The opposite leg to the adjacent one. Sine, cosine, tangent and cotangent - everything you need to know on the exam in mathematics (2020). Pythagorean theorem to find the leg of a right triangle
Unified State Exam for 4? Won't you burst with happiness?
The question, as they say, is interesting ... You can, you can pass at 4! And at the same time not to burst ... The main condition is to practice regularly. Here is the basic preparation for the exam in mathematics. With all the secrets and secrets of the 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 "That's enough for you!" does not cause any difficulties for you. But if suddenly ... Follow the links, do not be lazy!
And we'll 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 are "very even ...")
This topic presents 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? It is worth asking these harmless questions, as a person turns pale and tries to divert the conversation aside ... But in vain. These are simple concepts. And this topic is no more complicated than others. You just need to clearly understand the answers to these very questions from the very beginning. It 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 deep antiquity. Don't worry, we will cover all 20 centuries of trigonometry in 15 minutes. And, imperceptibly for ourselves, we will repeat a piece of geometry from the 8th grade.
Let's draw a right-angled triangle with sides a, b, c and angle NS... Here's one.
Let me remind you that the sides that form a right angle are called legs. a and b- legs. There are two of them. The remaining side is called the hypotenuse. with- 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. Measure the side in... In the figure, the cells are specially drawn, as it happens in the examinations. Side in is equal to four cells. OK. Measure the side but. Three cells.
Now let's divide the length of the side but by side length in... Or, as they say, take the attitude but To in. a / b= 3/4.
On the contrary, you can divide in on the but. We get 4/3. Can in divide into with. Hypotenuse with cannot be counted 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's the point in this interesting activity? None yet. Stupid occupation, frankly.)
Now let's do this. Let's enlarge the triangle. Extend sides in and with, but so that the triangle remains rectangular. Injection NS naturally does not change. To see this, move the mouse cursor over the picture, or tap 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 / b It was: a / b= 3/4, now m / n= 6/8 = 3/4. The relationship of other relevant parties is also will not change ... You can change the lengths of the sides in a right-angled triangle as you like, increase, decrease, without changing the angle x – the relationship of the parties concerned will not change ... You can check, but you can take the ancient people at their word.
But this is already very important! The ratios of the sides in a right-angled 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 special names. Their names, so to speak.) Meet.
What is the sine of angle x ? This is the ratio of the opposite leg to the hypotenuse:
sinx = a / s
What is the cosine of angle x ? This is the ratio of the adjacent leg to the hypotenuse:
withosx= a / c
What is the tangent of angle x ? This is the ratio of the opposite leg to the adjacent one:
tgx =a / b
What is the cotangent of angle x ? This is the ratio of the adjacent leg to the opposite one:
ctgx = in / a
Everything is very simple. Sine, cosine, tangent and cotangent are some of the numbers. Dimensionless. Just numbers. Each corner has its own.
Why am I repeating everything so boringly? Then what is it need to remember... It's hard to remember. Memorization can be made easier. Does the phrase "Let's start from afar ..." sound familiar? So start from afar.
Sinus angle is the ratio distant from the angle of the leg to the hypotenuse. Cosine- the ratio of the neighbor to the hypotenuse.
Tangent angle is the ratio distant from the corner of the leg to the nearest. Cotangent- vice versa.
It's easier now, right?
Well, if you remember that only 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 are also called trigonometric functions.
And now a question for consideration.
Why do we say sine, cosine, tangent and cotangent corner? It's about the relationship of the parties, like ... What does it have to do with injection?
We look at the second picture. Exactly the same as the first one.
Move the mouse over the picture. I changed the angle NS... Increased it from x to x. All relationships have changed! Attitude a / b was 3/4, and the corresponding ratio t / in became 6/4.
And all other relationships became different!
Therefore, the relationship of the sides does 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 firmly 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, it means that we know the angle.
There are special tables where trigonometric functions are described for each angle. Bradis tables are named. They were compiled a very long time ago. Before there were no calculators or computers ...
Of course, the trigonometric functions of all angles cannot be memorized. You are obliged to know them only for a few angles, more on that later. But the spell " I know the angle - it means 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's a typical exam from the exam. To solve which, 8th grade is enough. Given a picture:
Everything. No more data available. It is necessary to find the length of the BC leg.
The cells do not help much, the triangle is somehow incorrectly positioned .... Specially, go ... 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, which means that we know all of its trigonometric functions. Which of the four functions should you use? Let's see what we know? We know the hypotenuse, the angle, but we need to find adjacent to this corner of the legs! It's clear that the cosine needs to be put into operation! So we launch it. We just write, according to the definition of the cosine (the ratio adjacent leg to the hypotenuse):
cosC = BC / 8
Angle C is 60 degrees, its cosine is 1/2. You need to know this, without any tables! That is:
1/2 = BC / 8
Elementary linear equation. Unknown - Sun... If you have forgotten how to solve equations, follow the link, the rest decide:
BC = 4
When the ancient people realized that each angle has its own set of trigonometric functions, they had a reasonable question. Are not sine, cosine, tangent and cotangent related in some way? So that knowing one function of the angle, one can find the rest? Without calculating the angle itself?
