Sine, cosine, tangent: what is it? How to find sine, cosine and tangent? Trigonometry Finding an angle by cosine
Examples:
\(\cos(30^°)=\)\(\frac(\sqrt(3))(2)\)
\(\cos\)\(\frac(π)(3)\) \(=\)\(\frac(1)(2)\)
\(\cos2=-0.416…\)
Argument and value
Cosine of an acute angle
Cosine of an acute angle can be determined using a right triangle - it is equal to the ratio of the adjacent leg to the hypotenuse.
Example :
1) Let an angle be given and you need to determine the cosine of this angle.
2) Let's complete any right-angled triangle on this corner.
3) Having measured the necessary sides, we can calculate the cosine.
The cosine of an acute angle is greater than \(0\) and less than \(1\)
If, when solving the problem, the cosine of an acute angle turned out to be greater than 1 or negative, then somewhere in the solution there is an error.
Cosine of a number
The number circle allows you to determine the cosine of any number, but usually find the cosine of numbers somehow related to : \(\frac(π)(2)\) , \(\frac(3π)(4)\) , \(-2π\ ).
For example, for the number \(\frac(π)(6)\) - the cosine will be equal to \(\frac(\sqrt(3))(2)\) . And for the number \(-\)\(\frac(3π)(4)\) it will be equal to \(-\)\(\frac(\sqrt(2))(2)\) (approximately \(-0 ,71\)).
Cosine for other numbers often encountered in practice, see.
The cosine value always lies between \(-1\) and \(1\). In this case, the cosine can be calculated for absolutely any angle and number.
Cosine of any angle
Thanks to the numerical circle, it is possible to determine the cosine of not only an acute angle, but also an obtuse, negative, and even greater than \ (360 ° \) (full turn). How to do it - it's easier to see once than to hear \(100\) times, so look at the picture.
Now an explanation: let it be necessary to determine the cosine of the angle KOA with degree measure in \(150°\). We combine the point ABOUT with the center of the circle, and the side OK- with the \(x\) axis. After that, set aside \ (150 ° \) counterclockwise. Then the ordinate of the point BUT will show us the cosine of this angle.
If we are interested in an angle with a degree measure, for example, in \ (-60 ° \) (angle KOV), we do the same, but \(60°\) set aside clockwise.
And finally, the angle is greater than \(360°\) (the angle KOS) - everything is similar to blunt, only after passing a full turn clockwise, we go to the second round and “get the lack of degrees”. Specifically, in our case, the angle \(405°\) is plotted as \(360° + 45°\).
It is easy to guess that to set aside an angle, for example, in \ (960 ° \), you need to make two turns (\ (360 ° + 360 ° + 240 ° \)), and for an angle in \ (2640 ° \) - whole seven.
It is worth remembering that:
The cosine of a right angle is zero. The cosine of an obtuse angle is negative.
Cosine signs in quarters
Using the cosine axis (that is, the abscissa axis, highlighted in red in the figure), it is easy to determine the signs of the cosines along a numerical (trigonometric) circle:
Where the values on the axis are from \(0\) to \(1\), the cosine will have a plus sign (I and IV quarters are the green area),
- where the values on the axis are from \(0\) to \(-1\), the cosine will have a minus sign (II and III quarters - purple area).
Example.
Define the sign \(\cos 1\).
Solution:
Let's find \(1\) on the trigonometric circle. We will start from the fact that \ (π \u003d 3,14 \). This means that one is approximately three times closer to zero (the "start" point).
If we draw a perpendicular to the cosine axis, it becomes obvious that \(\cos1\) is positive.
Answer:
plus.
Relation to other trigonometric functions:
- the same angle (or number): the basic trigonometric identity \(\sin^2x+\cos^2x=1\)- the same angle (or number): by the formula \(1+tg^2x=\)\(\frac(1)(\cos^2x)\)
- and the sine of the same angle (or number): \(ctgx=\)\(\frac(\cos(x))(\sinx)\)
See other most commonly used formulas.
