Finding cos. Sine (sin x) and cosine (cos x) – properties, graphs, formulas
Unified State Exam for 4? Won't you burst with happiness?
The question, as they say, is interesting... It is possible, it is possible to pass with a 4! And at the same time not to burst... The main condition is to exercise regularly. Here is the basic preparation for the Unified State Exam in mathematics. With all the secrets and mysteries of the Unified State Exam, 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 "A C is enough for you!" it doesn't cause you any problems. But if suddenly... Follow the links, don’t be lazy!
And we will start with a great and terrible topic.
Trigonometry
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
This topic causes a lot of problems for students. It is considered one of the most severe. What are sine and cosine? What are tangent and cotangent? What is a number circle? As soon as you ask these harmless questions, the person turns pale and tries to divert the conversation... 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. This is very important. If you understand, you will like trigonometry. So,
What are sine and cosine? What are tangent and cotangent?
Let's start with ancient times. Don’t worry, we’ll go through all 20 centuries of trigonometry in about 15 minutes. And, without noticing it, we’ll repeat a piece of geometry from 8th grade.
Let's draw a right triangle with sides a, b, c and angle X. Here it is.
Let me remind you that the sides that form a right angle are called legs. a and c– legs. There are two of them. The remaining side is called the hypotenuse. With– hypotenuse.
Triangle and triangle, just think! What to do with it? But the ancient people knew what to do! Let's repeat their actions. Let's measure the side V. In the figure, the cells are specially drawn, as in Unified State Exam assignments It happens. Side V equal to four cells. OK. Let's measure the side A. Three cells.
Now let's divide the length of the side A per side length V. Or, as they also say, let’s take the attitude A To V. a/v= 3/4.
On the contrary, you can divide V on A. We get 4/3. Can V divide by With. Hypotenuse With It’s impossible to count by cells, but it is equal to 5. We get high quality= 4/5. In short, you can divide the lengths of the sides by each other and get some numbers.
So what? What is the point of this interesting activity? None yet. A pointless exercise, to put it bluntly.)
Now let's do this. Let's enlarge the triangle. Let's extend the sides in and with, but so that the triangle remains rectangular. Corner X, of course, does not change. To see this, hover your mouse over the picture, or touch it (if you have a tablet). Parties a, b and c will turn into m, n, k, and, of course, the lengths of the sides will change.
But their relationship is not!
Attitude a/v was: a/v= 3/4, became m/n= 6/8 = 3/4. The relationships of other relevant parties are also won't change . You can change the lengths of the sides in a right triangle as you like, increase, decrease, without changing the angle x – the relationship between the relevant parties will not change . You can check it, or you can take the ancient people’s word for it.
But this is already very important! The ratios of the sides in a right triangle do not depend in any way on the lengths of the sides (at the same angle). This is so important that the relationship between the parties has earned its own special name. Your names, so to speak.) Meet.
What is the sine of angle x ? This is the ratio of the opposite side to the hypotenuse:
sinx = a/c
What is the cosine of the angle x ? This is the ratio of the adjacent leg to the hypotenuse:
Withosx= high quality
What is tangent x ? This is the ratio of the opposite side to the adjacent:
tgx =a/v
What is the cotangent of angle x ? This is the ratio of the adjacent side to the opposite:
ctgx = v/a
It's very simple. Sine, cosine, tangent and cotangent are some numbers. Dimensionless. Just numbers. Each angle has its own.
Why am I repeating everything so boringly? Then what is this need to remember. It's important to remember. Memorization can be made easier. Is the phrase “Let’s start from afar…” familiar? So start from afar.
Sinus angle is a ratio distant from the leg angle to the hypotenuse. Cosine– the ratio of the neighbor to the hypotenuse.
Tangent angle is a ratio distant from the leg angle to the near one. Cotangent- vice versa.
It's easier, right?
Well, if you remember that in tangent and cotangent there are only legs, and in sine and cosine the hypotenuse appears, then everything will become quite simple.