They were so restless ...)
Relationship between trigonometric functions of one angle.
Of course, sine, cosine, tangent and cotangent of the same angle are related. Any connection between expressions is specified in mathematics by formulas. In trigonometry, there is a colossal amount of formulas. But here we will look at the most basic ones. These formulas are called: basic trigonometric identities. Here they are:
These formulas must be known ironically. Without them, there is nothing to do in trigonometry at all. Three more auxiliary identities follow from these basic identities:
I warn you right away that the last three formulas quickly fall out of memory. For some reason.) You can, of course, deduce these formulas from the first three. But, in difficult times ... you understand.)
In standard assignments such as the ones below, there is a way to do without these forgettable formulas. AND reduce errors dramatically for forgetfulness, and in calculations too. This practice is in Section 555, lesson "Relationship Between Trigonometric Functions of the Same Angle."
In what tasks and how are the basic trigonometric identities used? The most popular task is to find some function of an angle if another is given. In the exam, such a task is present from year to year.) For example:
Find 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. This is the formula:
sin 2 x + cos 2 x = 1
We substitute here the known value, namely, 0.8 instead of the 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, 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 is wrong. How to be !? Yes, as usual.) Read the assignment carefully. For some reason it says there: ... if x is an acute angle ... And in the tasks, each word has a meaning, yes ... This phrase - and there is additional information to the solution.
An acute angle is an angle less than 90 °. And at such corners 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-angled triangles, where the corners can only be sharp. And they don't know, happy ones, that there are negative angles and angles of 1000 ° ... And all these horrible angles have their own trigonometric functions with plus and minus ...
But high school students without taking into account the sign - in any way. Many knowledge multiplies sorrow, 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 such an entry:
Or something else. You will see in the examples below.) To solve such examples, you need to know in which quarter does the given angle x fall and what sign has the desired trigonometric function in this quarter.
These basics of trigonometry are discussed in the lessons about what a trigonometric circle is, counting 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 point out the most important thing:
Practical advice:
1. Memorize the definitions of sine, cosine, tangent and cotangent. Very useful.
2. Accurately learn: sine, cosine, tangent and cotangent are firmly connected with angles. We know one thing - it means we know another.
3. We clearly learn: sine, cosine, tangent and cotangent of one angle are connected with each other by basic trigonometric identities. If we know one function, then we can (if we have the necessary additional information) calculate all the others.
And now we will solve it, as usual. First, assignments in the scope of the 8th grade. But high school students can also ...)
1. Calculate the tgА value if ctgА = 0.4.
2. β is the angle in a right-angled triangle. Find the value of tgβ if sinβ = 12/13.
3. Determine the sine of an acute angle x if tgx = 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 (semicolon-separated, in a mess):
0,09; 3; 0,8; 2,4; 2,5
Happened? Excellent! Eighth graders can already get their A.)
Not everything worked out? Tasks 2 and 3 are somehow not very ...? No problem! There is one nice trick for such tasks. Everything is solved, practically, without formulas at all! Well, and, therefore, no mistakes. This technique in the lesson: "Relationship between trigonometric functions of the same angle" in Section 555 is described. All other tasks are also sorted out there.
These were tasks like the Unified State Exam, but in a truncated version. Unified State Exam - light). And now there are almost the same tasks, but in a full-fledged test form. For high school students burdened with knowledge.)
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 somehow not very unambiguous ... And 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!
And if you haven't decided? Um ... Well, Section 555 will help. There, the solutions to all these tasks are detailed, it is difficult not to understand.
This lesson introduces a very limited concept of trigonometric functions. Within 8th grade. And the elders still have questions ...
For example, if the angle NS(see the second picture on this page) - make it stupid !? The triangle will collapse altogether! And how to be? There will be no leg, no hypotenuse ... The sinus is gone ...
If the ancient people did not find a way out of this situation, we would not have mobile phones, TV, or electricity now. Yes Yes! The theoretical basis for all these things without trigonometric functions is zero without a stick. 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. Instant validation testing. Learning - with interest!)
you can get acquainted with functions and derivatives.
We will start the study of trigonometry with a right-angled triangle. Let's define what sine and cosine are, as well as tangent and cotangent of an acute angle. These are the basics of trigonometry.
Recall that right angle is the angle equal to. In other words, half of a flattened corner.
Sharp corner- smaller.
Obtuse angle- larger. When applied to such a corner, "dumb" is not an insult, but a mathematical term :-)
Let's draw a right-angled triangle. A right angle is usually indicated. Note that the side opposite the corner is denoted by the same letter, only small. So, the side opposite the corner is indicated.
The angle is indicated by the corresponding Greek letter.
Hypotenuse a right-angled triangle is the side opposite the right angle.
Legs- sides opposite sharp corners.