Function \(y=\cos(x)\)
If we plot the angles in radians along the \(x\) axis, and the cosine values \u200b\u200bcorresponding to these angles along the \(y\) axis, we get the following graph:
This graph is called and has the following properties:
The domain of definition is any value of x: \(D(\cos(x))=R\)
- range of values - from \(-1\) to \(1\) inclusive: \(E(\cos(x))=[-1;1]\)
- even: \(\cos(-x)=\cos(x)\)
- periodic with period \(2π\): \(\cos(x+2π)=\cos(x)\)
- points of intersection with the coordinate axes:
abscissa: \((\)\(\frac(π)(2)\) \(+πn\),\(;0)\), where \(n ϵ Z\)
y-axis: \((0;1)\)
- character intervals:
the function is positive on the intervals: \((-\)\(\frac(π)(2)\) \(+2πn;\) \(\frac(π)(2)\) \(+2πn)\), where \(n ϵ Z\)
the function is negative on the intervals: \((\)\(\frac(π)(2)\) \(+2πn;\)\(\frac(3π)(2)\) \(+2πn)\), where \(n ϵ Z\)
- intervals of increase and decrease:
the function increases on the intervals: \((π+2πn;2π+2πn)\), where \(n ϵ Z\)
the function decreases on the intervals: \((2πn;π+2πn)\), where \(n ϵ Z\)
- maxima and minima of the function:
the function has a maximum value \(y=1\) at the points \(x=2πn\), where \(n ϵ Z\)
the function has a minimum value \(y=-1\) at the points \(x=π+2πn\), where \(n ϵ Z\).
As you can see, this circle is built in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin, 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 of the circle corresponds to two numbers: the coordinate along the axis and the coordinate along the axis. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, remember about the considered right-angled 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 equal to from a triangle? That's right. In addition, we know that is the radius of the unit circle, and therefore, . Substitute this value into our cosine formula. Here's what happens:
And what is equal to from a triangle? Well, of course, ! Substitute the value of the radius into this formula and get:
So, can you tell me what are the coordinates of a point that belongs to the circle? Well, no way? And if you realize that and are just numbers? What coordinate does it correspond to? Well, of course, the coordinate! What coordinate does it correspond to? That's right, coordinate! Thus, the point.
And what then are equal and? That's right, let's use the appropriate definitions of tangent and cotangent and get that, a.
What if the angle is larger? Here, for example, as in this picture:
What has changed in this example? Let's figure it out. To do this, we again turn to a right-angled triangle. Consider a right triangle: an angle (as adjacent to an angle). What is the value of the sine, cosine, tangent and cotangent of 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 are applicable to any rotations 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 size, 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 the circle is or. Is it possible to rotate the radius vector by or by? Well, of course you can! In the first case, therefore, the radius vector will make one complete revolution and stop at position or.
In the second case, that is, the radius vector will make three complete 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, and so on. This list can be continued indefinitely. All these angles can be written with 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 \u200b\u200bare equal to:
Here's a unit circle to help you:
Any difficulties? Then let's figure it out. So we know that:
From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner 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, 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 \u200b\u200bof the trigonometric functions of the angles in and, given in the table below, must be remembered:
Do not be afraid, now we will show one of the examples rather simple memorization of the corresponding values:
To use this method, it is vital to remember the values of the sine for all three measures of the angle (), as well as the value of the tangent of the angle in. Knowing these values, it is quite easy 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 shown in the figure. If you understand this and remember the diagram with arrows, then it will be enough to remember the entire value 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 bring out general formula for finding the coordinates of a point.
Here, for example, we have such a circle:
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 the 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 to. The length of a segment can be expressed using the definition of cosine:
Then we have that for the point the coordinate.
By the same logic, we find the value of the y coordinate for the point. In this way,
So, in general terms, the coordinates of points are determined by the formulas:
Circle center coordinates,
circle radius,
Angle of rotation of the radius vector.
As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are zero, and the radius is equal to one:
Well, let's try these formulas for a taste, practicing finding points on a circle?
1. Find the coordinates of a point on a unit circle obtained by turning a point on.
2. Find the coordinates of a point on a unit circle obtained by rotating a point on.
3. Find the coordinates of a point on a unit circle obtained by turning a point on.
4. Point - 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. Point - 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 understand the solution well) and you will learn how to find them!
1.
It can be seen that. And we know what corresponds to a full turn of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:
2. The circle is unit with a center at a point, which means that we can use simplified formulas:
It can be seen that. We know what corresponds to two complete rotations of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:
Sine and cosine are tabular values. We remember their values and get:
Thus, the desired point has coordinates.
3. The circle is unit with a center at a point, which means that we can use simplified formulas:
It can be seen that. Let's depict the considered example in the figure:
The radius makes angles with the axis equal to and. Knowing that the tabular values of the cosine and sine are equal, and having determined that the cosine here takes a negative value, and the sine is positive, we have:
Similar examples are analyzed 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 vector (by condition)
To determine the corresponding signs of sine and cosine, we construct a unit circle and an 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
The coordinates of the center of the circle (in our example,
Circle radius (by condition)
Angle of rotation of the radius vector (by condition).