This whole glorious family - sine, cosine, tangent and cotangent is also called trigonometric functions.
Now a question for consideration.
Why do we say sine, cosine, tangent and cotangent corner? We are talking about the relationship between the parties, like... What does it have to do with it? corner?
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. Increased it from x to x. All relationships have changed! Attitude a/v was 3/4, and the corresponding ratio t/v became 6/4.
And all other relationships became different!
Therefore, the ratios of the sides do not depend in any way on their lengths (at one angle x), but depend sharply on this very angle! And only from him. Therefore, the terms sine, cosine, tangent and cotangent refer to corner. The angle here is the main one.
It must be clearly 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. This 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 sine, or any other trigonometric function- that means we know the angle.
There are special tables where for each angle its trigonometric functions are described. They are called Bradis tables. They were compiled a very long time ago. When there were no calculators or computers yet...
Of course, it is impossible to remember the trigonometric functions of all angles. You are required to know them only for a few angles, more on this later. But the spell I know an angle, which means I know its trigonometric functions” - always works!
So we repeated a piece of geometry from 8th grade. Do we need it for the Unified State Exam? Necessary. Here is a typical problem from the Unified State Exam. To solve this problem, 8th grade is enough. Given picture:
All. There is no more data. We need to find the length of the side of the aircraft.
The cells do not help much, the triangle is somehow incorrectly positioned.... On purpose, I guess... From the information there is the length of the hypotenuse. 8 cells. For some reason, the angle was given.
This is where you need to immediately remember about trigonometry. There is an angle, which means we know all its trigonometric functions. Which of the four functions should we use? Let's see, what do we know? We know the hypotenuse and the angle, but we need to find adjacent catheter to this corner! It’s clear, the cosine needs to be put into action! Here we go. We simply write, by the definition of cosine (the ratio adjacent leg to hypotenuse):
cosC = BC/8
Angle C is 60 degrees, its cosine is 1/2. You need to know this, without any tables! So:
1/2 = BC/8
Elementary linear equation. Unknown – Sun. For those who have forgotten how to solve equations, follow the link, the rest solve:
BC = 4
When ancient people realized that each angle has its own set of trigonometric functions, they had a reasonable question. Are sine, cosine, tangent and cotangent somehow related to each other? So that knowing one angle function, you can find the others? Without calculating the angle itself?
They were so restless...)
Relationship between trigonometric functions of one angle.
Of course, the sine, cosine, tangent and cotangent of the same angle are related to each other. Any connection between expressions is given in mathematics by formulas. In trigonometry there are a colossal number of formulas. But here we will look at the most basic ones. These formulas are called: basic trigonometric identities. Here they are:
You need to know these formulas thoroughly. Without them, there is generally nothing to do in trigonometry. Three more auxiliary identities follow from these basic identities:
I warn you right away that the last three formulas quickly fall out of your memory. For some reason.) You can, of course, derive these formulas from the first three. But, in difficult moment... You understand.)
In standard problems, like the ones below, there is a way to avoid these forgettable formulas. AND dramatically reduce errors due to forgetfulness, and in calculations too. This practical technique- in Section 555, lesson "Relationships 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 angle function if another is given. In the Unified State Examination 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 that contains sine and cosine. Here is the formula:
sin 2 x + cos 2 x = 1
We substitute here a known value, namely 0.8 instead of cosine:
sin 2 x + 0.8 2 = 1
Well, we count as usual:
sin 2 x + 0.64 = 1
sin 2 x = 1 - 0.64
That's practically all. We have calculated the square of the sine, all that remains is 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 there for a reason... The fact is that the answer sinx= - 0.6 is also suitable... (-0.6) 2 will also be 0.36.
There are two different answers. 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 corners All trigonometric functions - sine, cosine, and tangent with cotangent - positive. Those. We simply discard the negative answer here. We have the right.
Actually, eighth graders don’t 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 both negative angles and angles of 1000°... And all these terrible angles have their own trigonometric functions, both plus and minus...