The leg opposite the corner is called opposing(in relation to the corner). Another leg, which lies on one side of the corner, is called adjacent.
Sinus an acute angle in a right-angled triangle is the ratio of the opposite leg to the hypotenuse:
Cosine an acute angle in a right-angled triangle is the ratio of the adjacent leg to the hypotenuse:
Tangent an acute angle in a right-angled triangle - the ratio of the opposite leg to the adjacent one:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
Cotangent an acute angle in a right-angled triangle is the ratio of the adjacent leg to the opposite one (or, which is the same, the ratio of the cosine to the sine):
Note the basic relationships for sine, cosine, tangent and cotangent below. They will be useful to us when solving problems.
Let's prove some of them.
1. The sum of the angles of any triangle is. Means, the sum of two acute angles of a right-angled triangle is .
2. On the one hand, as the ratio of the opposite leg to the hypotenuse. On the other hand, since the leg will be adjacent for the angle.
We get that. In other words, .
3. Take the Pythagorean theorem:. Let's divide both parts into:
We got basic trigonometric identity:
Thus, knowing the sine of an angle, we can find its cosine, and vice versa.
4. Dividing both sides of the basic trigonometric identity by, we get:
This means that if we are given the tangent of an acute angle, then we can immediately find its cosine.
Similarly,
Okay, we have given definitions and written down formulas. And what are sine, cosine, tangent and cotangent for?
We know that the sum of the angles of any triangle is.
We know the relationship between parties right triangle. This is the Pythagorean theorem:.
It turns out that knowing the two angles in the triangle, you can find the third. Knowing the two sides in a right-angled triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what if in a right-angled triangle one angle is known (except for the right one) and one side, but you need to find other sides?
People faced this in the past, making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric functions of an angle- give the relationship between parties and corners triangle. Knowing the angle, you can find all of its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of sine, cosine, tangent and cotangent values for "good" angles from to.
Note the two red dashes in the table. The tangent and cotangent do not exist for the corresponding angles.
Let's analyze several trigonometry tasks from the FIPI Job Bank.
1. In a triangle, the angle is,. Find.
The problem is solved in four seconds.
Since, we have:.
2. In a triangle, the angle is,,. Find. , is equal to half of the hypotenuse.
A triangle with corners, and is isosceles. In it, the hypotenuse is times larger than the leg.
The ratio of the opposite leg to the hypotenuse is called sinus acute angle right triangle.
\ sin \ alpha = \ frac (a) (c)
Cosine of an acute angle of a right triangle
The ratio of the nearby leg to the hypotenuse is called cosine of an acute angle right triangle.
\ cos \ alpha = \ frac (b) (c)
Acute tangent of a right triangle
The ratio of the opposite leg to the adjacent leg is called tangent of an acute angle right triangle.
tg \ alpha = \ frac (a) (b)
Cotangent of an acute angle of a right triangle
The ratio of the adjacent leg to the opposite leg is called acute angle cotangent right triangle.
ctg \ alpha = \ frac (b) (a)
Sine of an arbitrary angle
The ordinate of the point on the unit circle to which the angle \ alpha corresponds is called sine of an arbitrary angle rotation \ alpha.
\ sin \ alpha = y
Cosine of an arbitrary angle
The abscissa of the point on the unit circle to which the angle \ alpha corresponds is called cosine of an arbitrary angle rotation \ alpha.
\ cos \ alpha = x
Arbitrary angle tangent
The ratio of the sine of an arbitrary angle of rotation \ alpha to its cosine is called tangent of an arbitrary angle rotation \ alpha.
tg \ alpha = y_ (A)
tg \ alpha = \ frac (\ sin \ alpha) (\ cos \ alpha)
Cotangent of an arbitrary angle
The ratio of the cosine of an arbitrary angle of rotation \ alpha to its sine is called cotangent of an arbitrary angle rotation \ alpha.
ctg \ alpha = x_ (A)
ctg \ alpha = \ frac (\ cos \ alpha) (\ sin \ alpha)
An example of finding an arbitrary angle
If \ alpha is some angle AOM, where M is a point of the unit circle, then
\ sin \ alpha = y_ (M), \ cos \ alpha = x_ (M), tg \ alpha = \ frac (y_ (M)) (x_ (M)), ctg \ alpha = \ frac (x_ (M)) (y_ (M)).
For example, if \ angle AOM = - \ frac (\ pi) (4), then: the ordinate of point M is equal to - \ frac (\ sqrt (2)) (2), the abscissa is \ frac (\ sqrt (2)) (2) and that's why
\ sin \ left (- \ frac (\ pi) (4) \ right) = - \ frac (\ sqrt (2)) (2);
\ cos \ left (\ frac (\ pi) (4) \ right) = \ frac (\ sqrt (2)) (2);
tg;
ctg \ left (- \ frac (\ pi) (4) \ right) = - 1.
