Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
First of all, let me remind you of a simple but very useful conclusion from the lesson “What are sine and cosine? What are tangent and cotangent?”
This is the output:
Sine, cosine, tangent and cotangent are tightly connected to their angles. We know one thing, which means we know another.
In other words, each angle has its own constant sine and cosine. And almost everyone has their own tangent and cotangent. Why almost? More on this below.
This knowledge helps a lot in your studies! There are a lot of tasks where you need to move from sines to angles and vice versa. For this there is table of sines. Similarly, for tasks with cosine - cosine table. And, as you may have guessed, there is tangent table And table of cotangents.)
Tables are different. Long ones, where you can see what, say, sin37°6’ is equal to. We open the Bradis tables, look for an angle of thirty-seven degrees six minutes and see the value of 0.6032. It’s clear that there is absolutely no need to remember this number (and thousands of other table values).
In fact, in our time, long tables of cosines, sines, tangents, cotangents are not really needed. One good calculator replaces them completely. But it doesn’t hurt to know about the existence of such tables. For general erudition.)
And why then this lesson?! - you ask.
But why. Among the infinite number of angles there are special, which you should know about All. All school geometry and trigonometry are built on these angles. This is a kind of "multiplication table" of trigonometry. If you don’t know what sin50° is equal to, for example, no one will judge you.) But if you don’t know what sin30° is equal to, be prepared to get a well-deserved two...
Such special The angles are also quite good. School textbooks usually kindly offer memorization sine table and cosine table for seventeen angles. And, of course, tangent table and cotangent table for the same seventeen angles... I.e. It is proposed to remember 68 values. Which, by the way, are very similar to each other, repeat themselves every now and then and change signs. For a person without perfect visual memory, this is quite a task...)
We'll take a different route. Let's replace rote memorization with logic and ingenuity. Then we will have to memorize 3 (three!) values for the table of sines and the table of cosines. And 3 (three!) values for the table of tangents and the table of cotangents. That's all. Six values are easier to remember than 68, it seems to me...)
We will obtain all other necessary values from these six using a powerful legal cheat sheet - trigonometric circle. If you have not studied this topic, follow the link, don’t be lazy. This circle is not only needed for this lesson. He is irreplaceable for all trigonometry at once. Not using such a tool is simply a sin! You do not want? That's your business. Memorize table of sines. Table of cosines. Table of tangents. Table of cotangents. All 68 values for a variety of angles.)
So, let's begin. First, let's divide all these special angles into three groups.
First group of angles.
Let's consider the first group seventeen angles special. These are 5 angles: 0°, 90°, 180°, 270°, 360°.
This is what the table of sines, cosines, tangents, and cotangents looks like for these angles:
Angle x
|
0 |
90 |
180 |
270 |
360 |
Angle x
|
0 |
||||
sin x |
0 |
1 |
0 |
-1 |
0 |
cos x |
1 |
0 |
-1 |
0 |
1 |
tg x |
0 |
noun |
0 |
noun |
0 |
ctg x |
noun |
0 |
noun |
0 |
noun |
Those who want to remember, remember. But I’ll say right away that all these ones and zeros get very confused in the head. Much stronger than you want.) Therefore, we turn on logic and the trigonometric circle.
We draw a circle and mark these same angles on it: 0°, 90°, 180°, 270°, 360°. I marked these corners with red dots:
It is immediately obvious what is special about these angles. Yes! These are the angles that fall exactly on the coordinate axis! Actually, that’s why people get confused... But we won’t get confused. Let's figure out how to find trigonometric functions of these angles without much memorization.
By the way, the angle position is 0 degrees completely coincides with a 360 degree angle position. This means that the sines, cosines, and tangents of these angles are exactly the same. I marked a 360 degree angle to complete the circle.
Suppose, in the difficult stressful environment of the Unified State Examination, you somehow doubted... What is the sine of 0 degrees? It seems like zero... What if it’s one?! Mechanical memorization is such a thing. In harsh conditions, doubts begin to gnaw...)
