This article contains tables of sines, cosines, tangents and cotangents. First, we will provide a table of the basic values of trigonometric functions, that is, a table of sines, cosines, tangents and cotangents of angles of 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
Bibliography.
- 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
Example 1.
Find the radian measure of an angle equal to a) 40°, b)120°, c)105°
a) 40° = 40 π / 180 = 2π/9
b) 120° = 120 π/180 = 2π/3
c) 105° = 105 π/180 = 7π/12
Example 2.
Find the degree measure of the angle expressed in radians a) π/6, b) π/9, c) 2 π/3
a) π/6 = 180°/6 = 30°
b) π/9 = 180°/9 = 20°
c) 2π/3 = 2 180°/6 = 120°
Definition of sine, cosine, tangent and cotangent
Sine of acute angle t right triangle equal to the ratio of the opposite side to the hypotenuse (Fig. 1):
The cosine of the acute angle t of a right triangle is equal to the ratio of the adjacent leg to the hypotenuse (Fig. 1):
These definitions apply to a right triangle and are special cases of the definitions presented in this section.
Let's place the same right triangle on the number circle (Fig. 2).
We see that the leg b equal to a certain value y on the Y axis (ordinate axis), leg A equal to a certain value x on the X-axis (x-axis). And the hypotenuse With equal to the radius of the circle (R).
Thus, our formulas take on a different form.
Since b = y, a = x, c = R, then:
y x
sin t = -- , cos t = --.
R R
By the way, then, naturally, the tangent and cotangent formulas take on a different form.
Since tg t = b/a, ctg t = a/b, then other equations are also true:
tg t = y/x,
ctg = x/y.
But let's return to sine and cosine. We are dealing with a number circle in which the radius is 1. This means:
y
sin t = -- = y,
1
x
cos t = -- = x.
1
So we come to the third, more simple view trigonometric formulas.
These formulas apply not only to acute, but also to any other angle (obtuse or developed).
Definitions and formulas cos t, sin t, tg t, ctg t.
From the tangent and cotangent formulas another formula follows:
Number circle equations.
Signs of sine, cosine, tangent and cotangent in quarter circles:
| 1st quarter | 2nd quarter | 3rd quarter | 4th quarter | |
| cos t | + | – | – | + |
| sint | + | + | – | – |
| tg t, ctg t | + | – | + | – |
Cosine and sine of the main points of the number circle:
How to remember the values of cosines and sines of the main points of the number circle.
First of all, you need to know that in each pair of numbers the cosine values come first, the sine values come second.
1) Please note: with all the many points on the number circle, we are dealing with only five numbers (per module):
1 √2 √3
0; -; --; --; 1.
2 2 2
Make this “discovery” for yourself - and you will remove the psychological fear of the abundance of numbers: there are actually only five of them.
2) Let's start with the integers 0 and 1. They are only on the coordinate axes.
There is no need to learn by heart where, for example, the cosine in the modulus has one and where it has 0.
At the ends of the axle cosines(axes X), of course, cosines equal modulo 1, and the sines are equal to 0.
At the ends of the axle sinuses(axes at) sines are equal to modulus 1, and the cosines are equal to 0.
Now about the signs. Zero has no sign. As for 1 - here you just need to remember the simplest thing: from the 7th grade course you know that on the axis X to the right of the center of the coordinate plane are positive numbers, to the left are negative; on the axis at positive numbers go up from the center, negative numbers go down. And then you won't be mistaken with sign 1.
3) Now let's move on to fractional values.
All denominators of fractions contain the same number 2. We will no longer be mistaken about what to write in the denominator.
In the middles of the quarters, cosine and sine have absolutely the same absolute value: √2/2. In which case they are with a plus or minus sign - see the table above. But you hardly need such a table: you know this from the same 7th grade course.
All closest to the axis X the points have absolutely identical cosine and sine values: (√3/2; 1/2).
Values of all closest to the axis at The points are also absolutely identical in modulus - and they have the same numbers, only they have “swapped” places: (1/2; √3/2).
Now about the signs - there is an interesting alternation here (although we believe you should be able to figure out the signs easily anyway).
If in the first quarter the values of both cosine and sine have a plus sign, then in the diametrically opposite (third) they have a minus sign.