Table of values of sines of cosines of tangents of cotangents
The values of the main common angles are given in the table:
| 0 ^ (\ circ) (0) | 30 ^ (\ circ) \ left (\ frac (\ pi) (6) \ right) | 45 ^ (\ circ) \ left (\ frac (\ pi) (4) \ right) | 60 ^ (\ circ) \ left (\ frac (\ pi) (3) \ right) | 90 ^ (\ circ) \ left (\ frac (\ pi) (2) \ right) | 180 ^ (\ circ) \ left (\ pi \ right) | 270 ^ (\ circ) \ left (\ frac (3 \ pi) (2) \ right) | 360 ^ (\ circ) \ left (2 \ pi \ right) | |
| \ sin \ alpha | 0 | \ frac12 | \ frac (\ sqrt 2) (2) | \ frac (\ sqrt 3) (2) | 1 | 0 | −1 | 0 |
| \ cos \ alpha | 1 | \ frac (\ sqrt 3) (2) | \ frac (\ sqrt 2) (2) | \ frac12 | 0 | −1 | 0 | 1 |
| tg \ alpha | 0 | \ frac (\ sqrt 3) (3) | 1 | \ sqrt3 | — | 0 | — | 0 |
| ctg \ alpha | — | \ sqrt3 | 1 | \ frac (\ sqrt 3) (3) | 0 | — | 0 | — |
Knowing one of the legs in a right-angled triangle, you can find the second leg and the hypotenuse using the trigonometric relationship - sine and tangent of a known angle. Since the ratio of the opposite leg angle to the hypotenuse is equal to the sine of this angle, therefore, to find the hypotenuse, the leg must be divided by the sine of the angle. a / c = sinα c = a / sinα
The second leg can be found from the tangent of a known angle, as the ratio of the known leg to the tangent. a / b = tanα b = a / tanα
To calculate the unknown angle in a right-angled triangle, you need to subtract the value of the angle α from 90 degrees. β = 90 ° -α
The perimeter and area of a right-angled triangle through the leg and the angle opposite to it can be expressed by substituting the previously obtained expressions for the second leg and the hypotenuse into the formulas. P = a + b + c = a + a / tanα + a / sinα = a tanα sinα + a sinα + a tanα S = ab / 2 = a ^ 2 / ( 2 tanα)
You can also calculate the height through trigonometric relations, but already in the inner right-angled triangle with side a, which it forms. To do this, you need the side a, how to multiply the hypotenuse of such a triangle by the sine of the angle β or the cosine α, since, according to trigonometric identities, they are equivalent. (fig. 79.2) h = a cosα
The median of the hypotenuse is equal to half of the hypotenuse or the known leg a divided by two sines α. To find the medians of the legs, we bring the formulas to the appropriate form for the known side and angles. (Fig. 79.3) m_с = c / 2 = a / (2 sinα) m_b = √ (2a ^ 2 + 2c ^ 2-b ^ 2) / 2 = √ (2a ^ 2 + 2a ^ 2 + 2b ^ 2-b ^ 2) / 2 = √ (4a ^ 2 + b ^ 2) / 2 = √ (4a ^ 2 + a ^ 2 / tan ^ 2α) / 2 = (a√ (4 tan ^ 2 α + 1)) / (2 tanα) m_a = √ (2c ^ 2 + 2b ^ 2-a ^ 2) / 2 = √ (2a ^ 2 + 2b ^ 2 + 2b ^ 2-a ^ 2) / 2 = √ (4b ^ 2 + a ^ 2) / 2 = √ (4b ^ 2 + c ^ 2-b ^ 2) / 2 = √ (3 a ^ 2 / tan ^ 2α + a ^ 2 / sin ^ 2α) / 2 = √ ((3a ^ 2 sin ^ 2α + a ^ 2 tan ^ 2α) / (tan ^ 2α sin ^ 2α)) / 2 = (a√ ( 3 sin ^ 2α + tan ^ 2α)) / (2 tanα sinα)
Since the bisector of a right angle in a triangle is the product of two sides and the root of two, divided by the sum of these sides, replacing one of the legs with the ratio of the known leg to the tangent, we get the following expression. Similarly, by substituting the ratio in the second and third formulas, you can calculate the bisectors of the angles α and β. (Fig. 79.4) l_с = (aa / tanα √2) / (a + a / tanα) = (a ^ 2 √2) / (a tanα + a) = (a√2) / (tanα + 1) l_a = √ (bc (a + b + c) (b + ca)) / (b + c) = √ (bc ((b + c) ^ 2-a ^ 2)) / (b + c) = √ (bc (b ^ 2 + 2bc + c ^ 2-a ^ 2)) / (b + c) = √ (bc (b ^ 2 + 2bc + b ^ 2)) / (b + c) = √ (bc (2b ^ 2 + 2bc)) / (b + c) = (b√ (2c (b + c))) / (b + c) = (a / tanα √ (2c (a / tanα + c))) / (a / tanα + c) = (a√ (2c (a / tanα + c))) / (a + c tanα) l_b = √ (ac (a + b + c) (a + cb)) / (a + c) = (a√ (2c (a + c))) / (a + c) = (a√ (2c (a + a / sinα))) / (a + a / sinα) = (a sinα √ (2c (a + a / sinα))) / (a sinα + a)
The middle line runs parallel to one of the sides of the triangle, while forming another similar right-angled triangle with the same angles, in which all sides are half the size of the original. Based on this, the midlines can be found by the following formulas, knowing only the leg and the angle opposite to it. (fig. 79.7) M_a = a / 2 M_b = b / 2 = a / (2 tanα) M_c = c / 2 = a / (2 sinα)
The radius of the inscribed circle is equal to the difference between the legs and the hypotenuse, divided by two, and to find the radius of the circumscribed circle, the hypotenuse must be divided by two. We replace the second leg and the hypotenuse with the ratio of leg a to the sine and tangent, respectively. (Fig. 79.5, 79.6) r = (a + bc) / 2 = (a + a / tanα -a / sinα) / 2 = (a tanα sinα + a sinα-a tanα) / (2 tanα sinα) R = c / 2 = a / 2sinα
Instructions