Substitute all the values into the formula and get:
and - table values. We remember and substitute them into the formula:
Thus, the desired point has coordinates.
SUMMARY AND BASIC FORMULA
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) leg to the adjacent (close).
The cotangent of an angle is the ratio of the adjacent (close) leg to the opposite (far).
USE for 4? Aren't you bursting with happiness?
The question, as they say, is interesting ... You can, you can pass on 4! And at the same time, do not burst ... The main condition is to practice regularly. Here is the basic preparation for the exam in mathematics. With all the secrets and mysteries of the Unified State Examination, which you will not read about in textbooks... Study this section, solve more tasks from various sources - and everything will work out! It is assumed that the basic section "Enough for you and three!" does not cause any problems for you. But if suddenly ... Follow the links, don't be lazy!
And we will begin with a great and terrible topic.
Trigonometry
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
This topic gives a lot of problems to students. It is considered one of the most severe. What is sine and cosine? What is tangent and cotangent? What is a number circle? It is worth asking these harmless questions, as a person turns pale and tries to divert the conversation to the side ... But in vain. These are simple concepts. And this topic is no more difficult than others. You just need to clearly understand the answers to these very questions from the very beginning. It is very important. If you figured it out, you will like trigonometry. So,
What is sine and cosine? What is tangent and cotangent?
Let's start from ancient times. Don't worry, we will go through all 20 centuries of trigonometry in 15 minutes. And, imperceptibly for ourselves, we will repeat a piece of geometry from grade 8.
Draw a right triangle with sides a, b, c and angle X. Here's one.
Let me remind you that the sides that form a right angle are called legs. a and c- skates. There are two of them. The other side is called the hypotenuse. from- hypotenuse.
Triangle and triangle, think about it! What to do with him? But the ancient people knew what to do! Let's repeat their actions. Let's measure the side in. In the figure, the cells are specially drawn, as it happens in the tasks of the exam. Side in is equal to four cells. OK. Let's measure the side but. Three cells.
Now let's divide the length of the side but per side length in. Or, as they say, let's take the ratio but to in. a/c= 3/4.
Alternatively, you can share in on the but. We get 4/3. Can in divide by from. hypotenuse from do not count by cells, but it is equal to 5. We get a/c= 4/5. In short, you can divide the lengths of the sides by each other and get some numbers.
So what? What is the meaning of this interesting activity? So far none. A stupid job, to be honest.)
And now let's do this. Let's enlarge the triangle. Let's extend the sides to and from, but so that the triangle remains right-angled. Injection X, of course, does not change. To see it, hover your mouse over the picture, or touch it (if you have a tablet). Parties a, b and c turn into m, n, k, and, of course, the lengths of the sides will change.
But their relationship is not!
Attitude a/c It was: a/c= 3/4, became m/n= 6/8 = 3/4. Relationships of other relevant parties also will not change . You can arbitrarily change the lengths of the sides in a right triangle, increase, decrease, without changing the angle x – the relationship of the respective parties will not change . You can check, or you can take the word of ancient people.
Now this is very important! The ratios of the sides in a right triangle do not depend in any way on the lengths of the sides (for the same angle). This is so important that the relations of the parties have earned their special names. Their names, so to speak.) Get acquainted.
What is the sine of angle x ? This is the ratio of the opposite leg to the hypotenuse:
sinx = a/c
What is the cosine of angle x ? This is the ratio of the adjacent leg to the hypotenuse:
fromosx= a/c
What is the tangent of the angle x ? This is the ratio of the opposite leg to the adjacent:
tgx=a/c
What is the cotangent of angle x ? This is the ratio of the adjacent leg to the opposite:
ctgx = in/a
Everything is very simple. Sine, cosine, tangent and cotangent are some numbers. Dimensionless. Just numbers. For each corner - their own.
Why do I repeat myself so boringly? Then what is it need to remember. Ironically remember. Memorization can be made easier. The phrase "Let's start from afar ..." is familiar? So start from afar.
Sinus angle is the ratio distant from the angle of the leg to the hypotenuse. Cosine is the ratio of the nearest to the hypotenuse.
Tangent angle is the ratio distant from the angle of the catheter to the nearest. Cotangent- vice versa.
Already easier, right?
Well, if you remember that only the legs sit in the tangent and cotangent, and the hypotenuse appears in the sine and cosine, then everything will become quite simple.
This whole glorious family - sine, cosine, tangent and cotangent is also called trigonometric functions.
And now a question for consideration.
Why do we say sine, cosine, tangent and cotangent corner? We are talking about the relationship of the parties, like ... What does it have to do with injection?
Let's look at the second picture. Exactly the same as the first one.