If in the second quarter with a minus sign there are only cosines, then in the diametrically opposite (fourth) there are only sines.
It remains only to recall that in each combination of cosine and sine values, the first number is the cosine value, the second number is the sine value.
Pay attention to one more regularity: the sine and cosine of all diametrically opposite points of the circle are absolutely equal in magnitude. Take, for example, the opposite points π/3 and 4π/3:
cos π/3 = 1/2, sin π/3 = √3/2
cos 4π/3 = -1/2, sin 4π/3 = -√3/2
The values of the cosines and sines of two opposite points differ only in sign. But here, too, there is a pattern: the sines and cosines of diametrically opposite points always have opposite signs.
It is important to know:
The values of the cosines and sines of points on the number circle sequentially increase or decrease in a strictly defined order: from the smallest value to the largest and vice versa (see the section “Increasing and decreasing trigonometric functions” - however, this is easy to verify by just looking at the number circle higher).
In descending order, the following alternation of values is obtained:
√3 √2 1 1 √2 √3
1; --; --; -; 0; – -; – --; – --; –1
2 2 2 2 2 2
They increase strictly in the reverse order.
Once you understand this simple pattern, you will learn how to determine the values of sine and cosine quite easily.
In general, this question deserves special attention, but here everything is simple: at the angle of degrees both the sine and cosine are positive (see figure), then we take the “plus” sign.
Now try, based on the above, to find the sine and cosine of the angles: and
You can cheat: in particular for an angle in degrees. Since if one angle of a right triangle is equal to degrees, then the second is equal to degrees. Now the familiar formulas come into force:
Then since, then and. Since, then and. With degrees it’s even simpler: if one of the angles of a right triangle is equal to degrees, then the other is also equal to degrees, which means the triangle is isosceles.
This means that its legs are equal. This means that its sine and cosine are equal.
Now, using the new definition (using X and Y!), find the sine and cosine of angles in degrees and degrees. You won’t be able to draw any triangles here! They will be too flat!
You should have gotten:
You can find the tangent and cotangent yourself using the formulas:
Please note that you cannot divide by zero!!
Now all the obtained numbers can be tabulated:
Here are the values of sine, cosine, tangent and cotangent of angles 1st quarter. For convenience, angles are given in both degrees and radians (but now you know the relationship between them!). Pay attention to the 2 dashes in the table: namely, the cotangent of zero and the tangent of degrees. This is no accident!
In particular:
Now let's generalize the concept of sine and cosine to a completely arbitrary angle. I will consider two cases here:
- The angle ranges from to degrees
- Angle greater than degrees
Generally speaking, I twisted my heart a little when I spoke about “absolutely all” angles. They can also be negative! But we will consider this case in another article. Let's look at the first case first.
If the angle lies in the 1st quarter, then everything is clear, we have already considered this case and even drew tables.
Now let our angle be more than degrees and not more than. This means that it is located either in the 2nd, 3rd or 4th quarter.
What do we do? Yes, exactly the same!
Let's take a look instead of something like this...
...like this:
That is, consider the angle lying in the second quarter. What can we say about him?
The point that is the intersection point of the ray and the circle still has 2 coordinates (nothing supernatural, right?). These are the coordinates and.
Moreover, the first coordinate is negative, and the second is positive! It means that at the corners of the second quarter, the cosine is negative and the sine is positive!
Amazing, right? Before this, we had never encountered a negative cosine.
And in principle this could not be the case when we considered trigonometric functions as the ratio of the sides of a triangle. By the way, think about which angles have the same cosine? Which ones have the same sine?
Similarly, you can consider the angles in all other quarters. Let me just remind you that the angle is counted counterclockwise! (as shown in the last picture!).
Of course, you can count in the other direction, but the approach to such angles will be somewhat different.
Based on the above reasoning, we can arrange the signs of sine, cosine, tangent (as sine divided by cosine) and cotangent (as cosine divided by sine) for all four quarters.
But once again, there is no point in memorizing this drawing. Everything you need to know:
Let's practice a little with you. Very simple tasks:
Find out what sign the following quantities have:
Shall we check?