Method 1. Using the Pythagorean theorem. The theorem says: the square of the hypotenuse is equal to the sum of the squares of the legs. It follows that any of the sides of a right-angled triangle can be calculated by knowing its other two sides (Fig. 2)
Method 2. It follows from the fact that the median drawn from to the hypotenuse forms 3 similar triangles among themselves (Fig. 3). In this figure, triangles ABC, BCD, and ACD are similar.
Example 6: Using Unit Circles to Find Coordinates
First, we find the reference angle corresponding to the given angle. Then we take the sine and cosine value of the reference angle, and give them the signs corresponding to the y- and x-values of the quadrant. Next, we will find the cosine and sine of the given angle.
Sieve angle, angle triangle and cube root
Polygons that can be drawn with a compass and ruler include.Note: The sieve angle cannot be plotted using a compass and a ruler. Multiplying the side length of a cube by the cube root of 2 gives the side length of a double volume cube. With the help of the innovative theory of the French mathematician Évariste Galois, it can be shown that for all three classical problems, construction with a circle and a ruler is impossible.
The hypotenuse is the side in a right-angled triangle that is opposite an angle of 90 degrees. In order to calculate its length, it is enough to know the length of one of the legs and the size of one of the acute angles of the triangle.
Keep in mind: a three-piece angle and cube root design is not possible with a compass and ruler. 
On the other hand, the solution to a third-degree equation by Cardano's formula can be represented by dividing the angle and the cubic root. In what follows, we build a certain angle with a circle and a ruler. However, after the triangle of this angle and the determination of the cube root, the completion of the construction of the square sieve can be done using a compass and a ruler.
Build a lattice deck according to this calculation
The algebraic formulation of the construction problem leads to an equation, the structural analysis of which will provide additional information on the construction of the ternary structure. Here, a one-to-one ratio of the angle to its cosine is used: if the value of the angle is known, the length of the cosine of the angle can be uniquely plotted on the unit circle and vice versa.
Instructions
With a known leg and an acute angle of a right-angled triangle, the size of the hypotenuse can be equal to the ratio of the leg to the cosine / sine of this angle, if this angle is opposite / adjacent to it:
h = C1 (or C2) / sinα;
h = C1 (or C2) / cosα.
Example: Let a right-angled triangle ABC be given with a hypotenuse AB and a right angle C. Let the angle B be 60 degrees, and the angle A 30 degrees. The length of the leg BC is 8 cm. It is necessary to find the length of the hypotenuse AB. To do this, you can use any of the above methods:
This one-to-one assignment allows you to go from determining the angle to determining the cosine of the angle. In what follows, 3 φ denotes the angle to be divided. Thus, φ is the angle, the value of which should be determined for given 3 φ. Starting with compounds known from trigonometry.
Follows at a given angle of 3 φ. An algebraic consideration of the solvability of a three-dimensional equation leads directly to the question of the possibility of constructing solutions and, consequently, to the question of the possibility or impossibility of a constructive triple angle of a given angle.
AB = BC / cos60 = 8 cm.
AB = BC / sin30 = 8 cm.
The hypotenuse is the side of a right-angled triangle that lies opposite the right angle. It is the largest side of a right-angled triangle. You can calculate it using the Pythagorean theorem or using the formulas of trigonometric functions.
The magnitude of the exit angle has a great influence on the possibility of linking the third angle, since this, as an absolute term, decisively determines the type of solutions in the three-dimensional equation. If the triangulation equation has at least one real solution that can be obtained by rational operations or by drawing square roots for a given starting angle, this solution is constructive.
Breidenbach formulated as a criterion that the three-second angle can be interpreted only in the rational solution of the equation of three parts. If no such solution is available, the problem of three-part construction is irreconcilable with compass and ruler. Cluster analysis is a general method for assembling small groups from a large dataset. Similar to discriminant analysis, cluster analysis is also used to classify cases into groups. On the other hand, discriminatory analysis requires knowledge of group memberships in the cases used to derive the classification rule.