Hover your mouse over the picture. I changed the angle X. enlarged it from x to x. All relationships have changed! Attitude a/c was 3/4, and the corresponding ratio t/in became 6/4.
And all other relationships have become different!
Therefore, the ratios of the sides do not depend in any way on their lengths (at one angle x), but sharply depend on this very angle! And only from him. Therefore, the terms sine, cosine, tangent and cotangent refer to corner. The corner here is the main one.
It must be ironically understood that the angle is inextricably linked with its trigonometric functions. Each angle has its own sine and cosine. And almost everyone has their own tangent and cotangent. It is important. It is believed that if we are given an angle, then its sine, cosine, tangent and cotangent we know ! And vice versa. Given a sine, or any other trigonometric function, then we know the angle.
There are special tables where for each angle its trigonometric functions are written. The Bradys tables are called. They have been made for a very long time. Back when there were no calculators or computers...
Of course, the trigonometric functions of all angles cannot be memorized. You only need to know them for a few angles, more on that later. But the spell I know an angle, so I know its trigonometric functions" - always works!
So we repeated a piece of geometry from the 8th grade. Do we need it for the exam? Necessary. Here is a typical problem from the exam. For the solution of which the 8th grade is enough. Picture given:
Everything. There is no more data. We need to find the length of leg BC.
The cells help little, the triangle is somehow incorrectly located .... On purpose, I guess ... From the information there is the length of the hypotenuse. 8 cells. For some reason, an angle is given.
Here we must immediately remember about trigonometry. There is an angle, so we know all its trigonometric functions. Which function out of the four should be put into action? Let's see what we know, shall we? We know the hypotenuse, the angle, but we need to find adjacent to this corner catet! Clearly, the cosine needs to be put into action! Here we are launching. We just write, by definition of cosine (ratio adjacent leg to hypotenuse):
cosC = BC/8
Angle C is 60 degrees and its cosine is 1/2. You need to know this, without any tables! That is:
1/2 = sun/8
Elementary linear equation. Unknown - sun. Who forgot how to solve equations, take a walk on the link, the rest solve:
sun = 4
When the ancient people realized that each angle has its own set of trigonometric functions, they had a reasonable question. Isn't the sine, cosine, tangent and cotangent somehow related to each other? So that knowing one function of the angle, you can find the rest? Without calculating the angle itself?
That's how they were restless ...)
Connection between trigonometric functions of one angle.
Of course, the sine, cosine, tangent and cotangent of the same angle are related. Any connection between expressions is given in mathematics by formulas. In trigonometry, there are a huge number of formulas. But here we will look at the most basic ones. These formulas are called: basic trigonometric identities. Here they are:
These formulas need to know iron. Without them, there is nothing to do in trigonometry at all. Three more auxiliary identities follow from these basic identities:
I immediately warn you that the last three formulas quickly fall out of memory. For some reason.) You can, of course, derive these formulas from the first three. But, in a difficult moment ... You understand.)
In standard tasks such as the ones below, there is a way to get around these forgettable formulas. AND drastically reduce errors out of forgetfulness, and in calculations too. This practice is in Section 555, lesson "Relationship between trigonometric functions of one angle."
In what tasks and how are the basic trigonometric identities used? The most popular task is to find some function of the angle, if another is given. In the exam, such a task is present from year to year.) For example:
Find the value of sinx if x is an acute angle and cosx=0.8.
The task is almost elementary. We are looking for a formula where there are sine and cosine. Here is that formula:
sin 2 x + cos 2 x = 1
We substitute here a known value, namely, 0.8 instead of the cosine:
sin 2 x + 0.8 2 = 1
Well, we consider, as usual:
sin 2 x + 0.64 = 1
sin 2 x \u003d 1 - 0.64
Here, almost everything. We have calculated the square of the sine, it remains to extract the square root and the answer is ready! The root of 0.36 is 0.6.
The task is almost elementary. But the word "almost" is not in vain here ... The fact is that the answer sinx = - 0.6 is also suitable ... (-0.6) 2 will also be 0.36.
Two different answers are obtained. And you need one. The second one is wrong. How to be!? Yes, as usual.) Read the assignment carefully. For some reason it says... if x is an acute angle... And in tasks, every word has a meaning, yes ... This phrase is additional information for the solution.
An acute angle is an angle less than 90°. And at such angles all trigonometric functions - both sine and cosine, and tangent with cotangent - positive. Those. we simply discard the negative answer here. We have the right.
Actually, eighth graders do not need such subtleties. They only work with right triangles, where the corners can only be acute. And they don’t know, happy ones, that there are negative angles, and angles of 1000 ° ... And all these nightmarish angles have their own trigonometric functions with both plus and minus ...