Calm, just calm!) I will tell you a practical technique that will give you a 100% correct answer and completely remove all doubts.
As an example, let's figure out how to clearly and reliably determine, say, the sine of 0 degrees. And at the same time, cosine 0. It is in these values, oddly enough, that people often get confused.
To do this, draw on a circle arbitrary corner X. In the first quarter, it was close to 0 degrees. Let us mark the sine and cosine of this angle on the axes X, everything is fine. Like this:
And now - attention! Let's reduce the angle X, bring the moving side closer to the axis OH. Hover your cursor over the picture (or tap the picture on your tablet) and you’ll see everything.
Now let's turn on elementary logic! Let's look and think: How does sinx behave as the angle x decreases? As the angle approaches zero? It's shrinking! And cosx increases! It remains to figure out what will happen to the sine when the angle collapses completely? When does the moving side of the angle (point A) settle down on the OX axis and the angle becomes equal to zero? Obviously, the sine of the angle will go to zero. And the cosine will increase to... to... What is the length of the moving side of the angle (the radius of the trigonometric circle)? One!
Here is the answer. The sine of 0 degrees is equal to 0. The cosine of 0 degrees is equal to 1. Absolutely ironclad and without any doubt!) Simply because otherwise it can not be.
In exactly the same way, you can find out (or clarify) the sine of 270 degrees, for example. Or cosine 180. Draw a circle, arbitrary an angle in a quarter next to the coordinate axis of interest to us, mentally move the side of the angle and grasp what the sine and cosine will become when the side of the angle falls on the axis. That's all.
As you can see, there is no need to memorize anything for this group of angles. Not needed here table of sines... Yes and cosine table- too.) By the way, after several uses of the trigonometric circle, all these values will be remembered by themselves. And if they forget, I drew a circle in 5 seconds and clarified it. Much easier than calling a friend from the toilet and risking your certificate, right?)
As for tangent and cotangent, everything is the same. We draw a tangent (cotangent) line on the circle - and everything is immediately visible. Where they are equal to zero, and where they do not exist. What, you don’t know about tangent and cotangent lines? This is sad, but fixable.) We visited Section 555 Tangent and cotangent on the trigonometric circle - and there are no problems!
If you have figured out how to clearly define sine, cosine, tangent and cotangent for these five angles, congratulations! Just in case, I inform you that you can now define functions any angles falling on the axes. And this is 450°, and 540°, and 1800°, and an infinite number of others...) I counted (correctly!) the angle on the circle - and there are no problems with the functions.
But it’s precisely with the measurement of angles that problems and errors occur... How to avoid them is written in the lesson: How to draw (count) any angle on a trigonometric circle in degrees. Elementary, but very helpful in the fight against errors.)
Here's a lesson: How to draw (measure) any angle on a trigonometric circle in radians - it will be cooler. In terms of possibilities. Let's say, determine which of the four semi-axes the angle falls on
you can do it in a couple of seconds. I am not kidding! Just in a couple of seconds. Well, of course, not only 345 pi...) And 121, and 16, and -1345. Any integer coefficient is suitable for an instant answer.
And if the corner
Just think! The correct answer is obtained in 10 seconds. For any fractional value of radians with a two in the denominator.
Actually, this is what is good about the trigonometric circle. Because the ability to work with some corners it automatically expands to infinite set corners
So, we’ve sorted out five corners out of seventeen.
Second group of angles.
The next group of angles are the angles 30°, 45° and 60°. Why exactly these, and not, for example, 20, 50 and 80? Yes, somehow it turned out this way... Historically.) Further it will be seen why these angles are good.
The table of sines cosines tangents cotangents for these angles looks like this:
Angle x
|
0 |
30 |
45 |
60 |
90 |
Angle x
|
0 |
||||
sin x |
0 |
1 |
|||
cos x |
1 |
0 |
|||
tg x |
0 |
1 |
noun |
||
ctg x |
noun |
1 |
0 |
I left the values for 0° and 90° from the previous table to complete the picture.) So that you can see that these angles lie in the first quarter and increase. From 0 to 90. This will be useful to us later.
