no matter how many rooms are occupied. If all the rooms in the endless visitor corridor are occupied, there is another endless corridor with the guest rooms. There will be an endless number of such corridors. Moreover, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, however, are not able to distance themselves from commonplace everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Here are mathematicians and are trying to manipulate the serial numbers of hotel rooms, convincing us that it is possible to "shove the stuff in."
I will demonstrate the logic of my reasoning to you on the example of an infinite set of natural numbers. First, you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves, in Nature there are no numbers. Yes, Nature is excellent at counting, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers there are. Consider both options, as befits a real scientist.
Option one. "Let us be given" a single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. And if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:
I recorded the actions in algebraic system notation and in the notation system adopted in set theory, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one subtracts from it and adds the same unit.
Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:
Subscripts "one" and "two" indicate that these items belonged to different sets. Yes, if you add one to the infinite set, the result will also be an infinite set, but it will not be the same as the original set. If we add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.
Lots of natural numbers are used for counting in the same way as a ruler for measurements. Now imagine adding one centimeter to the ruler. This will already be a different line, not equal to the original.
You can accept or not accept my reasoning - it's your own business. But if you ever run into mathematical problems, think about whether you are not following the path of false reasoning trodden by generations of mathematicians. After all, doing mathematics, first of all, form a stable stereotype of thinking in us, and only then add mental abilities to us (or vice versa, deprive us of free thought).
pozg.ru
Sunday, 4 August 2019
I was writing a postscript to an article about and saw this wonderful text on Wikipedia:
We read: "... rich theoretical basis mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques devoid of common system and the evidence base. "
Wow! How smart we are and how well we can see the shortcomings of others. Is it hard for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:
The rich theoretical basis of modern mathematics is not holistic and is reduced to a set of disparate sections devoid of a common system and evidence base.
I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious blunders of modern mathematics. See you soon.
Saturday, 3 August 2019
How do you divide a set into subsets? To do this, it is necessary to enter a new unit of measurement that is present for some of the elements of the selected set. Let's look at an example.
Let us have many A consisting of four people. This set was formed on the basis of "people" Let us denote the elements of this set by the letter a, a subscript with a digit will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sex" and denote it by the letter b... Since sexual characteristics are inherent in all people, we multiply each element of the set A by gender b... Note that now our multitude of "people" has become a multitude of "people with sex characteristics." After that, we can divide the sex characteristics into masculine bm and women bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sex characteristics, it does not matter which one is male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.
After multiplication, reduction and rearrangement, we got two subsets: the subset of men Bm and a subset of women Bw... Mathematicians think about the same when they apply set theory in practice. But they do not devote us to the details, but give a finished result - "a lot of people consist of a subset of men and a subset of women." Naturally, you may wonder how correctly the mathematics is applied in the above transformations? I dare to assure you, in fact, the transformations were done correctly, it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I'll tell you about it.
As for supersets, you can combine two sets into one superset by choosing the unit of measurement that is present for the elements of these two sets.
As you can see, units of measurement and common mathematics make set theory a thing of the past. An indication that set theory is not all right is that for set theory, mathematicians have come up with own language and own designations. Mathematicians did what shamans once did. Only shamans know how to "correctly" apply their "knowledge". They teach us this "knowledge".
Finally, I want to show you how mathematicians manipulate with.
Monday, January 7, 2019
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". This is how it sounds:
Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.
This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... the discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them has become a generally accepted solution to the question ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.
If we turn over the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly catch up with the turtle."
How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:
During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. For the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". We still have to study, rethink and solve this problem. And the solution must not be looked for endlessly large numbers, and in units of measurement.
Another interesting aporia Zeno tells about a flying arrow:
The flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but it is impossible to determine the distance from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Let me show you the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, but there are no bows. After that we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.
Now let's do a little dirty trick. Take "solid in a pimple with a bow" and combine these "wholes" by color, selecting the red elements. We got a lot of "red". Now a question to fill in: the resulting sets "with a bow" and "red" are the same set or are they two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.
This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We have formed a set of "red solid into a bump with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a pimple), ornaments (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics... This is what it looks like.
The letter "a" with different indices denotes different units of measurement. Units of measurement are highlighted in brackets, by which the "whole" is allocated at the preliminary stage. The unit of measurement, by which the set is formed, is taken out of the brackets. The last line shows the final result - the element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, not the dancing of shamans with tambourines. Shamans can "intuitively" arrive at the same result, arguing it "by the obviousness", because units of measurement are not included in their "scientific" arsenal.
