TABLE OF VALUES OF TRIGONOMETRIC FUNCTIONS
The table of values of trigonometric functions is compiled for angles of 0, 30, 45, 60, 90, 180, 270 and 360 degrees and the corresponding angle values in vradians. Of the trigonometric functions, the table shows sine, cosine, tangent, cotangent, secant and cosecant. For the convenience of solving school examples, the values of trigonometric functions in the table are written in the form of a fraction while preserving the signs for extracting the square root of numbers, which very often helps to reduce complex mathematical expressions. For tangent and cotangent, the values of some angles cannot be determined. For the values of tangent and cotangent of such angles, there is a dash in the table of values of trigonometric functions. It is generally accepted that the tangent and cotangent of such angles is equal to infinity. On a separate page there are formulas for reducing trigonometric functions.
The table of values for the trigonometric sine function shows the values for the following angles: sin 0, sin 30, sin 45, sin 60, sin 90, sin 180, sin 270, sin 360 in degrees, which corresponds to sin 0 pi, sin pi/6 , sin pi/4, sin pi/3, sin pi/2, sin pi, sin 3 pi/2, sin 2 pi in radian measure of angles. School table of sines.
For the trigonometric cosine function, the table shows the values for the following angles: cos 0, cos 30, cos 45, cos 60, cos 90, cos 180, cos 270, cos 360 in degrees, which corresponds to cos 0 pi, cos pi by 6, cos pi by 4, cos pi by 3, cos pi by 2, cos pi, cos 3 pi by 2, cos 2 pi in radian measure of angles. School table of cosines.
The trigonometric table for the trigonometric tangent function gives values for the following angles: tg 0, tg 30, tg 45, tg 60, tg 180, tg 360 in degree measure, which corresponds to tg 0 pi, tg pi/6, tg pi/4, tg pi/3, tg pi, tg 2 pi in radian measure of angles. The following values of the trigonometric tangent functions are not defined tan 90, tan 270, tan pi/2, tan 3 pi/2 and are considered equal to infinity.
For the trigonometric function cotangent in the trigonometric table the values of the following angles are given: ctg 30, ctg 45, ctg 60, ctg 90, ctg 270 in degree measure, which corresponds to ctg pi/6, ctg pi/4, ctg pi/3, tg pi/ 2, tan 3 pi/2 in radian measure of angles. The following values of the trigonometric cotangent functions are not defined ctg 0, ctg 180, ctg 360, ctg 0 pi, ctg pi, ctg 2 pi and are considered equal to infinity.
The values of the trigonometric functions secant and cosecant are given for the same angles in degrees and radians as sine, cosine, tangent, cotangent.
The table of values of trigonometric functions of non-standard angles shows the values of sine, cosine, tangent and cotangent for angles in degrees 15, 18, 22.5, 36, 54, 67.5 72 degrees and in radians pi/12, pi/10, pi/ 8, pi/5, 3pi/8, 2pi/5 radians. The values of trigonometric functions are expressed in terms of fractions and square roots to make it easier to reduce fractions in school examples.
Three more trigonometry monsters. The first is the tangent of 1.5 one and a half degrees or pi divided by 120. The second is the cosine of pi divided by 240, pi/240. The longest is the cosine of pi divided by 17, pi/17.
The trigonometric circle of values of the functions sine and cosine visually represents the signs of sine and cosine depending on the magnitude of the angle. Especially for blondes, the cosine values are underlined with a green dash to reduce confusion. The conversion of degrees to radians is also very clearly presented when radians are expressed in terms of pi.
This trigonometric table presents the values of sine, cosine, tangent, and cotangent for angles from 0 zero to 90 ninety degrees at one-degree intervals. For the first forty-five degrees, the names of trigonometric functions should be looked at at the top of the table. The first column contains degrees, the values of sines, cosines, tangents and cotangents are written in the next four columns.
For angles from forty-five degrees to ninety degrees, the names of the trigonometric functions are written at the bottom of the table. The last column contains degrees; the values of cosines, sines, cotangents and tangents are written in the previous four columns. You should be careful because the names of the trigonometric functions at the bottom of the trigonometric table are different from the names at the top of the table. Sines and cosines are interchanged, just like tangent and cotangent. This is due to the symmetry of the values of trigonometric functions.
The signs of trigonometric functions are shown in the figure above. Sine has positive values from 0 to 180 degrees, or 0 to pi. Sine has negative values from 180 to 360 degrees or from pi to 2 pi. Cosine values are positive from 0 to 90 and 270 to 360 degrees, or 0 to 1/2 pi and 3/2 to 2 pi. Tangent and cotangent have positive values from 0 to 90 degrees and from 180 to 270 degrees, corresponding to values from 0 to 1/2 pi and pi to 3/2 pi. Negative values of tangent and cotangent are from 90 to 180 degrees and from 270 to 360 degrees, or from 1/2 pi to pi and from 3/2 pi to 2 pi. When determining the signs of trigonometric functions for angles greater than 360 degrees or 2 pi, you should use the periodicity properties of these functions.
The trigonometric functions sine, tangent and cotangent are odd functions. The values of these functions for negative angles will be negative. Cosine is an even trigonometric function - the cosine value for a negative angle will be positive. Sign rules must be followed when multiplying and dividing trigonometric functions.
The table of values for the trigonometric sine function shows the values for the following angles
DocumentThere are reduction formulas on a separate page trigonometricfunctions. IN tablevaluesFortrigonometricfunctionssinusgivenvaluesForthe followingcorners: sin 0, sin 30, sin 45 ...
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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:
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!
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.
Well, the topic is over. If you are reading these lines, it means you are very cool.
Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!
Now the most important thing.
You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.
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For what?
For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.
I won’t convince you of anything, I’ll just say one thing...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?
GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.
You won't be asked for theory during the exam.
You will need solve problems against time.
And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.
It's like in sports - you need to repeat it many times to win for sure.
Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!
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Find problems and solve them!
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 defined.
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
in radians(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
Centered at a point A.
α - angle expressed in radians.Definition
Sine (sin α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the hypotenuse |AC|.Cosine (cos α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the hypotenuse |AC|.
Accepted notations
;
;
.;
;
.Graph of the sine function, y = sin x
Graph of the cosine function, y = cos x
Properties of sine and cosine
Periodicity
Functions y = sin x and y = cos x periodic with period 2π.
Parity
The sine function is odd. The cosine function is even.
Domain of definition and values, extrema, increase, decrease
The sine and cosine functions are continuous in their domain of definition, that is, for all x (see proof of continuity). Their main properties are presented in the table (n - integer).
y = sin x y = cos x Scope and continuity - ∞ < x < + ∞ - ∞ < x < + ∞ Range of values -1 ≤ y ≤ 1 -1 ≤ y ≤ 1 Increasing Descending Maxima, y = 1 Minima, y = - 1 Zeros, y = 0 Intercept points with the ordinate axis, x = 0 y = 0 y = 1 Basic formulas
Sum of squares of sine and cosine
Formulas for sine and cosine from sum and difference
;
;Formulas for the product of sines and cosines
Sum and difference formulas
Expressing sine through cosine
;
;
;
.Expressing cosine through sine
;
;
;
.Expression through tangent
; .
When , we have:
; .At :
; .Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for certain values of the argument.
Expressions through complex variables
;Euler's formula
Expressions through hyperbolic functions
;
;Derivatives
; . Deriving formulas > > >
Derivatives of nth order:
{ -∞ < x < +∞ }Secant, cosecant
Inverse functions
The inverse functions of sine and cosine are arcsine and arccosine, respectively.
Arcsine, arcsin
Arccosine, arccos
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