The table values for angles of 30°, 45° and 60° must be remembered. Memorize it if you want. But here, too, there is an opportunity to make your life easier.) Pay attention to sine table values these angles. And compare with cosine table values...
Yes! They same! Just arranged in reverse order. Angles increase (0, 30, 45, 60, 90) - and sine values increase from 0 to 1. You can check with a calculator. And the cosine values are are decreasing from 1 to zero. Moreover, the values themselves same. For angles of 20, 50, 80 this would not work...
This is a useful conclusion. Enough to learn three values for angles of 30, 45, 60 degrees. And remember that for the sine they increase, and for the cosine they decrease. Towards the sine.) They meet halfway (45°), that is, the sine of 45 degrees is equal to the cosine of 45 degrees. And then they diverge again... Three meanings can be learned, right?
With tangents - cotangents the picture is exactly the same. One to one. Only the meanings are different. These values (three more!) also need to be learned.
Well, almost all the memorization is over. You have (hopefully) understood how to determine the values for the five angles falling on the axis and learned the values for the angles of 30, 45, 60 degrees. Total 8.
It remains to deal with the last group of 9 corners.
These are the angles:
120°; 135°; 150°; 210°; 225°; 240°; 300°; 315°; 330°. For these angles, you need to know the table of sines, the table of cosines, etc.
Nightmare, right?)
And if you add angles here, such as: 405°, 600°, or 3000° and many, many equally beautiful ones?)
Or angles in radians? For example, about angles:
and many others you should know All.
The funniest thing is to know this All - impossible in principle. If you use mechanical memory.
And it’s very easy, in fact elementary - if you use a trigonometric circle. Once you get the hang of working with the trigonometric circle, all those dreaded angles in degrees can be easily and elegantly reduced to the good old fashioned ones:
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.
The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concept of angle. In order to have a good understanding of these, at first glance, complex concepts (which cause a state of horror in many schoolchildren), and to make sure that “the devil is not as terrible as he is painted,” let’s start from the very beginning and understand the concept of an angle.
Angle concept: radian, degree
Let's look at the picture. The vector has “turned” relative to the point by a certain amount. So the measure of this rotation relative to the initial position will be corner.
What else do you need to know about the concept of angle? Well, of course, angle units!
Angle, in both geometry and trigonometry, can be measured in degrees and radians.
Angle (one degree) is the central angle in a circle subtended by a circular arc equal to part of the circle. Thus, the entire circle consists of “pieces” of circular arcs, or the angle described by the circle is equal.
That is, the figure above shows an angle equal to, that is, this angle rests on a circular arc the size of the circumference.
An angle in radians is the central angle in a circle subtended by a circular arc whose length is equal to the radius of the circle. Well, did you figure it out? If not, then let's figure it out from the drawing.
So, the figure shows an angle equal to a radian, that is, this angle rests on a circular arc, the length of which is equal to the radius of the circle (the length is equal to the length or the radius is equal to the length of the arc). Thus, the arc length is calculated by the formula:
Where is the central angle in radians.
Well, knowing this, can you answer how many radians are contained in the angle described by the circle? Yes, for this you need to remember the formula for circumference. Here she is:
Well, now let’s correlate these two formulas and find that the angle described by the circle is equal. That is, by correlating the value in degrees and radians, we get that. Respectively, . As you can see, unlike "degrees", the word "radian" is omitted, since the unit of measurement is usually clear from the context.
How many radians are there? That's right!
Got it? Then go ahead and fix it:
Having difficulties? Then look answers:
Right triangle: sine, cosine, tangent, cotangent of angle
So, we figured out the concept of an angle. But what is sine, cosine, tangent, and cotangent of an angle? Let's figure it out. To do this, a right triangle will help us.