- degrees is an angle, greater and lesser, which means it lies in 3 quarters. Draw any corner in the 3rd quarter and see what kind of player it has. It will turn out to be negative. Then.
degrees - 2 quarter angle. The sine there is positive, and the cosine is negative. Plus divided by minus equals minus. Means.
degrees - angle, greater and lesser. This means it lies in the 4th quarter. For any angle of the fourth quarter, the “x” will be positive, which means - We work with radians in the same way: this is the angle of the second quarter (since and. The sine of the second quarter is positive.
.
, this is the fourth quarter corner. There the cosine is positive.
- corner of the fourth quarter again. There the cosine is positive and the sine is negative. Then the tangent will be less than zero:
Perhaps it is difficult for you to determine quarters in radians. In that case, you can always go to degrees. The answer, of course, will be exactly the same.
Now I would like to very briefly dwell on another point. Let's remember the basic trigonometric identity again.
As I already said, from it we can express the sine through the cosine or vice versa:
The choice of sign will be influenced only by the quarter in which our alpha angle is located. There are a lot of problems on the last two formulas in the Unified State Exam, for example, these:
Task
Find if and.
In fact, this is a quarter task! Look how it is solved:
Solution
So, let's substitute the value here, then. Now the only thing left to do is deal with the sign. What do we need for this? Know which quarter our corner is in. According to the conditions of the problem: . What quarter is this? Fourth. What is the sign of the cosine in the fourth quarter? The cosine in the fourth quarter is positive. Then all we have to do is select the plus sign in front. , Then.
I will not dwell in detail on such tasks now, but they detailed analysis you can find in the article "". I just wanted to point out to you the importance of what sign this or that trigonometric function takes depending on the quarter.
Angles greater than degrees
The last thing I would like to point out in this article is what to do with angles greater than degrees?
What is it and what can you eat it with to avoid choking? Let's take, let's say, an angle in degrees (radians) and go counterclockwise from it...
In the picture I drew a spiral, but you understand that in reality we do not have any spiral: we only have a circle.
So where will we end up if we start from a certain angle and walk the entire circle (degrees or radians)?
Where will we go? And we will come to the same corner!
The same is, of course, true for any other angle:
Taking an arbitrary angle and walking the entire circle, we will return to the same angle.
What will this give us? Here's what: if, then
From where we finally get:
For any whole. It means that sine and cosine are periodic functions with period.
Thus, there is no problem in finding the sign of a now arbitrary angle: we just need to discard all the “whole circles” that fit in our angle and find out in which quarter the remaining angle lies.
For example, find a sign:
We check:
- In degrees fits times by degrees (degrees):
degrees left. This is a 4 quarter angle. There the sine is negative, which means - . degrees. This is a 3 quarter angle. There the cosine is negative. Then
- . . Since, then - the angle of the first quarter. There the cosine is positive. Then cos
- . . Since, our angle lies in the second quarter, where the sine is positive.
We can do the same for tangent and cotangent. However, in fact, they are even simpler: they are also periodic functions, only their period is 2 times less:
So, you understand what a trigonometric circle is and what it is needed for.
But we still have a lot of questions:
- What are negative angles?
- How to calculate trigonometric functions at these angles
- How to use the known values of trigonometric functions of the 1st quarter to look for the values of functions in other quarters (is it really necessary to cram the table?!)
- How can you use a circle to simplify solutions to trigonometric equations?
AVERAGE LEVEL
Well, in this article we will continue our study of the trigonometric circle and discuss the following points:
- What are negative angles?
- How to calculate the values of trigonometric functions at these angles?
- How to use the known values of trigonometric functions of 1 quarter to look for the values of functions in other quarters?
- What is the tangent axis and cotangent axis?
We don’t need any additional knowledge other than basic skills in working with a unit circle (previous article). Well, let's get to the first question: what are negative angles?
Negative angles
Negative angles in trigonometry are plotted on the trigonometric circle down from the beginning, in the direction of clockwise movement:
Let's remember how we previously plotted angles on a trigonometric circle: We started from the positive direction of the axis counterclock-wise:
Then in our drawing an angle equal to is constructed. We built all the corners in the same way.
However, nothing prevents us from moving from the positive direction of the axis clockwise.
We will also get different angles, but they will be negative:
The following picture shows two angles, equal in absolute value, but opposite in sign:
In general, the rule can be formulated like this:
- We go counterclockwise - we get positive angles
- We go clockwise - we get negative angles
The rule is shown schematically in this figure: 
You could ask me a completely reasonable question: well, we need angles in order to measure their sine, cosine, tangent and cotangent values.