Instructions
The legs are called the sides of a right-angled triangle adjacent to a right angle. In the figure, the legs are designated as AB and BC. Let the lengths of both legs be given. Let's designate them as | AB | and | BC |. In order to find the length of the hypotenuse | AC |, we use the Pythagorean theorem. According to this theorem, the sum of the squares of the legs is equal to the square of the hypotenuse, i.e. in the notation of our figure | AB | ^ 2 + | BC | ^ 2 = | AC | ^ 2. From the formula we obtain that the length of the hypotenuse AC is found as | AC | = √ (| AB | ^ 2 + | BC | ^ 2).
Cluster analysis is more primitive because it does not make assumptions about the number of groups or group membership. Classification Cluster analysis provides a way to discover potential relationships and create a systematic structure across a large number of variables and observations. Hierarchical cluster analysis is the main statistical method for finding relatively homogeneous clusters of cases based on measured characteristics. It starts with each case as a separate cluster.
The clusters are then combined sequentially, the number of clusters decreasing with each step until there is only one cluster left. The clustering method uses differences between objects to form clusters. Hierarchical cluster analysis is best suited for small samples.
Let's look at an example. Let the lengths of the legs | AB | = 13, | BC | = 21. By the Pythagorean theorem, we obtain that | AC | ^ 2 = 13 ^ 2 + 21 ^ 2 = 169 + 441 = 610. In order to obtain the length of the hypotenuse, it is necessary to extract the square root of the sum of the squares of the legs, ie from among 610: | AC | = √610. Using the table of squares of integers, we find out that the number 610 is not a complete square of any integer. In order to get the final value of the length of the hypotenuse, let's try to take out the full square from the root sign. To do this, factor the number 610. 610 = 2 * 5 * 61. According to the table of prime numbers, we see that 61 is a prime number. Therefore, further reduction of the number √610 is impossible. We get the final answer | AC | = √610.
If the square of the hypotenuse were equal, for example, 675, then √675 = √ (3 * 25 * 9) = 5 * 3 * √3 = 15 * √3. If such a reduction is possible, perform the reverse check - square the result and compare with the original value.
Hierarchical cluster analysis is just one way to observe the formation of homogeneous variable groups. There is no specific way to set the number of clusters for your analysis. Perhaps you need to look at the dendrogram as well as the characteristics of the clusters and then adjust the number in stages to get a good clustering solution.
When variables are measured at different scales, you have three ways to standardize the variables. As a result, all variables with roughly equal proportions contribute to the measurement of distance, even if you might lose information about the variance of the variables.
Let us know one of the legs and the corner adjacent to it. For definiteness, let it be leg | AB | and angle α. Then we can use the formula for the trigonometric function cosine - the cosine of the angle is equal to the ratio of the adjacent leg to the hypotenuse. Those. in our notation cos α = | AB | / | AC |. From this we obtain the length of the hypotenuse | AC | = | AB | / cos α.
If we know the leg | BC | and angle α, then we will use the formula to calculate the sine of the angle - the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse: sin α = | BC | / | AC |. We get that the length of the hypotenuse is found as | AC | = | BC | / cos α.
Euclidean Distance: Euclidean distance is the most common measurement method. Square Euclidean Distance: The square of Euclidean distance focuses attention on objects that are farther apart. Distance to City Block: Both the city block and the Euclidean distance are special cases of the Minkowski metric. While Euclidean distance corresponds to the length of the shortest path between two points, the distance along a city block is the sum of the distances along each dimension. Pearson Correlation Distance Difference between 1 and the cosine factor of two observations The cosine factor is the cosine of the angle between two vectors. Jacard Distance The difference between 1 and Jacard's coefficient for two observations For binary data, Jacard's coefficient is the ratio of the overlap and the sum of the two observations. Nearest Neighbor This method assumes that the distance between two clusters corresponds to the distance between the features in their nearest neighborhood. Best Neighbor In this method, the distance between two clusters corresponds to the maximum distance between two objects in different clusters. Group Average: With this method, the distance between two clusters corresponds to the average distance between all pairs of objects in different clusters. This method is generally recommended because it contains a higher amount of information. Median This method is identical to the centroid method, except that it is unweighted. Then, for each case, the quadratic Euclidean distance to the cluster means is calculated. The cluster to be merged is the one that increases the amount at least. That is, this method minimizes the increase in the total sum of the squared distances within the clusters. This method tends to create smaller clusters.
- This is the geometric distance in multidimensional space.
- It is only suitable for continuous variables.
- Cosine Distance The cosine of the angle between two vectors of values.
- This method is recommended when drawing painted clusters.
- If the clusters drawn form unique clumps, the method is appropriate.
- The cluster centroid is the midpoint in multidimensional space.
- It should not be used if the cluster sizes are very different.