But for high school students, without taking into account the sign - no way. Much knowledge multiplies sorrows, yes...) And for the correct solution, additional information is necessarily present in the task (if it is necessary). For example, it can be given by the following entry:
Or some other way. You will see in the examples below.) To solve such examples you need to know Which quarter does the given angle x fall into and what sign does the desired trigonometric function have in this quarter?
These basics of trigonometry are discussed in the lessons on what a trigonometric circle is, the measurement of angles on this circle, the radian measure of an angle. Sometimes you need to know the table of sines, cosines of tangents and cotangents.
So, let's note the most important thing:
Practical tips:
1. Remember the definitions of sine, cosine, tangent and cotangent. It will be very useful.
2. We clearly understand: sine, cosine, tangent and cotangent are tightly connected with angles. We know one thing, which means we know another.
3. We clearly understand: sine, cosine, tangent and cotangent of one angle are related to each other by basic trigonometric identities. We know one function, which means we can (if we have the necessary additional information) calculate all the others.
Now let’s decide, as usual. First, tasks in the scope of 8th grade. But high school students can do it too...)
1. Calculate the value of tgA if ctgA = 0.4.
2. β is an angle in a right triangle. Find the value of tanβ if sinβ = 12/13.
3. Define sine acute angle x if tgх = 4/3.
4. Find the meaning of the expression:
6sin 2 5° - 3 + 6cos 2 5°
5. Find the meaning of the expression:
(1-cosx)(1+cosx), if sinx = 0.3
Answers (separated by semicolons, in disarray):
0,09; 3; 0,8; 2,4; 2,5
Did it work? Great! Eighth graders can already go get their A's.)
Didn't everything work out? Tasks 2 and 3 are somehow not very good...? No problem! There is one beautiful reception for similar tasks. Everything can be solved practically without formulas at all! And, therefore, without errors. This technique is described in the lesson: “Relationships between trigonometric functions of one angle” in Section 555. All other tasks are also dealt with there.
These were problems like the Unified State Exam, but in a stripped-down version. Unified State Exam - light). And now almost the same tasks, but in a full-fledged format. For knowledge-burdened high school students.)
6. Find the value of tanβ if sinβ = 12/13, and
7. Determine sinх if tgх = 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 not specified very clearly... But in problem 8 it is not specified at all! This is 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? Hmm... Well, Section 555 will help here. There the solutions to all these tasks are described in detail, it is difficult not to understand.
This lesson provides a very limited understanding of trigonometric functions. Within 8th grade. And the elders still have questions...
For example, if the angle X(look at the second picture on this page) - make it stupid!? The triangle will completely fall apart! So what should we do? There will be no leg, no hypotenuse... The sine has disappeared...
If ancient people had not found a way out of this situation, we would not have cell phones, TV, or electricity now. Yes, yes! Theoretical basis all these things without trigonometric functions are zero without a stick. But the ancient people did not disappoint. How they got out is 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. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Cosine– one of the basic trigonometric functions. Cosine ohm spicy angle in a right triangle the ratio of the adjacent side to the hypotenuse is called. The definition of cosine is tied to right triangle, but often the angle whose cosine needs to be determined is not placed in a right triangle. How to find out the value of the cosine of any angle ?
Instructions
1. angle in a right triangle, you need to use the definition of cosine and find the ratio of the adjacent leg to the hypotenuse: cos? = a/c, where a is the length of the leg, c is the length of the hypotenuse.
2. If you need to detect cosine angle in an arbitrary triangle, you need to use the cosine theorem: if the angle is acute: cos? = (a2 + b2 – c2)/(2ab); if the angle is obtuse: cos? = (c2 – a2 – b2)/(2ab), where a, b are the lengths of the sides adjacent to the corner, c is the length of the side opposite the corner.
3. If you need to detect cosine angle in free geometric figure, you need to determine the value angle in degrees or radians, and cosine angle detect by its value with the support of an engineering calculator, Bradis tables or any other mathematical application.