So is there a difference when our angle is positive and when it is negative? I will answer you: as a rule, there is.
However, you can always reduce the calculation trigonometric function from negative angle to function calculation in angle positive.
Look at the following picture:
I constructed two angles, they are equal in absolute value, but have the opposite sign. For each angle, mark its sine and cosine on the axes.
What do you and I see? Here's what:
- The sines are at the angles and are opposite in sign! Then if
- The cosines of the angles coincide! Then if
- Since then:
- Since then:
Thus, we can always get rid of the negative sign inside any trigonometric function: either by simply eliminating it, as with cosine, or by placing it in front of the function, as with sine, tangent and cotangent.
By the way, remember the name of the function that executes for any valid value: ?
Such a function is called odd.
But if for any admissible one the following is true: ? Then in this case the function is called even.
So, you and I have just shown that:
| Sine, tangent and cotangent are odd functions, and cosine is an even function. |
Thus, as you understand, it makes no difference whether we are looking for the sine of a positive angle or a negative one: dealing with a minus is very simple. So we don't need tables separately for negative angles.
On the other hand, you must agree that it would be very convenient, knowing only the trigonometric functions of the angles of the first quarter, to be able to calculate similar functions for the remaining quarters. Is it possible to do this? Of course you can! You have at least 2 ways: the first is to build a triangle and apply the Pythagorean theorem (that’s how you and I found the values of trigonometric functions for the main angles of the first quarter), and the second is to remember the values of the functions for angles in the first quarter and some simple rule, to be able to calculate trigonometric functions for all other quarters. The second method will save you a lot of fuss with triangles and Pythagoras, so I see it as more promising:
So, this method(or rule) is called reduction formulas.
Reduction formulas
Roughly speaking, these formulas will help you not to remember this table (by the way, it contains 98 numbers!):
if you remember this one (only 20 numbers):
That is, you can not bother yourself with completely unnecessary 78 numbers! Let, for example, we need to calculate. It is clear that this is not the case in a small table. What do we do? Here's what:
First, we will need the following knowledge:
- Sine and cosine have a period (degrees), that is
Tangent (cotangent) have a period (degrees)
Any integer
- Sine and tangent are odd functions, and cosine is an even function:
We have already proven the first statement with you, and the validity of the second was established quite recently.
The actual casting rule looks like this:
- If we calculate the value of a trigonometric function from a negative angle, we make it positive using a group of formulas (2). For example:
- We discard its periods for sine and cosine: (in degrees), and for tangent - (in degrees). For example:
- If the remaining “corner” is less than degrees, then the problem is solved: we look for it in the “small table”.
- Otherwise, we are looking for which quarter our corner lies in: it will be the 2nd, 3rd or 4th quarter. Let's look at the sign of the required function in the quadrant. Remember this sign!!!
- We represent the angle in one of the following forms:
(if in the second quarter)
(if in the second quarter)
(if in the third quarter)
(if in the third quarter)
(if in the fourth quarter)so that the remaining angle is greater than zero and less than degrees. For example:
In principle, it does not matter in which of the two alternative forms for each quarter you represent the angle. This will not affect the final result.
- Now let’s see what we got: if you chose to write in terms of or degrees plus minus something, then the sign of the function will not change: you simply remove or and write the sine, cosine or tangent of the remaining angle. If you chose notation in or degrees, then change sine to cosine, cosine to sine, tangent to cotangent, cotangent to tangent.
- We put the sign from point 4 in front of the resulting expression.
