Trigonometric circle. Unit circle. Number circle. What is it?
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
Very often terms trigonometric circle, unit circle, number circle poorly understood by students. And completely in vain. These concepts are a powerful and universal assistant in all areas of trigonometry. In fact, this is a legal cheat sheet! I drew a trigonometric circle and immediately saw the answers! Tempting? So let's learn, it would be a sin not to use such a thing. Moreover, it is not at all difficult.
To successfully work with the trigonometric circle, you need to know only three things.
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You can get acquainted with functions and derivatives.
This article contains tables of sines, cosines, tangents and cotangents. First we will provide a table of basic values trigonometric functions, that is, a table of sines, cosines, tangents and cotangents of angles 0, 30, 45, 60, 90, ..., 360 degrees ( 0, π/6, π/4, π/3, π/2, …, 2π radian). After this, we will give a table of sines and cosines, as well as a table of tangents and cotangents by V. M. Bradis, and show how to use these tables when finding the values of trigonometric functions.
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Table of sines, cosines, tangents and cotangents for angles of 0, 30, 45, 60, 90, ... degrees
References.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
- Bradis V. M. Four-digit math tables: For general education. textbook establishments. - 2nd ed. - M.: Bustard, 1999.- 96 p.: ill. ISBN 5-7107-2667-2
Counting angles on a trigonometric circle.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
It is almost the same as in the previous lesson. There are axes, a circle, an angle, everything is in order. Added quarter numbers (in the corners of the large square) - from the first to the fourth. What if someone doesn’t know? As you can see, the quarters (they are also called the beautiful word “quadrants”) are numbered counterclockwise. Added angle values on axes. Everything is clear, no problems.
And a green arrow is added. With a plus. What does it mean? Let me remind you that the fixed side of the angle Always nailed to the positive semi-axis OX. So, if we rotate the movable side of the angle along the arrow with a plus, i.e. in ascending order of quarter numbers, the angle will be considered positive. For example, the picture shows positive angle+60°.
If we put aside the corners in the opposite direction, clockwise, the angle will be considered negative. Hover your cursor over the picture (or touch the picture on your tablet), you will see a blue arrow with a minus sign. This is the direction of negative angle reading. For example, a negative angle (- 60°) is shown. And you will also see how the numbers on the axes have changed... I also converted them to negative angles. The numbering of the quadrants does not change.
This is where the first misunderstandings usually begin. How so!? What if a negative angle on a circle coincides with a positive one!? And in general, it turns out that the same position of the moving side (or point on number circle) can be called both a negative angle and a positive one!?
Yes. That's right. Let's say a positive angle of 90 degrees takes on a circle exactly the same position as a negative angle of minus 270 degrees. A positive angle, for example, +110° degrees takes exactly the same position as negative angle -250°.
No question. Anything is correct.) The choice of positive or negative angle calculation depends on the conditions of the task. If the condition says nothing in clear text about the sign of the angle, (like "determine the smallest positive angle", etc.), then we work with values that are convenient for us.
An exception (and how could we live without them?!) are trigonometric inequalities, but there we will master this trick.
And now a question for you. How did I know that the position of the 110° angle is the same as the position of the -250° angle?
Let me hint that this is connected with a complete revolution. In 360°... Not clear? Then we draw a circle. We draw it ourselves, on paper. Marking the corner approximately 110°. AND we think, how much time remains until a full revolution. Just 250° will remain...
Got it? And now - attention! If angles 110° and -250° occupy a circle same thing
situation, then what? Yes, the angles are 110° and -250° exactly the same
sine, cosine, tangent and cotangent!
Those. sin110° = sin(-250°), ctg110° = ctg(-250°) and so on. Now this is really important! And in itself, there are a lot of tasks where you need to simplify expressions, and as a basis for the subsequent mastery of reduction formulas and other intricacies of trigonometry.
Of course, I took 110° and -250° at random, purely as an example. All these equalities work for any angles occupying the same position on the circle. 60° and -300°, -75° and 285°, and so on. Let me note right away that the angles in these pairs are different. But they have trigonometric functions - identical.