What are the sides of a right triangle called? That's right, hypotenuse and legs: the hypotenuse is the side that lies opposite the right angle (in our example this is the side); the legs are the two remaining sides and (those adjacent to the right angle), and if we consider the legs relative to the angle, then the leg is the adjacent leg, and the leg is the opposite. So, now let’s answer the question: what are sine, cosine, tangent and cotangent of an angle?
Sine of angle- this is the ratio of the opposite (distant) leg to the hypotenuse.
In our triangle.
Cosine of angle- this is the ratio of the adjacent (close) leg to the hypotenuse.
In our triangle.
Tangent of the angle- this is the ratio of the opposite (distant) side to the adjacent (close).
In our triangle.
Cotangent of angle- this is the ratio of the adjacent (close) leg to the opposite (far).
In our triangle.
These definitions are necessary remember! To make it easier to remember which leg to divide into what, you need to clearly understand that in tangent And cotangent only the legs sit, and the hypotenuse appears only in sinus And cosine. And then you can come up with a chain of associations. For example, this one:
Cosine→touch→touch→adjacent;
Cotangent→touch→touch→adjacent.
First of all, you need to remember that sine, cosine, tangent and cotangent as the ratios of the sides of a triangle do not depend on the lengths of these sides (at the same angle). Do not believe? Then make sure by looking at the picture:
Consider, for example, the cosine of an angle. By definition, from a triangle: , but we can calculate the cosine of an angle from a triangle: . You see, the lengths of the sides are different, but the value of the cosine of one angle is the same. Thus, the values of sine, cosine, tangent and cotangent depend solely on the magnitude of the angle.
If you understand the definitions, then go ahead and consolidate them!
For the triangle shown in the figure below, we find.
Well, did you get it? Then try it yourself: calculate the same for the angle.
Unit (trigonometric) circle
Understanding the concepts of degrees and radians, we considered a circle with a radius equal to. Such a circle is called single. It will be very useful when studying trigonometry. Therefore, let's look at it in a little more detail.
As you can see, this circle is constructed in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin of coordinates, the initial position of the radius vector is fixed along the positive direction of the axis (in our example, this is the radius).
Each point on the circle corresponds to two numbers: the axis coordinate and the axis coordinate. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, we need to remember about the considered right triangle. In the figure above, you can see two whole right triangles. Consider a triangle. It is rectangular because it is perpendicular to the axis.
What is the triangle equal to? That's right. In addition, we know that is the radius of the unit circle, which means . Let's substitute this value into our formula for cosine. Here's what happens:
What is the triangle equal to? Well, of course, ! Substitute the radius value into this formula and get:
So, can you tell what coordinates a point belonging to a circle has? Well, no way? What if you realize that and are just numbers? Which coordinate does it correspond to? Well, of course, the coordinates! And what coordinate does it correspond to? That's right, coordinates! Thus, period.
What then are and equal to? That's right, let's use the corresponding definitions of tangent and cotangent and get that, a.
What if the angle is larger? For example, like in this picture:
What has changed in this example? Let's figure it out. To do this, let's turn again to a right triangle. Consider a right triangle: angle (as adjacent to an angle). What are the values of sine, cosine, tangent and cotangent for an angle? That's right, we adhere to the corresponding definitions of trigonometric functions:
Well, as you can see, the value of the sine of the angle still corresponds to the coordinate; the value of the cosine of the angle - the coordinate; and the values of tangent and cotangent to the corresponding ratios. Thus, these relations apply to any rotation of the radius vector.
It has already been mentioned that the initial position of the radius vector is along the positive direction of the axis. So far we have rotated this vector counterclockwise, but what happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain value, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles, and when rotating clockwise - negative.
So, we know that a whole revolution of the radius vector around a circle is or. Is it possible to rotate the radius vector to or to? Well, of course you can! In the first case, therefore, the radius vector will make one full revolution and stop at position or.
In the second case, that is, the radius vector will make three full revolutions and stop at position or.