It is very easy to use units to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.
Sine values are enclosed in the interval [-1; 1], ie -1 ≤ sin α ≤ 1. Therefore, if | a | > 1, then the equation sin x = a has no roots. For example, the equation sin x = 2 has no roots.
Let's turn to some tasks.
Solve the equation sin x = 1/2.
Solution.
Note that sin x is the ordinate of a point on the unit circle, which is obtained by rotating the point P (1; 0) by an angle x around the origin.
The ordinate equal to ½ is present at two points of the circle M 1 and M 2.
Since 1/2 = sin π / 6, the point M 1 is obtained from the point P (1; 0) by turning through an angle x 1 = π / 6, and also by angles x = π / 6 + 2πk, where k = +/- 1, +/- 2, ...
Point М 2 is obtained from point Р (1; 0) as a result of rotation through an angle х 2 = 5π / 6, as well as through angles х = 5π / 6 + 2πk, where k = +/- 1, +/- 2, ... , i.e. at the angles х = π - π / 6 + 2πk, where k = +/- 1, +/- 2,….
So, all the roots of the equation sin x = 1/2 can be found by the formulas x = π / 6 + 2πk, x = π - π / 6 + 2πk, where k € Z.
These formulas can be combined into one: x = (-1) n π / 6 + πn, where n € Z (1).
Indeed, if n is an even number, i.e. n = 2k, then from formula (1) we obtain х = π / 6 + 2πk, and if n is an odd number, i.e. n = 2k + 1, then from formula (1) we obtain х = π - π / 6 + 2πk.
Answer. х = (-1) n π / 6 + πn, where n € Z.
Solve the equation sin x = -1/2.
Solution.
The ordinate -1/2 has two points of the unit circle M 1 and M 2, where x 1 = -π / 6, x 2 = -5π / 6. Therefore, all roots of the equation sin x = -1/2 can be found by the formulas x = -π / 6 + 2πk, x = -5π / 6 + 2πk, k € Z.
We can combine these formulas into one: x = (-1) n (-π / 6) + πn, n € Z (2).
Indeed, if n = 2k, then by formula (2) we obtain x = -π / 6 + 2πk, and if n = 2k - 1, then by formula (2) we find x = -5π / 6 + 2πk.
Answer. x = (-1) n (-π / 6) + πn, n € Z.
Thus, each of the equations sin x = 1/2 and sin x = -1/2 has an infinite number of roots.
On the interval -π / 2 ≤ x ≤ π / 2, each of these equations has only one root:
x 1 = π / 6 is the root of the equation sin x = 1/2 and x 1 = -π / 6 is the root of the equation sin x = -1/2.
The number π / 6 is called the arcsine of the number 1/2 and is written: arcsin 1/2 = π / 6; the number -π / 6 is called the arcsine of the number -1/2 and is written: arcsin (-1/2) = -π / 6.
In general, the equation sin x = a, where -1 ≤ a ≤ 1, has only one root in the interval -π / 2 ≤ x ≤ π / 2. If a ≥ 0, then the root is contained in the interval; if a< 0, то в промежутке [-π/2; 0). Этот корень называют арксинусом числа а и обозначают arcsin а.
Thus, the arcsine of the number a € [–1; 1] such a number is called a € [–π / 2; π / 2], whose sine is equal to a.
arcsin а = α, if sin α = а and -π / 2 ≤ х ≤ π / 2 (3).
For example, arcsin √2 / 2 = π / 4, since sin π / 4 = √2 / 2 and - π / 2 ≤ π / 4 ≤ π / 2;
arcsin (-√3 / 2) = -π / 3, since sin (-π / 3) = -√3 / 2 and - π / 2 ≤ 
- π / 3 ≤ π / 2.
Similarly to how it was done in solving problems 1 and 2, it can be shown that the roots of the equation sin x = a, where | a | ≤ 1, are expressed by the formula
х = (-1) n аrcsin а + πn, n € Z (4).
We can also prove that for any a € [-1; 1] the formula arcsin (-a) = -arcsin a is valid.
It follows from formula (4) that the roots of the equation
sin x = a for a = 0, a = 1, a = -1 can be found using simpler formulas:
sin х = 0 х = πn, n € Z (5)
sin x = 1 x = π / 2 + 2πn, n € Z (6)
sin x = -1 x = -π / 2 + 2πn, n € Z (7)
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Simplest solution trigonometric equations.
The solution of trigonometric equations of any level of complexity ultimately comes down to solving the simplest trigonometric equations. And in this best helper again it turns out to be a trigonometric circle.