I think you understand what negative angles are. It's quite simple. Counterclockwise - positive counting. Along the way - negative. Consider the angle positive or negative depends on us. From our desire. Well, and also from the task, of course... I hope you understand how to move in trigonometric functions from negative angles to positive ones and back. Draw a circle, an approximate angle, and see how much is missing to complete a full revolution, i.e. up to 360°.
Angles greater than 360°.
Let's deal with angles that are greater than 360°. Are there such things? There are, of course. How to draw them on a circle? No problem! Let's say we need to understand which quarter an angle of 1000° will fall into? Easily! We make one full turn counterclockwise (the angle we were given is positive!). We rewinded 360°. Well, let's move on! One more turn - it’s already 720°. How many are left? 280°. It’s not enough for a full turn... But the angle is more than 270° - and this is the border between the third and fourth quarter. Therefore, our angle of 1000° falls into the fourth quarter. All.
As you can see, it's quite simple. Let me remind you once again that the angle of 1000° and the angle of 280°, which we obtained by discarding the “extra” full revolutions, are, strictly speaking, different corners. But the trigonometric functions of these angles exactly the same! Those. sin1000° = sin280°, cos1000° = cos280°, etc. If I were a sine, I wouldn't notice the difference between these two angles...
Why is all this needed? Why do we need to convert angles from one to another? Yes, all for the same thing.) In order to simplify expressions. Simplifying expressions is, in fact, the main task of school mathematics. Well, and, along the way, the head is trained.)
Well, let's practice?)
We answer questions. Simple ones first.
1. Which quarter does the -325° angle fall into?
2. Which quarter does the 3000° angle fall into?
3. Which quarter does the angle -3000° fall into?
Any problems? Or uncertainty? Go to Section 555, Trigonometric Circle Practice. There, in the first lesson of this very " Practical work..." all in detail... In such questions of uncertainty to be shouldn't!
4. What sign does sin555° have?
5. What sign does tg555° have?
Have you determined? Great! Do you have any doubts? You need to go to Section 555... By the way, there you will learn to draw tangent and cotangent on a trigonometric circle. A very useful thing.
And now the questions are more sophisticated.
6. Reduce the expression sin777° to the sine of the smallest positive angle.
7. Reduce the expression cos777° to the cosine of the largest negative angle.
8. Reduce the expression cos(-777°) to the cosine of the smallest positive angle.
9. Reduce the expression sin777° to the sine of the largest negative angle.
Are questions 6-9 puzzling? Get used to it, on the Unified State Exam you don’t find such formulations... So be it, I’ll translate it. Just for you!
The words "bring an expression to..." mean to transform the expression so that its meaning hasn't changed A appearance changed according to the assignment. So, in tasks 6 and 9 we must get a sine, inside of which there is smallest positive angle. Everything else doesn't matter.
I will give out the answers in order (in violation of our rules). But what to do, there are only two signs, and there are only four quarters... You won’t be spoiled for choice.
6. sin57°.
7. cos(-57°).
8. cos57°.
9. -sin(-57°)
I assume that the answers to questions 6-9 confused some people. Especially -sin(-57°), really?) Indeed, in the elementary rules for calculating angles there is room for errors... That is why I had to do a lesson: “How to determine the signs of functions and give angles on a trigonometric circle?” In Section 555. Tasks 4 - 9 are covered there. Well sorted, with all the pitfalls. And they are here.)
In the next lesson we will deal with the mysterious radians and the number "Pi". Let's learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to discover that this basic information on the site enough already to solve some custom trigonometry problems!
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Coordinates x points lying on the circle are equal to cos(θ), and the coordinates y correspond to sin(θ), where θ is the magnitude of the angle.
- If you find it difficult to remember this rule, just remember that in the pair (cos; sin) “the sine comes last.”
- This rule can be derived by considering right triangles and the definition of these trigonometric functions (the sine of an angle is equal to the ratio of the length of the opposite side, and the cosine of the adjacent side to the hypotenuse).