Let's demonstrate all of the above with examples:
- Calculate
- Calculate
- Find your meaning:
Let's start in order:
- We act according to our algorithm. Select an integer number of circles for:
In general, we conclude that the entire corner fits 5 times, but how much is left? Left. Then
Well, we have discarded the excess. Now let's look at the sign. lies in the 4th quarter. The sine of the fourth quarter has a minus sign, and I shouldn’t forget to put it in the answer. Next, we present according to one of the two formulas of paragraph 5 of the reduction rules. I will choose:
Now let’s look at what happened: we have a case with degrees, then we discard it and change the sine to cosine. And we put a minus sign in front of it!
degrees - the angle in the first quarter. We know (you promised me to learn a small table!!) its meaning:
Then we get the final answer:
Answer:
- everything is the same, but instead of degrees - radians. It's OK. The main thing to remember is that
But you don’t have to replace radians with degrees. It's a matter of your taste. I won't change anything. I'll start again by discarding entire circles:
Let's discard - these are two whole circles. All that remains is to calculate. This angle is in the third quarter. The cosine of the third quarter is negative. Don't forget to put a minus sign in the answer. you can imagine how. Let us remember the rule again: we have the case of an “integer” number (or), then the function does not change:
Then.
Answer: . - . You need to do the same thing, but with two functions. I'll be a little more brief: and degrees - the angles of the second quarter. The cosine of the second quarter has a minus sign, and the sine has a plus sign. can be represented as: , and how, then
Both cases are “halves of the whole”. Then the sine changes to a cosine, and the cosine changes to a sine. Moreover, there is a minus sign in front of the cosine:
Answer: .
Now practice on your own using the following examples:
And here are the solutions:
First, let's get rid of the minus by placing it in front of the sine (since sine is an odd function!!!). Next let's look at the angles:We discard an integer number of circles - that is, three circles ().
It remains to calculate: .
We do the same with the second corner:We delete an integer number of circles - 3 circles () then:
Now we think: in which quarter does the remaining angle lie? He “falls short” of everything. Then what quarter is it? Fourth. What is the sign of the cosine of the fourth quarter? Positive. Now let's imagine. Since we are subtracting from a whole quantity, we do not change the sign of the cosine:
We substitute all the obtained data into the formula:
Answer: .
Standard: remove the minus from the cosine, using the fact that.
All that remains is to calculate the cosine of degrees. Let's remove whole circles: . ThenThen.
Answer: .- We proceed as in the previous example.
Since you remember that the period of the tangent is (or) unlike the cosine or sine, for which it is 2 times larger, then we will remove the integer quantity.
degrees - the angle in the second quarter. The tangent of the second quarter is negative, then let’s not forget about the “minus” at the end! can be written as. The tangent changes to cotangent. Finally we get:
Then.
Answer: .
Well, there's just a little left!
Tangent axis and cotangent axis
The last thing I would like to touch on here is the two additional axes. As we already discussed, we have two axes:
- Axis - cosine axis
- Axis - axis of sines
In fact, we've run out of coordinate axes, haven't we? But what about tangents and cotangents?
Is there really no graphic interpretation for them?
In fact, it exists, you can see it in this picture:
In particular, from these pictures we can say this:
- Tangent and cotangent have the same quarter signs
- They are positive in the 1st and 3rd quarters
- They are negative in the 2nd and 4th quarters
- Tangent is not defined at angles
- Cotangent not defined at corners
What else are these pictures for? You'll learn at an advanced level, where I'll tell you how you can use a trigonometric circle to simplify solutions to trigonometric equations!
ADVANCED LEVEL
In this article I will describe how unit circle (trigonometric circle) may be useful in solving trigonometric equations.
I can think of two cases where it might be useful:
- In the answer we don’t get a “beautiful” angle, but nevertheless we need to select the roots
- The answer contains too many series of roots
You don’t need any specific knowledge other than knowledge of the topic:
I tried to write the topic “trigonometric equations” without resorting to circles. Many would not praise me for such an approach.
But I prefer the formula, so what can I do? However, in some cases there are not enough formulas. The following example motivated me to write this article:
Solve the equation:
Well then. Solving the equation itself is not difficult.
Reverse replacement:
Hence, our original equation is equivalent to as many as four simple equations! Do we really need to write down 4 series of roots:
In principle, we could stop there. But not for the readers of this article, which claims to be some kind of “complexity”!
Let's look at the first series of roots first. So, we take the unit circle, now let's apply these roots to the circle (separately for and for):
Pay attention: what angle is between the corners and? This is the corner. Now let's do the same for the series: .
The angle between the roots of the equation is again . Now let's combine these two pictures:
What do we see? Otherwise, all angles between our roots are equal. What does it mean?