- Ward Averages for all variables are calculated for each cluster.
- These distances are added up for all cases.
For clarity, consider an example. Let the length of the leg | AB | = 15. And the angle α = 60 °. We get | AC | = 15 / cos 60 ° = 15 / 0.5 = 30.
Consider how you can check your result using the Pythagorean theorem. To do this, we need to calculate the length of the second leg | BC |. Using the formula for the tangent of the angle tan α = | BC | / | AC |, we obtain | BC | = | AB | * tan α = 15 * tan 60 ° = 15 * √3. Then we apply the Pythagorean theorem, we get 15 ^ 2 + (15 * √3) ^ 2 = 30 ^ 2 => 225 + 675 = 900. The check is completed.
The sine function is determined from the concept of sine, bearing in mind that the angle must always be expressed in radians. We can observe several characteristics of a sinusoidal function.
- Your domain contains all the real ones.
- In this case, the function is said to be periodic, of period 2π.
We can observe several characteristics of the cosine function. Thus, this is a periodic period of 2π. ... The limitation does not remove the generality of the formula, because we can always reduce the angles of the second, third and fourth quadrants to the first. An exercise. - Calculate 15º sine without using a calculator.
After calculating the hypotenuse, check whether the resulting value satisfies the Pythagorean theorem.
Sources:
- A table of prime numbers from 1 to 10000
Legs call the two short sides of a right-angled triangle that make up that apex, the value of which is 90 °. The third side in such a triangle is called the hypotenuse. All these sides and angles of the triangle are related to each other by certain ratios that allow you to calculate the length of the leg, if several other parameters are known.
Cosine of the sum of two angles
The cosine of the difference of two angles
To get the formula, we can proceed in the same way as in the previous section, but we will see another very simple demonstration based on the Pythagorean theorem. Simplifying and changing the sign, we have. The tangent sum and the difference of two angles.An exercise. In today's article, we'll take a look at a very specific subset: trigonometric functions. To enjoy everything that mathematics has to offer, we have to import it. In the next article, we will see other import styles, each with their own advantages and disadvantages. But with this simple instruction, you already have access to the entire namespace of the math module, filled with dozens of functions, among which are the ones we will be dealing with today.
Instructions
Use the Pythagorean theorem to calculate the length of the leg (A) if you know the length of the other two sides (B and C) of a right triangle. This theorem states that the sum of the squared leg lengths is equal to the square of the hypotenuse. It follows from this that the length of each of the legs is equal to the square root of the difference between the squares of the lengths of the hypotenuse and the second leg: A = √ (C²-B²).
Basically, we will need to calculate the sine, cosine and tangent of an angle, as well as its inverse functions. In addition, we would like to be able to work in both radians and degrees so that we can also use the appropriate conversion functions.
You should keep in mind that these functions expect the argument to be supplied in radians, not degrees. To this end, you will be interested to know that you have the following constant. So we can use this expression instead of a numeric value.
There is no direct function for cosecant, secant and cotangent, as this is not necessary since they are simply inverse of sine, cosine and tangent respectively. As before, the returned angle is also in radians. Another useful math function allows us to find out the value of the hypotenuse of a right triangle, given its legs, which allows us to calculate the square root of the sum of the squares of them.
Use the definition of the direct trigonometric function "sine" for an acute angle, if you know the value of the angle (α), which lies opposite the calculated leg, and the length of the hypotenuse (C). This definition states that the sine of this known angle is equal to the ratio of the length of the desired leg to the length of the hypotenuse. This means that the length of the desired leg is equal to the product of the length of the hypotenuse and the sine of the known angle: A = C ∗ sin (α). For the same known values, you can use the definition of the cosecant function and calculate the required length by dividing the length of the hypotenuse by the cosecant of the known angle A = C / cosec (α).
Use the definition of the direct trigonometric cosine function if, in addition to the length of the hypotenuse (C), the value of the acute angle (β) adjacent to the desired leg is also known. The cosine of this angle is defined as the ratio of the lengths of the desired leg and the hypotenuse, and from this we can conclude that the length of the leg is equal to the product of the length of the hypotenuse by the cosine of the known angle: A = C ∗ cos (β). You can use the definition of the secant function and calculate the desired value by dividing the length of the hypotenuse by the secant of the known angle A = C / sec (β).
Derive the desired formula from a similar definition for the derivative of the trigonometric function of the tangent, if, in addition to the acute angle (α), which lies opposite the desired leg (A), the length of the second leg (B) is known. The tangent of the angle opposite to the desired leg is the ratio of the length of this leg to the length of the second leg. This means that the required value will be equal to the product of the length of the known leg and the tangent of the known angle: A = B ∗ tg (α). Another formula can be derived from the same known quantities if we use the definition of the cotangent function. In this case, to calculate the length of the leg, it will be necessary to find the ratio of the length of the known leg to the cotangent of the known angle: A = B / ctg (α).