Cosine is a basic trigonometric function of angle. Knowing how to determine cosine will come in handy in vector algebra when determining the projections of vectors onto different axes.
Instructions
1. Cosine The ohm of an angle is the ratio of the leg adjacent to the angle to the hypotenuse. This means that in a right triangle ABC (ABC is a right angle), the cosine of angle BAC is equal to the ratio of AB to AC. For angle ACB: cos ACB = BC/AC.
2. But an angle does not always belong to a triangle; in addition, there are obtuse angles that obviously cannot be part of a right triangle. Let's consider the case when the angle is specified by rays. In order to calculate the cosine of the angle in this case, proceed as follows. A coordinate system is attached to the corner, the coordinates are calculated from the vertex of the corner, the X axis goes along one side of the corner, the Y axis is built perpendicular to the X axis. After this, a circle of unit radius is built with the center at the vertex of the corner. The second side of the angle intersects the circle at point A. Drop a perpendicular from point A to the X axis, mark the point of intersection of the perpendicular with the Ax axis. Then you get a right triangle AAxO, and the cosine of the angle is AAx/AO. Since the circle is of unit radius, then AO = 1 and the cosine of the angle is primitively equal to AAx.
3. In the case of an obtuse angle, the same constructions are carried out. Cosine The obtuse angle is negative, but it is also equal to Ax.
Video on the topic
Pay attention!
The cosines of some angles are presented in the Bradis tables.
Concepts such as sine, cosine, tangent are unlikely to be encountered often in everyday life. However, if you sat down to solve mathematical problems with your high school son, it would be good to remember what these representations are and how to detect, say, a cosine.
Instructions
Video on the topic
Often in geometric (trigonometric) problems it is required to find cosine angle in triangle, because cosine angle allows you to unambiguously determine the size of the angle itself.
Instructions
1. In order to discover cosine angle in triangle, the lengths of the sides are known, we can use the theorem cosine ov. According to this theorem, the square of the length of a side of an arbitrary triangle is equal to the sum of the squares of its 2 other sides without twice the product of the lengths of these sides by cosine angle between them: a?=b?+c?-2*b*c*cos?, where: a, b, c are the sides of the triangle (or rather their lengths),? - corner, opposite side a (its value). From the above equality it is easy to find сos?:сos?=(b?+c?-а?)/(2*b*c) Example 1. There is a triangle with sides a, b, c equal to 3 , 4, 5 mm, respectively. Detect cosine the angle enclosed between the large sides. Solution: According to the conditions of the problem, we have: a = 3, b = 4, c = 5. Let us denote the angle opposite to side a by ?, then, according to the formula derived above, we have: cos? = (b? + c?-a?)/(2*b*c)=(4?+5?-3?)/(2*4*5)=(16+25-9)/40=32/40=0, 8Answer: 0.8.
2. If the triangle is right angled, then to find cosine and it is enough for an angle to know the lengths of each 2 sides ( cosine right angle is 0). Let there be a right triangle with sides a, b, c, where c is the hypotenuse. Let's look at all the options: Example 2. Find cos? if the lengths of sides a and b (the legs of the triangle) are known. Let's additionally use the Pythagorean theorem: c?=b?+a?,c=v(b?+a?)cos?=(b?+c?-a?)/(2*b*c)=(b?+b?+a? -a?)/(2*b*v(b?+a?))=(2*b?)/(2*b*v(b?+a?))=b/v(b?+a ?)In order to check the correctness of the resulting formula, we substitute into it the values from example 1, i.e. a = 3, b = 4. Having done basic calculations, we get: cos? = 0.8.
3. Similar is located cosine in a rectangular triangle in other cases: Example 3. Famous a and c (hypotenuse and opposite leg), find сos?b?=с?-а?,b=v(c?-а?)сos?=(b?+c?- a?)/(2*b*c)=(с?-а?+с?-а?)/(2*с*v(с?-а?))=(2*с?-2*а ?)/(2*c*v(c?-a?))=v(c?-a?)/c. Substituting the values a=3 and c=5 from the first example, we get: cos?=0.8 .