Write down the coordinates of four points on the circle. A “unit circle” is a circle whose radius is equal to one. Use this to determine the coordinates x And y at four points of intersection of the coordinate axes with the circle. Above, for clarity, we designated these points as “east”, “north”, “west” and “south”, although they do not have established names.
- "East" corresponds to the point with coordinates (1; 0) .
- "North" corresponds to the point with coordinates (0; 1) .
- "West" corresponds to the point with coordinates (-1; 0) .
- "South" corresponds to the point with coordinates (0; -1) .
- This is similar to a regular graph, so there is no need to memorize these values, just remember the basic principle.
Remember the coordinates of the points in the first quadrant. The first quadrant is located in the upper right part of the circle, where the coordinates x And y take positive values. These are the only coordinates you need to remember:
Draw straight lines and determine the coordinates of the points of their intersection with the circle. If you draw straight horizontal and vertical lines from the points of one quadrant, the second points of intersection of these lines with the circle will have the coordinates x And y with the same absolute values, but different signs. In other words, you can draw horizontal and vertical lines from the points of the first quadrant and label the points of intersection with the circle with the same coordinates, but at the same time leave space on the left for the correct sign ("+" or "-").
To determine the sign of the coordinates, use the rules of symmetry. There are several ways to determine where to place the "-" sign:
- Remember the basic rules for regular charts. Axis x negative on the left and positive on the right. Axis y negative below and positive above;
- start with the first quadrant and draw lines to other points. If the line crosses the axis y, coordinate x will change its sign. If the line crosses the axis x, the sign of the coordinate will change y;
- remember that in the first quadrant all functions are positive, in the second quadrant only the sine is positive, in the third quadrant only the tangent is positive, and in the fourth quadrant only the cosine is positive;
- Whichever method you use, you should get (+,+) in the first quadrant, (-,+) in the second, (-,-) in the third, and (+,-) in the fourth.
Check if you made a mistake. Below is full list coordinates of “special” points (except for four points on the coordinate axes), if you move along the unit circle counterclockwise. Remember that to determine all these values, it is enough to remember the coordinates of the points only in the first quadrant:
- first quadrant: ( 3 2 , 1 2 (\displaystyle (\frac (\sqrt (3))(2)),(\frac (1)(2)))); (2 2 , 2 2 (\displaystyle (\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (1 2 , 3 2 (\displaystyle (\frac (1)(2)),(\frac (\sqrt (3))(2))));
- second quadrant: ( − 1 2 , 3 2 (\displaystyle -(\frac (1)(2)),(\frac (\sqrt (3))(2)))); (− 2 2 , 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (− 3 2 , 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),(\frac (1)(2))));
- third quadrant: ( − 3 2 , − 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),-(\frac (1)(2)))); (− 2 2 , − 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (− 1 2 , − 3 2 (\displaystyle -(\frac (1)(2)),-(\frac (\sqrt (3))(2))));
- fourth quadrant: ( 1 2 , − 3 2 (\displaystyle (\frac (1)(2)),-(\frac (\sqrt (3))(2)))); (2 2 , − 2 2 (\displaystyle (\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (3 2 , − 1 2 (\displaystyle (\frac (\sqrt (3))(2)),-(\frac (1)(2)))).
Diverse. Some of them are about in which quarters the cosine is positive and negative, in which quarters the sine is positive and negative. Everything turns out to be simple if you know how to calculate the value of these functions in different angles and are familiar with the principle of plotting functions on a graph.
What are the cosine values?
If we consider it, we have the following aspect ratio, which determines it: the cosine of the angle A is the ratio of the adjacent leg BC to the hypotenuse AB (Fig. 1): cos a= BC/AB.
Using the same triangle you can find the sine of an angle, tangent and cotangent. The sine will be the ratio of the opposite side of the angle AC to the hypotenuse AB. The tangent of an angle is found if the sine of the desired angle is divided by the cosine of the same angle; Substituting the corresponding formulas for finding sine and cosine, we obtain that tg a= AC/BC. Cotangent, as a function inverse to tangent, will be found like this: ctg a= BC/AC.
That is, with the same angle values, it was discovered that in a right triangle the aspect ratio is always the same. It would seem that it has become clear where these values come from, but why do we get negative numbers?