But for high school students without taking into account the sign - no way. A lot of knowledge multiplies sorrows, yes...) And for the correct solution, the task must contain additional information (if necessary). For example, it could be given as:
Or some other way. You will see in the examples below.) To solve such examples, you need to know in which quarter the given angle x falls and what sign the desired trigonometric function has in this quarter.
These basics of trigonometry are discussed in the lessons what is a trigonometric circle, the counting of angles on this circle, the radian measure of an angle. Sometimes you also need to know the table of sines of cosines of tangents and cotangents.
So, let's note the most important:
Practical Tips:
1. Remember the definitions of sine, cosine, tangent and cotangent. Very useful.
2. We clearly assimilate: sine, cosine, tangent and cotangent are firmly connected with angles. We know one thing, so we know something else.
3. We clearly assimilate: the sine, cosine, tangent and cotangent of one angle are interconnected by basic trigonometric identities. We know one function, which means that we can (if we have the necessary additional information) calculate all the others.
And now let's decide, as usual. First, tasks in the volume of the 8th grade. But high school students can also ...)
1. Calculate the value of tgA if ctgA = 0.4.
2. β - angle in a right triangle. Find the value of tgβ if sinβ = 12/13.
3. Determine the sine of an acute angle x if tgx \u003d 4/3.
4. Find the value of an expression:
6sin 2 5° - 3 + 6cos 2 5°
5. Find the value of an expression:
(1-cosx)(1+cosx), if sinx = 0.3
Answers (separated by semicolons, in disarray):
0,09; 3; 0,8; 2,4; 2,5
Happened? Fine! Eighth graders can already follow their A's.)
Didn't everything work out? Tasks 2 and 3 are somehow not very ...? No problem! There is one beautiful technique for such tasks. Everything is decided, practically, without formulas at all! And, therefore, without errors. This technique is described in the lesson: "Relationship between trigonometric functions of one angle" in Section 555. All other tasks are also disassembled there.
These were problems like the Unified State Examination, but in a stripped-down version. USE - light). And now almost the same tasks, but in a full-fledged form. For knowledge-burdened high school students.)
6. Find the value of tgβ if sinβ = 12/13 and
7. Determine sinx if tgx = 4/3, and x belongs to the interval (- 540°; - 450°).
8. Find the value of the expression sinβ cosβ if ctgβ = 1.
Answers (in disarray):
0,8; 0,5; -2,4.
Here, in problem 6, the angle is given somehow not very unambiguously... But in problem 8, it is not set at all! It's on purpose). Additional information is taken not only from the task, but also from the head.) But if you decide - one correct task is guaranteed!
What if you haven't decided? Um... Well, Section 555 will help here. There, the solutions to all these tasks are described in detail, it is difficult not to understand.
In this lesson, a very limited concept of trigonometric functions is given. Within 8th grade. Seniors have questions...
For example, if the angle X(see the second picture on this page) - make it dumb!? The triangle will fall apart! And how to be? There will be no leg, no hypotenuse ... The sine is gone ...
If the ancient people had not found a way out of this situation, we would not have mobile phones, TV, or electricity now. Yes Yes! The theoretical basis of all these things without trigonometric functions is zero without a wand. But the ancient people did not disappoint. How they got out - in the next lesson.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
|BD| - arc length centered on a point A.
α
is an angle expressed in radians.
sine ( sinα) 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 ( cosα) 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 designations
;
;
.
;
;
.
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 a period 2 π.
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 on their domain of definition, that is, for all x (see Fig. continuity proof). 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 |
| Ascending | ||
| Descending | ||
| Maximums, y= 1 | ||
| Minima, y = - 1 | ||
| Zeros, y= 0 | ||
| Points of intersection with the y-axis, x = 0 | y= 0 | y= 1 |
Basic Formulas
Sum of squared sine and cosine
Sine and cosine formulas for sum and difference
;
;
Formulas for the product of sines and cosines
Sum and difference formulas
Expression of sine through cosine
;
;
;
.
Expression of cosine through sine
;
;
;
.
Expression in terms of tangent
; .
For , we have:
;
.
At :
;
.
Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for some values of the argument. 
Expressions through complex variables
;
Euler formula
Expressions in terms of hyperbolic functions
;
;
Derivatives
; . Derivation of formulas > > >
Derivatives of the nth order:
{ -∞ <
x < +∞ }
Secant, cosecant
Inverse functions
The inverse functions to 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 Students of Higher Educational Institutions, Lan, 2009.