4. Example 4. Vestims b and c (hypotenuse and adjacent leg). Detect cos? Having carried out similar reformations (shown in examples 2 and 3), we find that in this case cosine V triangle is calculated using a very easy formula: cos? = b/c. The simplicity of the derived formula is explained simply: really, adjacent to the corner? the leg is a projection of the hypotenuse, therefore its length is equal to the length of the hypotenuse multiplied by cos?. Substituting the values b = 4 and c = 5 from the first example, we get: cos? = 0.8 This means that all our formulas are correct.
Tip 5: How to detect an acute angle in a right triangle
Directly carbonic the triangle, apparently, is one of the most famous, from a historical point of view, geometric figures. Pythagorean “pants” can only compete with “Eureka!” Archimedes.
You will need
- – drawing of a triangle;
- - ruler;
- – protractor
Instructions
1. As usual, the vertices of the corners of a triangle are designated by capital Latin letters (A, B, C), and the opposite sides by small Latin letters (a, b, c) or by the names of the vertices of the triangle forming this side (AC, BC, AB).
2. The sum of the angles of a triangle is 180 degrees. In a rectangular triangle one angle (straight) will invariably be 90 degrees, and the rest acute, i.e. less than 90 degrees all the way. In order to determine what angle in a rectangular triangle is straight, measure the sides of the triangle with the support of a ruler and determine the largest one. It is called the hypotenuse (AB) and is located opposite the right angle (C). The remaining two sides form a right angle and are called legs (AC, BC).
3. Once you have determined which angle is acute, you can either measure the angle using a protractor or calculate it using mathematical formulas.
4. In order to determine the size of the angle with the help of a protractor, align its vertex (denoted by the letter A) with a special mark on the ruler in the center of the protractor; the leg AC should coincide with its upper edge. Mark on the semicircular part of the protractor the point through which the hypotenuse AB passes. The value at this point corresponds to the angle in degrees. If there are 2 values indicated on the protractor, then for an acute angle you need to choose the smaller one, for an obtuse angle - the larger one.
6. Find the resulting value in the Bradis reference tables and determine which angle the resulting numerical value corresponds to. Our grandmothers used this method.
7. Nowadays, it is enough to take a calculator with a function for calculating trigonometric formulas. Let's say the built-in Windows calculator. Launch the “Calculator” application, in the “View” menu item, select the “Engineering” item. Calculate the sine of the desired angle, say sin(A) = BC/AB = 2/4 = 0.5
8. Switch the calculator to inverse functions, by clicking on the INV button on the calculator display, then click on the button for calculating the arcsine function (indicated on the display as sin to the minus first power). A further inscription will appear in the calculation window: asind (0.5) = 30. I.e. the desired angle is 30 degrees.
The cosine theorem in mathematics is most often used in the case when you need to detect a third side from an angle and two sides. However, sometimes the condition of the problem is set opposite: it is required to detect an angle with given 3 sides.
Instructions
1. Imagine that you are given a triangle whose lengths of 2 sides and the value of one angle are known. All the angles of this triangle are not equal to each other, and its sides are also different in size. Corner? lies opposite the side of the triangle, designated AB, which is the base of this figure. Through this angle, as well as through the remaining sides AC and BC, it is possible to detect that side of the triangle that is unknown, using the cosine theorem, deriving on its basis the formula presented below: a^2=b^2+c^2-2bc*cos?, where a=BC, b=AB, c=ACThe cosine theorem, on the contrary, is called the generalized Pythagorean theorem.
2. Now imagine that all three sides of the figure are given, but at the same time its angle? unknown Knowing that the formula has the form a^2=b^2+c^2-2bc*cos?, transform this expression so that the desired value becomes the angle?: b^2+c^2=2bc*cos?+a ^2.After this, bring the equation shown above to a slightly different form: b^2+c^2-a^2=2bc*cos?.After this, this expression should be converted to the one below: cos?=?b^2+c ^2-a^2/2bc. All that remains is to substitute the numbers into the formula and carry out the calculations.