Thus, from the above examples we can conclude that angles that differ by or (where is any integer) correspond to the same position of the radius vector.
The figure below shows an angle. The same image corresponds to the corner, etc. This list can be continued indefinitely. All these angles can be written by the general formula or (where is any integer)
Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values are:
Here's a unit circle to help you:
Having difficulties? Then let's figure it out. So we know that:
From here, we determine the coordinates of the points corresponding to certain angle measures. Well, let's start in order: the angle at corresponds to a point with coordinates, therefore:
Does not exist;
Further, adhering to the same logic, we find out that the corners in correspond to points with coordinates, respectively. Knowing this, it is easy to determine the values of trigonometric functions at the corresponding points. Try it yourself first, and then check the answers.
Answers:
Does not exist
Does not exist
Does not exist
Does not exist
Thus, we can make the following table:
There is no need to remember all these values. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values of trigonometric functions:
But the values of the trigonometric functions of angles in and, given in the table below, must be remembered:
Don't be scared, now we'll show you one example quite simple to remember the corresponding values:
To use this method, it is vital to remember the values of the sine for all three measures of angle (), as well as the value of the tangent of the angle. Knowing these values, it is quite simple to restore the entire table - the cosine values are transferred in accordance with the arrows, that is:
Knowing this, you can restore the values for. The numerator " " will match and the denominator " " will match. Cotangent values are transferred in accordance with the arrows indicated in the figure. If you understand this and remember the diagram with the arrows, then it will be enough to remember all the values from the table.
Coordinates of a point on a circle
Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation?
Well, of course you can! Let's get it out general formula for finding the coordinates of a point.
For example, here is a circle in front of us:
We are given that the point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of a point obtained by rotating the point by degrees.
As can be seen from the figure, the coordinate of the point corresponds to the length of the segment. The length of the segment corresponds to the coordinate of the center of the circle, that is, it is equal. The length of a segment can be expressed using the definition of cosine:
Then we have that for the point coordinate.
Using the same logic, we find the y coordinate value for the point. Thus,
So, in general, the coordinates of points are determined by the formulas:
Coordinates of the center of the circle,
Circle radius,
The rotation angle of the vector radius.
As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are equal to zero and the radius is equal to one:
Well, let's try out these formulas by practicing finding points on a circle?
1. Find the coordinates of a point on the unit circle obtained by rotating the point on.
2. Find the coordinates of a point on the unit circle obtained by rotating the point on.
3. Find the coordinates of a point on the unit circle obtained by rotating the point on.
4. The point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
5. The point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
Having trouble finding the coordinates of a point on a circle?
Solve these five examples (or get good at solving them) and you will learn to find them!
1.
You can notice that. But we know what corresponds to a full revolution of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the required coordinates of the point:
2. The unit circle is centered at a point, which means we can use simplified formulas:
You can notice that. We know what corresponds to two full revolutions of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the required coordinates of the point:
Sine and cosine are table values. We recall their meanings and get:
Thus, the desired point has coordinates.
3. The unit circle is centered at a point, which means we can use simplified formulas:
You can notice that. Let's depict the example in question in the figure:
The radius makes angles equal to and with the axis. Knowing that the table values of cosine and sine are equal, and having determined that the cosine here takes a negative value and the sine takes a positive value, we have:
Such examples are discussed in more detail when studying the formulas for reducing trigonometric functions in the topic.
Thus, the desired point has coordinates.
4.
Angle of rotation of the radius of the vector (by condition)
To determine the corresponding signs of sine and cosine, we construct a unit circle and angle:
As you can see, the value, that is, is positive, and the value, that is, is negative. Knowing the tabular values of the corresponding trigonometric functions, we obtain that:
Let's substitute the obtained values into our formula and find the coordinates:
Thus, the desired point has coordinates.
5. To solve this problem, we use formulas in general form, where
Coordinates of the center of the circle (in our example,
Circle radius (by condition)
Angle of rotation of the radius of the vector (by condition).