If we start from a corner and take equal angles (for any integer), then we will always end up at one of the four points on the upper circle! Thus, 2 series of roots:
Can be combined into one:
Alas, for the root series:
These arguments will no longer be valid. Make a drawing and understand why this is so. However, they can be combined as follows:
Then the original equation has roots:
Which is a pretty short and succinct answer. What does brevity and conciseness mean? About the level of your mathematical literacy.
This was the first example in which the use of the trigonometric circle produced useful results.
The second example is equations that have “ugly roots.”
For example:
- Solve the equation.
- Find its roots belonging to the interval.
The first part is not difficult at all.
Since you are already familiar with the topic, I will allow myself to be brief in my statements.
then or
This is how we found the roots of our equation. Nothing complicated.
It is more difficult to solve the second part of the task without knowing exactly what the arc cosine of minus one quarter is (this is not a table value).
However, we can depict the found series of roots on the unit circle:
What do we see? Firstly, the figure made it clear to us within what limits the arc cosine lies:
This visual interpretation will help us find the roots belonging to the segment: .
Firstly, the number itself falls into it, then (see figure).
also belongs to the segment.
Thus, the unit circle helps determine where the “ugly” angles fall.
You should have at least one more question: But what should we do with tangents and cotangents?
In fact, they also have their own axes, although they have a slightly specific appearance:
Otherwise, the way to handle them will be the same as with sine and cosine.
Example
The equation is given.
- Solve this equation.
- Indicate the roots of this equation that belong to the interval.
Solution:
We draw a unit circle and mark our solutions on it:
From the figure you can understand that:
Or even more: since, then
Then we find the roots belonging to the segment.
, (because)
I leave it to you to verify for yourself that our equation has no other roots belonging to the interval.
SUMMARY AND BASIC FORMULAS
The main tool of trigonometry is trigonometric circle, it allows you to measure angles, find their sines, cosines, etc.
There are two ways to measure angles.
- Through degrees
- Through radians
And vice versa: from radians to degrees:
To find the sine and cosine of an angle you need:
- Draw a unit circle with the center coinciding with the vertex of the angle.
- Find the point of intersection of this angle with the circle.
- Its “X” coordinate is the cosine of the desired angle.
- Its “game” coordinate is the sine of the desired angle.
Reduction formulas
These are formulas to simplify complex expressions trigonometric function.
These formulas will help you not to remember this table:
Summarizing
You learned how to make a universal spur using trigonometry.
You have learned to solve problems much easier and faster and, most importantly, without mistakes.
You realized that you don’t need to cram any tables and don’t need to cram anything at all!
Now I want to hear you!
Did you manage to figure this one out? complex topic?
What did you like? What didn't you like?
Maybe you found a mistake?
Write in the comments!
And good luck on the exam!
Allows you to establish a number of characteristic results - properties of sine, cosine, tangent and cotangent. In this article we will look at three main properties. The first of them indicates the signs of the sine, cosine, tangent and cotangent of the angle α depending on the angle of which coordinate quarter is α. Next we will consider the property of periodicity, which establishes the invariance of the values of sine, cosine, tangent and cotangent of the angle α when this angle changes by an integer number of revolutions. The third property expresses the relationship between the values of sine, cosine, tangent and cotangent of opposite angles α and −α.
If you are interested in the properties of the functions sine, cosine, tangent and cotangent, then you can study them in the corresponding section of the article.
Page navigation.
Signs of sine, cosine, tangent and cotangent by quarters
Below in this paragraph the phrase “angle of I, II, III and IV coordinate quarter” will appear. Let's explain what these angles are.
Let's take a unit circle, mark the starting point A(1, 0) on it, and rotate it around the point O by an angle α, and we will assume that we will get to the point A 1 (x, y).
They say that angle α is the angle of the I, II, III, IV coordinate quadrant, if point A 1 lies in the I, II, III, IV quarters, respectively; if the angle α is such that point A 1 lies on any of the coordinate lines Ox or Oy, then this angle does not belong to any of the four quarters.
For clarity, here is a graphic illustration. The drawings below show rotation angles of 30, −210, 585, and −45 degrees, which are the angles of the I, II, III, and IV coordinate quarters, respectively.
Angles 0, ±90, ±180, ±270, ±360, … degrees do not belong to any of the coordinate quarters.