3. In order to find the cosine of the angle of a triangle, denoted as ?, it must be expressed through the inverse trigonometric function called arc cosine. The arc cosine of the number m is the value of the angle? for which the cosine of the angle? equals m. The function y=arccos m is decreasing. Imagine, say, what is the cosine of the angle? equal to one 2nd. Then the angle? can be defined through arc cosine as follows:? = arccos, m = arccos 1/2 = 60°, where m = 1/2. In a similar way, it is possible to detect the remaining angles of a triangle with 2 other unknown sides.
4. If angles are presented in radians, convert them to degrees using the following ratio:? radian = 180 degrees. Remember that the vast majority engineering calculators equipped with the probability of switching units of measurement of angles.
Sine and cosine are two trigonometric functions that are called “direct”. It is they who have to be calculated more often than others, and to solve this problem today each of us has large selection options. Below are a few particularly primitive methods.
Instructions
1. Use a protractor, a pencil, and a piece of paper if no other means of calculation is available. One of the definitions of cosine is given in terms of acute angles in a right triangle - its value is equal to the ratio between the length of the leg opposite this angle and the length of the hypotenuse. Draw a triangle in which one of the angles is right (90°), and the other is equal to the angle whose cosine you want to calculate. The length of the sides does not matter - draw them the way you feel most comfortable measuring. Measure the length of the required leg and hypotenuse and divide the first by the second using any convenient method.
2. Take advantage of the ability to determine the values of trigonometric functions with the support of the built-in calculator search engine Nigma, if you have internet access. Let's say, if you need to calculate the cosine of an angle of 20°, then by loading the main service page http://nigma.ru, type “cosine 20 degrees” into the search query field and click the “Detect!” button. You can omit the word “degrees” and replace the word “cosine” with cos - in any case, the search engine will show the result accurate to 15 decimal places (0.939692620785908).
3. Open the standard calculator program installed with the Windows operating system if you do not have access to the Internet. This can be done, say, by pressing the win and r keys at the same time, then entering the calc command and clicking on the OK button. To calculate trigonometric functions, there is a pre-designed interface called “engineer” or “scientist” (depending on the OS version) - select the required item in the “View” section of the calculator menu. Later, enter the angle value in degrees and click on the cos button in the program interface.
Video on the topic
Tip 8: How to Determine Angles in a Right Triangle
A right triangle is characterized by certain relationships between the angles and sides. Knowing the values of some of them, it is possible to calculate others. For this purpose, formulas are used, based, in turn, on the axioms and theorems of geometry.
Instructions
1. From the very name of a right triangle it is clear that one of its angles is right. Regardless of whether a right triangle is isosceles or not, it invariably has one angle equal to 90 degrees. If you are given a right triangle that is simultaneously and isosceles, then, based on the fact that there is a right angle in the figure, find two angles at its base. These angles are equal to each other, therefore each of them has a value equal to:? = 180° - 90°/2 = 45°
2. In addition to the one discussed above, we also allow another case when the triangle is right-angled, but not isosceles. In many problems, the angle of a triangle is 30°, and in others it is 60°, so the sum of all angles in a triangle must be equal to 180°. If the hypotenuse of a right triangle and its leg are given, then the angle can be found from the correspondence of these 2 sides: sin ?=a/c, where a is the leg opposite to the hypotenuse of the triangle, c is the hypotenuse of the triangle. Accordingly, ?=arcsin(a/c )Angle can also be determined using the formula for finding the cosine: cos ?=b/c, where b is the adjacent leg to the hypotenuse of the triangle
3. If only two legs are known, then the angle? can be found using the tangent formula. The tangent of this angle is equal to the ratio of the opposite side to the adjacent one: tg ? = a/b From this it follows that? = arctg (a/b) When a right angle and one of the angles found by the above method are given, the 2nd is found as follows:? = 180°-(90°+?)