Let's substitute all the values into the formula and get:
and - table values. Let’s remember and substitute them into the formula:
Thus, the desired point has coordinates.
SUMMARY AND BASIC FORMULAS
The sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.
The cosine of an angle is the ratio of the adjacent (close) leg to the hypotenuse.
The tangent of an angle is the ratio of the opposite (far) side to the adjacent (close) side.
The cotangent of an angle is the ratio of the adjacent (close) side to the opposite (far) side.
Table of basic trigonometric functions for angles of 0, 30, 45, 60, 90, ... degrees
From the trigonometric definitions of the functions $\sin$, $\cos$, $\tan$ and $\cot$, you can find out their values for angles $0$ and $90$ degrees:
$\sin0°=0$, $\cos0°=1$, $\tan 0°=0$, $\cot 0°$ not defined;
$\sin90°=1$, $\cos90°=0$, $\cot90°=0$, $\tan 90°$ is not determined.
In a school geometry course, when studying right triangles, one finds the trigonometric functions of the angles $0°$, $30°$, $45°$, $60°$ and $90°$.
Found values of trigonometric functions for the indicated angles in degrees and radians, respectively ($0$, $\frac(\pi)(6)$, $\frac(\pi)(4)$, $\frac(\pi)(3) $, $\frac(\pi)(2)$) for ease of memorization and use are entered into a table called trigonometric table, table of basic values of trigonometric functions and so on.
When using reduction formulas, the trigonometric table can be expanded to an angle of $360°$ and, accordingly, $2\pi$ radians:
Using the periodicity properties of trigonometric functions, each angle, which will differ from the already known one by $360°$, can be calculated and recorded in a table. For example, the trigonometric function for angle $0°$ will have the same value for angle $0°+360°$, and for angle $0°+2 \cdot 360°$, and for angle $0°+3 \cdot 360°$ and etc.
Using a trigonometric table, you can determine the values of all angles of a unit circle.
In a school geometry course, you are supposed to memorize the basic values of trigonometric functions collected in a trigonometric table for the convenience of solving trigonometric problems.
Using a table
In the table, it is enough to find the required trigonometric function and the value of the angle or radians for which this function needs to be calculated. At the intersection of the row with the function and the column with the value, we obtain the desired value of the trigonometric function of the given argument.
In the figure you can see how to find the value of $\cos60°$, which is equal to $\frac(1)(2)$.
The extended trigonometric table is used in the same way. The advantage of using it is, as already mentioned, the calculation of the trigonometric function of almost any angle. For example, you can easily find the value $\tan 1 380°=\tan (1 380°-360°)=\tan(1 020°-360°)=\tan(660°-360°)=\tan300°$:
Bradis tables of basic trigonometric functions
The ability to calculate the trigonometric function of absolutely any angle value for an integer value of degrees and an integer value of minutes is provided by the use of Bradis tables. For example, find the value of $\cos34°7"$. The tables are divided into 2 parts: a table of values of $\sin$ and $\cos$ and a table of values of $\tan$ and $\cot$.
Bradis tables make it possible to obtain approximate values of trigonometric functions with an accuracy of up to 4 decimal places.
Using Bradis tables
Using the Bradis tables for sines, we find $\sin17°42"$. To do this, in the left column of the table of sines and cosines we find the value of degrees - $17°$, and in the top line we find the value of minutes - $42"$. At their intersection we obtain the desired value:
$\sin17°42"=0.304$.
To find the value $\sin17°44"$ you need to use the correction on the right side of the table. In this case, to the value $42"$, which is in the table, you need to add a correction for $2"$, which is equal to $0.0006$. We get:
$\sin17°44"=0.304+0.0006=0.3046$.
To find the value $\sin17°47"$ we also use the correction on the right side of the table, only in this case we take the value $\sin17°48"$ as a basis and subtract the correction for $1"$:
$\sin17°47"=0.3057-0.0003=0.3054$.