Let's recall the definitions of cosine and sine.
The cosine of an angle is the abscissa (that is, the coordinate along the axis) of a point on the unit circle corresponding to a rotation by a given angle.
The sine of an angle is the ordinate (that is, the coordinate along the axis) of a point on the unit circle corresponding to a rotation by a given angle.
The positive direction of movement in the trigonometric circle is counterclockwise movement. A rotation of 0 degrees (or 0 radians) corresponds to a point with coordinates (1; 0)
We will use these definitions to solve the simplest trigonometric equations.
1. Let's solve the equation
This equation is satisfied by all such values of the angle of rotation, which correspond to the points of the circle, the ordinate of which is equal to.
Let's mark on the ordinate axis a point with an ordinate:
Let's draw a horizontal line parallel to the abscissa axis until it intersects with the circle. We get two points lying on a circle and having an ordinate. These points correspond to the angles of rotation by and radians:
If we, leaving the point corresponding to the angle of rotation by radians, go around a full circle, then we will come to the point corresponding to the angle of rotation by radians and having the same ordinate. That is, this angle of rotation also satisfies our equation. We can do as many "idle" revolutions as we want, returning to the same point, and all these values of the angles will satisfy our equation. The number of "idle" revolutions will be denoted by the letter (or). Since we can make these revolutions both in the positive and in the negative direction, (or) can take any integer values.
That is, the first series of solutions to the original equation has the form:
,, is the set of integers (1)
Similarly, the second series of solutions is:
, where , . (2)
As you may have guessed, this series of solutions is based on the point of the circle corresponding to the angle of rotation by.
These two series of solutions can be combined into one entry:
If we take in this record (that is, even), then we get the first series of solutions.
If we take in this record (that is, odd), then we get the second series of solutions.
2. Now let's solve the equation
Since is the abscissa of the point of the unit circle obtained by turning through an angle, mark the point with the abscissa on the axis:
Draw a vertical line parallel to the axis until it intersects with the circle. We get two points lying on a circle and having an abscissa. These points correspond to the angles of rotation by and radians. Recall that when moving clockwise, we get a negative rotation angle:
Let's write down two series of solutions:
,
,
(We get to the desired point, passing from the main full circle, that is.
Let's combine these two series into one entry:
3. Solve the equation
The tangent line passes through the point with coordinates (1,0) of the unit circle parallel to the OY axis
We mark a point on it with an ordinate equal to 1 (we are looking for the tangent of which angles is 1):
Let's connect this point with the origin of coordinates with a straight line and mark the points of intersection of the straight line with the unit circle. The intersection points of the straight line and the circle correspond to the angles of rotation on and:
Since the points corresponding to the angles of rotation that satisfy our equation lie at a distance of radians from each other, we can write the solution in this way:
4. Solve the equation
The cotangent line passes through the point with the coordinates of the unit circle parallel to the axis.
Let's mark on the line of cotangents a point with abscissa -1:
Let's connect this point with the origin of coordinates of a straight line and continue it to the intersection with the circle. This line will intersect the circle at the points corresponding to the angles of rotation by and radians:
Since these points are separated from each other by a distance equal to, then common decision we can write this equation as follows:
In the given examples, illustrating the solution of the simplest trigonometric equations, tabular values of trigonometric functions were used.
However, if there is not a tabular value on the right side of the equation, then we substitute the value in the general solution of the equation:
SPECIAL SOLUTIONS:
Note on the circle the points whose ordinate is equal to 0:
Let us mark on the circle a single point, the ordinate of which is equal to 1:
Let's mark on the circle the only point, the ordinate of which is equal to -1:
Since it is customary to indicate the values that are closest to zero, we write the solution as follows:
Note on the circle the points whose abscissa is equal to 0:
5.
Let's mark on the circle the only point, the abscissa of which is equal to 1:
Let's mark on the circle the only point, the abscissa of which is equal to -1:
And a little more complex examples:
1.
The sine is one if the argument is
The argument of our sine is equal, so we get:
Divide both sides of the equality by 3:
Answer:
2.
Cosine is zero if the argument of the cosine is
The argument of our cosine is equal, so we get:
Let us express, for this we first move to the right with the opposite sign:
Let's simplify the right side:
Divide both parts by -2:
Note that the sign does not change in front of the term, since k can take any integer values.
Answer:
And finally, watch the video tutorial "Selecting roots in a trigonometric equation using a trigonometric circle"
This concludes the conversation about solving the simplest trigonometric equations. Next time we'll talk about how to solve.