The word “cosine” refers to one of the trigonometric functions, which when written is denoted as cos. It is especially common to deal with it when solving problems of finding the parameters of correct figures in geometry. In such problems, the values of the angles at the vertices of polygons are, as usual, denoted in capital letters Greek alphabet. If we are talking about a right triangle, then from this one letter you can sometimes find out which of the corners is meant.
Instructions
1. If the value of the angle, indicated by the letter ?, is known from the conditions of the problem, then to find the value corresponding to the cosine alpha, you can use a standard Windows OS calculator. It is launched through the main menu of the operating system - press the Win button, expand the “All programs” section in the menu, go to the “Typical” subsection, and then to the “Utility” section. There you will find the line “Calculator” - click on it to launch the application.
2. Press the key combination Alt + 2 to switch the application interface to the “engineering” (in other versions of the OS – “scientist”) option. After that, enter the angle value? and click the button marked with the letters cos with the mouse pointer - the calculator will calculate the function and display the result.
3. If you calculate the cosine of an angle? necessary in a right triangle, then it is probably one of the 2 acute angles. If the sides of such a triangle are correctly designated, the hypotenuse (the longest side) is denoted by the letter c, and the right angle lying opposite it is denoted by the Greek letter ?. The other two sides (legs) are designated by the letters a and b, and the acute angles lying opposite them are ? And?. For the values of the acute angles of a right triangle, there are relationships that will allow you to calculate the cosine, even without knowing the value of the angle itself.
4. If in a right triangle the lengths of the sides b (the leg adjacent to the angle?) and c (the hypotenuse) are known, then to calculate the cosine? divide the length of this leg by the length of the hypotenuse: cos(?)=b/c.
5. In an arbitrary triangle, what is the cosine value of the angle? An unknown quantity can be calculated if the lengths of all sides are given in the conditions. To do this, first square the lengths of all sides, then the resulting values for 2 sides adjacent to the corner? add, and subtract the resulting value for the opposite side from the total. After this, divide the resulting value by twice the product of the lengths adjacent to the corner? sides - this will be the desired cosine of the angle?: cos(?)=(b?+c?-a?)/(2*b*c). This solution follows from the cosine theorem.
Useful advice
The mathematical notation for cosine is cos. The cosine value cannot be greater than 1 and less than -1.
As you can see, this circle is constructed in the Cartesian coordinate system. The radius of the circle is equal to one, and the center of the circle lies at the origin, starting position 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 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:
Doesn't 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:
Doesn't exist
Doesn't exist
Doesn't exist
Doesn't 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 view 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 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.
Cosine is a well-known trigonometric function, which is also one of the main functions of trigonometry. The cosine of an angle in a right-angled triangle is the ratio of the adjacent side of the triangle to the hypotenuse of the triangle. Most often, the definition of cosine is associated with a triangle of the rectangular type. But it also happens that the angle for which it is necessary to calculate the cosine in a rectangular triangle is not located in this very rectangular triangle. What to do then? How to find the cosine of an angle of a triangle?
If you need to calculate the cosine of an angle in a right-angled triangle, then everything is very simple. You just need to remember the definition of cosine, which contains the solution to this problem. You just need to find the same relationship between the adjacent side, as well as the hypotenuse of the triangle. Indeed, it is not difficult to express the cosine of the angle here. The formula is as follows: - cosα = a/c, here “a” is the length of the leg, and side “c”, respectively, is the length of the hypotenuse. For example, the cosine of an acute angle of a right triangle can be found using this formula.
If you are interested in what the cosine of an angle in an arbitrary triangle is equal to, then the cosine theorem comes to the rescue, which should be used in such cases. The cosine theorem states that the square of a side of a triangle is a priori equal to the sum of the squares of the remaining sides of the same triangle, but without doubling the product of these sides by the cosine of the angle located between them.
- If you need to find the cosine of an acute angle in a triangle, then you need to use the following formula: cosα = (a 2 + b 2 – c 2)/(2ab).