Now let's figure out what signs have the values of sine, cosine, tangent and cotangent of the angle of rotation α, depending on which quadrant angle α is.
For sine and cosine this is easy to do.
By definition, the sine of angle α is the ordinate of point A 1. Obviously, in the I and II coordinate quarters it is positive, and in the III and IV quarters it is negative. Thus, the sine of angle α has a plus sign in the 1st and 2nd quarters, and a minus sign in the 3rd and 6th quarters.
In turn, the cosine of the angle α is the abscissa of point A 1. In the I and IV quarters it is positive, and in the II and III quarters it is negative. Consequently, the values of the cosine of the angle α in the I and IV quarters are positive, and in the II and III quarters they are negative.
To determine the signs of the quarters of tangent and cotangent, you need to remember their definitions: tangent is the ratio of the ordinate of point A 1 to the abscissa, and cotangent is the ratio of the abscissa of point A 1 to the ordinate. Then from rules for dividing numbers with the same and different signs it follows that tangent and cotangent have a plus sign when the abscissa and ordinate signs of point A 1 are the same, and have a minus sign when the abscissa and ordinate signs of point A 1 are different. Consequently, the tangent and cotangent of the angle have a + sign in the I and III coordinate quarters, and a minus sign in the II and IV quarters.
Indeed, for example, in the first quarter both the abscissa x and the ordinate y of point A 1 are positive, then both the quotient x/y and the quotient y/x are positive, therefore, tangent and cotangent have + signs. And in the second quarter, the abscissa x is negative, and the ordinate y is positive, therefore both x/y and y/x are negative, hence the tangent and cotangent have a minus sign.
Let's move on to the next property of sine, cosine, tangent and cotangent.
Periodicity property
Now we will look at perhaps the most obvious property of sine, cosine, tangent and cotangent of an angle. It is as follows: when the angle changes by an integer number of full revolutions, the values of the sine, cosine, tangent and cotangent of this angle do not change.
This is understandable: when the angle changes by an integer number of revolutions, we starting point And we will always get to point A 1 on the unit circle, therefore, the values of sine, cosine, tangent and cotangent remain unchanged, since the coordinates of point A 1 remain unchanged.
Using formulas, the considered property of sine, cosine, tangent and cotangent can be written as follows: sin(α+2·π·z)=sinα, cos(α+2·π·z)=cosα, tan(α+2·π· z)=tgα , ctg(α+2·π·z)=ctgα , where α is the angle of rotation in radians, z is any , absolute value which indicates the number of full revolutions by which the angle α changes, and the sign of the number z indicates the direction of rotation.
If the rotation angle α is specified in degrees, then the indicated formulas will be rewritten as sin(α+360° z)=sinα , cos(α+360° z)=cosα , tg(α+360° z)=tgα , ctg(α+360°·z)=ctgα .
Let's give examples of using this property. For example, 
, because 
, A 
. Here's another example: or .
This property, together with reduction formulas, is very often used when calculating the values of sine, cosine, tangent and cotangent of “large” angles.
The considered property of sine, cosine, tangent and cotangent is sometimes called the property of periodicity.
Properties of sines, cosines, tangents and cotangents of opposite angles
Let A 1 be the point obtained by rotating the initial point A(1, 0) around point O by an angle α, and point A 2 be the result of rotating point A by an angle −α, opposite to angle α.
The property of sines, cosines, tangents and cotangents of opposite angles is based on a fairly obvious fact: the points A 1 and A 2 mentioned above either coincide (at) or are located symmetrically relative to the Ox axis. That is, if point A 1 has coordinates (x, y), then point A 2 will have coordinates (x, −y). From here, using the definitions of sine, cosine, tangent and cotangent, we write the equalities and .
Comparing them, we come to relationships between sines, cosines, tangents and cotangents of opposite angles α and −α of the form.
This is the property under consideration in the form of formulas.
Let's give examples of using this property. For example, the equalities and 
.
It only remains to note that the property of sines, cosines, tangents and cotangents of opposite angles, like the previous property, is often used when calculating the values of sine, cosine, tangent and cotangent, and allows you to completely avoid negative angles.
Bibliography.
- 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
- 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.
- 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.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Trigonometric circle. Unit circle. Number circle. What it is?
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.
For successful work With the trigonometric circle you only need to know three things.
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