To do this, you need to consider the triangle in a Cartesian coordinate system, where there are both positive and negative values.
Clearly about the quarters, where is which
What are Cartesian coordinates? If we talk about two-dimensional space, we have two directed lines that intersect at point O - these are the abscissa axis (Ox) and the ordinate axis (Oy). From point O in the direction of the straight line there are positive numbers, and in the opposite direction - negative numbers. Ultimately, this directly determines in which quarters the cosine is positive and in which, accordingly, negative.
First quarter
If you place right triangle in the first quarter (from 0 o to 90 o), where the x and y axes have positive values (segments AO and BO lie on the axes where the values have a “+” sign), then both sine and cosine will also have positive values , and they are assigned a value with a plus sign. But what happens if you move the triangle to the second quarter (from 90 o to 180 o)?
Second quarter
We see that along the y-axis the legs AO received a negative value. Cosine of angle a now has this side in relation to a minus, and therefore its final value becomes negative. It turns out that in which quarter the cosine is positive depends on the placement of the triangle in the Cartesian coordinate system. And in this case, the cosine of the angle receives a negative value. But for the sine nothing has changed, because to determine its sign you need the OB side, which in this case remained with the plus sign. Let's summarize the first two quarters.
To find out in which quarters the cosine is positive and in which it is negative (as well as sine and other trigonometric functions), you need to look at what sign is assigned to which side. For cosine of angle a The side AO is important, for the sine - OB.
The first quarter has so far become the only one that answers the question: “In which quarters are sine and cosine positive at the same time?” Let's see further whether there will be further coincidences in the sign of these two functions.
In the second quarter, the side AO began to have a negative value, which means the cosine also became negative. The sine is kept positive.
Third quarter
Now both sides AO and OB have become negative. Let us recall the relations for cosine and sine:
Cos a = AO/AB;
Sin a = VO/AV.
AB always has a positive sign in a given coordinate system, since it is not directed in either of the two directions defined by the axes. But the legs became negative, which means the result for both functions is also negative, because if you perform multiplication or division operations with numbers, among which one and only one has a minus sign, then the result will also be with this sign.
The result at this stage:
1) In which quarter is the cosine positive? In the first of three.
2) In which quarter is the sine positive? In the first and second of three.
Fourth quarter (from 270 o to 360 o)
Here the side AO again acquires a plus sign, and therefore the cosine too.
For sine, things are still “negative”, because leg OB remains below the starting point O.
Conclusions
In order to understand in which quarters the cosine is positive, negative, etc., you need to remember the relationship for calculating the cosine: the leg adjacent to the angle divided by the hypotenuse. Some teachers suggest remembering this: k(osine) = (k) angle. If you remember this “cheat”, then you automatically understand that sine is the ratio of the opposite leg of the angle to the hypotenuse.
It is quite difficult to remember in which quarters the cosine is positive and in which it is negative. There are many trigonometric functions, and they all have their own meanings. But still, as a result: positive values for the sine are 1.2 quarters (from 0 o to 180 o); for cosine 1.4 quarters (from 0 o to 90 o and from 270 o to 360 o). In the remaining quarters the functions have minus values.
Perhaps it will be easier for someone to remember which sign is which by depicting the function.
For the sine it is clear that from zero to 180 o the ridge is above the line of sin(x) values, which means the function here is positive. For the cosine it’s the same: in which quarter the cosine is positive (photo 7), and in which it is negative, you can see by moving the line above and below the cos(x) axis. As a result, we can remember two ways to determine the sign of the sine and cosine functions:
1. On an imaginary circle with radius equal to one(although, in fact, it does not matter what the radius of the circle is, this is the example most often given in textbooks; this makes it easier to understand, but at the same time, if you do not make it clear that this is not important, children may get confused).
2. By depicting the dependence of the function along (x) on the argument x itself, as in the last figure.
Using the first method, you can UNDERSTAND what exactly the sign depends on, and we explained this in detail above. Figure 7, constructed from these data, visualizes the resulting function and its sign in the best possible way.