- If you need to find the cosine of an obtuse angle in a triangle, then you need to use the following formula: cosα = (c 2 – a 2 – b 2)/(2ab). The designations in the formula - a and b - are the lengths of the sides that are adjacent to the desired angle, c - is the length of the side that is opposite to the desired angle.
The cosine of an angle can also be calculated using the sine theorem. It states that all sides of a triangle are proportional to the sines of the angles that are opposite. Using the theorem of sines, you can calculate the remaining elements of a triangle, having information only about two sides and an angle that is opposite to one side, or from two angles and one side. Consider this with an example. Problem conditions: a=1; b=2; c=3. The angle that is opposite to side “A” is denoted by α, then, according to the formulas, we have: cosα=(b²+c²-a²)/(2*b*c)=(2²+3²-1²)/(2*2 *3)=(4+9-1)/12=12/12=1. Answer: 1.
If the cosine of an angle needs to be calculated not in a triangle, but in some other arbitrary geometric figure, then everything becomes a little more complicated. The magnitude of the angle must first be determined in radians or degrees, and only then the cosine must be calculated from this value. Cosine by numerical value determined using Bradis tables, engineering calculators or special mathematical applications.
Special mathematical applications may have functions such as automatically calculating the cosines of angles in a particular figure. The beauty of such applications is that they give the correct answer, and the user does not waste his time solving sometimes quite complex problems. On the other hand, when constantly using applications exclusively to solve problems, all skills in working with the solution are lost mathematical problems to find the cosines of angles in triangles, as well as other arbitrary figures.
One of the areas of mathematics that students struggle with the most is trigonometry. It is not surprising: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.
Origins of trigonometry
Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.
Historically, the main object of study in this section mathematical science were right triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values of all parameters of the figure in question using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy and even in art.
Initial stage
Initially, people talked about the relationship between angles and sides exclusively using the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in everyday life of this branch of mathematics.
The study of trigonometry in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract trigonometric equations, which begin in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, and cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence at least because earth's surface, and the surface of any other planet is convex, which means that any surface marking will be “arc-shaped” in three-dimensional space.
Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.
Right triangle
Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.
The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. It is the longest. We remember that according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.
For example, if the two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.
The two remaining sides, which form a right angle, are called legs. In addition, we must remember that the sum of the angles in a triangle is rectangular system coordinates is 180 degrees.
Definition
Finally, with a firm understanding of the geometric basis, one can turn to the definition of sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.
Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.
So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.
The simplest formulas
In trigonometry you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.
The first formula that you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you need to know the size of the angle rather than the side.
Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation does trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, transformation rules and several basic formulas, you can at any time derive the required more complex formulas on a sheet of paper.
Formulas for double angles and addition of arguments
Two more formulas that you need to learn are related to the values of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of sine and cosine is added.
There are also formulas associated with arguments in the form double angle. They are completely derived from the previous ones - as a practice, try to get them yourself by taking the alpha angle equal to the beta angle.
Finally, note that double angle formulas can be rearranged to reduce the power of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of the figure, and the size of each side, etc.
The sine theorem states that dividing the length of each side of a triangle by the opposite angle results in the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a given triangle.
The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the adjacent angle - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.
Careless mistakes
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.
Firstly, you shouldn't convert fractions to decimals until you get the final result - you can leave the answer as common fraction, unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will waste your time on unnecessary mathematical operations. This is especially true for values such as the root of three or the root of two, because they are found in problems at every step. The same goes for rounding “ugly” numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but you will also demonstrate a complete lack of understanding of the subject. This is worse than a careless mistake.
Thirdly, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to confuse them, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to start studying trigonometry because they do not understand its practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
In conclusion
So you're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: length three sides and the sizes of the three angles. The only difference in the tasks lies in the fact that different input data are given.
You now know how to find sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a ratio, and a ratio is a fraction, main goal The trigonometric problem becomes finding the roots of an ordinary equation or a system of equations. And here regular school mathematics will help you.