When calculating cosines, we perform similar actions, but we look at the degrees in the right column, and the minutes in the bottom column of the table. For example, $\cos20°=0.9397$.
There are no corrections for tangent values up to $90°$ and small angle cotangent. For example, let's find $\tan 78°37"$, which according to the table is equal to $4.967$.
Table of values of trigonometric functions
Note. This table of trigonometric function values uses the √ sign to represent the square root. To indicate a fraction, use the symbol "/".
see also useful materials:
For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, sine 30 degrees - we look for the column with the heading sin (sine) and find the intersection of this table column with the row “30 degrees”, at their intersection we read the result - one half. Similarly we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin column and the 60 degree line we find the value sin 60 = √3/2), etc. The values of sines, cosines and tangents of other “popular” angles are found in the same way.
Sine pi, cosine pi, tangent pi and other angles in radians
The table below of cosines, sines and tangents is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the angle of 60 degrees in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.
The number pi unambiguously expresses the dependence of the circumference on the degree measure of the angle. Thus, pi radians are equal to 180 degrees.
Any number expressed in terms of pi (radians) can be easily converted to degrees by replacing pi (π) with 180.
Examples:
1. Sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and it is equal to zero.
2. Cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and it is equal to minus one.
3. Tangent pi
tg π = tg 180 = 0
thus, tangent pi is the same as tangent 180 degrees and it is equal to zero.
Table of sine, cosine, tangent values for angles 0 - 360 degrees (common values)
|
angle α value (degrees) |
angle α value (via pi) |
sin (sinus) |
cos (cosine) |
tg (tangent) |
ctg (cotangent) |
sec (secant) |
cosec (cosecant) |
| 0 | 0 | 0 | 1 | 0 | - | 1 | - |
| 15 | π/12 | 2 - √3 | 2 + √3 | ||||
| 30 | π/6 | 1/2 | √3/2 | 1/√3 | √3 | 2/√3 | 2 |
| 45 | π/4 | √2/2 | √2/2 | 1 | 1 | √2 | √2 |
| 60 | π/3 | √3/2 | 1/2 | √3 | 1/√3 | 2 | 2/√3 |
| 75 | 5π/12 | 2 + √3 | 2 - √3 | ||||
| 90 | π/2 | 1 | 0 | - | 0 | - | 1 |
| 105 | 7π/12 |
- |
- 2 - √3 | √3 - 2 | |||
| 120 | 2π/3 | √3/2 | -1/2 | -√3 | -√3/3 | ||
| 135 | 3π/4 | √2/2 | -√2/2 | -1 | -1 | -√2 | √2 |
| 150 | 5π/6 | 1/2 | -√3/2 | -√3/3 | -√3 | ||
| 180 | π | 0 | -1 | 0 | - | -1 | - |
| 210 | 7π/6 | -1/2 | -√3/2 | √3/3 | √3 | ||
| 240 | 4π/3 | -√3/2 | -1/2 | √3 | √3/3 | ||
| 270 | 3π/2 | -1 | 0 | - | 0 | - | -1 |
| 360 | 2π | 0 | 1 | 0 | - | 1 | - |
If in the table of values of trigonometric functions a dash is indicated instead of the function value (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle the function does not have a specific value. If there is no dash, the cell is empty, which means we have not yet entered the required value. We are interested in what queries users come to us for and supplement the table with new values, despite the fact that current data on the values of cosines, sines and tangents of the most common angle values is quite sufficient to solve most problems.
Table of values of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numeric values “as per Bradis tables”)
| angle α value (degrees) | angle α value in radians | sin (sine) | cos (cosine) | tg (tangent) | ctg (cotangent) |
|---|---|---|---|---|---|
| 0 | 0 | ||||
| 15 |
0,2588 |
0,9659
|
0,2679 |
||
| 30 |
0,5000 |
0,5774 |
|||
| 45 |
0,7071 |
||||
|
0,7660 |
|||||
| 60 |
0,8660 |
0,5000
|
1,7321 |
||
|
7π/18 |
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