Related Videos
The word "cathet" came into Russian from Greek. In exact translation, it means a plumb line, that is, a perpendicular to the surface of the earth. In mathematics, legs are called sides that form a right angle of a right-angled triangle. The side opposite to this corner is called the hypotenuse. The term "leg" is also used in architecture and welding technology.
Draw a right-angled triangle ACB. Label its legs as a and b, and the hypotenuse as c. All sides and corners of a right-angled triangle are connected with each other by certain relationships. The ratio of the leg, opposite one of the acute angles, to the hypotenuse is called the sine of the given angle. In this triangle sinCAB = a / c. Cosine is the ratio to the hypotenuse of the adjacent leg, i.e. cosCAB = b / c. Reverse relations are called secant and cosecant.
The secant of a given angle is obtained by dividing the hypotenuse by the adjacent leg, that is, secCAB = c / b. It turns out the inverse of the cosine, that is, it can be expressed by the formula secCAB = 1 / cosSAB.
The cosecant is equal to the quotient of dividing the hypotenuse by the opposite leg and this is the reciprocal of the sine. It can be calculated using the formula cosecCAB = 1 / sinCAB
Both legs are connected by tangent and cotangent. In this case, the tangent will be the ratio of side a to side b, that is, the opposite leg to the adjacent leg. This ratio can be expressed by the formula tgCAB = a / b. Accordingly, the inverse relation will be the cotangent: ctgCAB = b / a.
The ratio between the dimensions of the hypotenuse and both legs was determined by the ancient Greek mathematician Pythagoras. People still use the theorem named after him. It says that the square of the hypotenuse is equal to the sum of the squares of the legs, that is, c2 = a2 + b2. Accordingly, each leg will be equal to the square root of the difference between the squares of the hypotenuse and the other leg. This formula can be written as b = √ (c2-a2).
The length of the leg can also be expressed through the relations known to you. According to the theorems of sines and cosines, the leg is equal to the product of the hypotenuse and one of these functions. You can also express it in terms of tangent or cotangent. Leg a can be found, for example, by the formula a = b * tan CAB. In the same way, depending on the specified tangent or cotangent, the second leg is also determined.
In architecture, the term "leg" is also used. It applies to an Ionic capital and denotes a plumb line through the middle of its back. That is, in this case, this term denotes a perpendicular to a given line.
In the technology of welding, there is the concept of "fillet weld legs". As in other cases, this is the shortest distance. Here we are talking about the gap between one of the parts to be welded to the border of the seam located on the surface of the other part.
Related Videos
Sources:
- what is leg and hypotenuse
Related Videos
note
When calculating the sides of a right-angled triangle, knowledge of its features can play:
1) If the leg of a right angle lies opposite an angle of 30 degrees, then it is equal to half of the hypotenuse;
2) The hypotenuse is always longer than any of the legs;
3) If a circle is described around a right-angled triangle, then its center should lie in the middle of the hypotenuse.
Where problems were considered for solving a right-angled triangle, I promised to outline a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which leg belongs to the hypotenuse (adjacent or opposite). I decided not to put it on the back burner, the necessary material is below, please read 😉
The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember perfectly well that the leg belongs to the hypotenuse, but which of them - they forget and confused. The cost of a mistake, as you know in the exam, is a lost point.
The information that I will present directly to mathematics has nothing to do with it. It is associated with figurative thinking, and with the methods of verbal-logical communication. That's right, I myself, once and for all, remember definition data. If you do forget them, then with the help of the presented techniques it is always easy to remember.
Let me remind you of the definitions of sine and cosine in a right-angled triangle:
Cosine an acute angle in a right-angled triangle is the ratio of the adjacent leg to the hypotenuse:
Sinus an acute angle in a right-angled triangle is the ratio of the opposite leg to the hypotenuse:
So, what associations do you have with the word cosine?
Probably everyone has their own 😉 Remember the bunch:
Thus, you will immediately have an expression in your memory -
«… the ratio of the ADJUSTING leg to the hypotenuse».
The problem with determining the cosine is solved.
If you need to recall the definition of the sine in a right-angled triangle, then recalling the definition of the cosine, you can easily establish that the sine of an acute angle in a right-angled triangle is the ratio of the opposite leg to the hypotenuse. After all, there are only two legs, if the adjacent leg is "occupied" by the cosine, then only the opposite sine remains.
What about tangent and cotangent? The confusion is the same. Students know that this is the relationship of the legs, but the problem is to remember which one it belongs to - either the opposite to the adjacent one, or vice versa.
Definitions:
Tangent an acute angle in a right-angled triangle is the ratio of the opposite leg to the adjacent one:
Cotangent an acute angle in a right-angled triangle is the ratio of the adjacent leg to the opposite one:
How to remember? There are two ways. One also uses a verbal-logical connection, the other - a mathematical one.
METHOD MATHEMATICAL
There is such a definition - the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
* Having memorized the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.
Likewise. The cotangent of an acute angle is the ratio of the cosine of an angle to its sine:
So! Having memorized the indicated formulas, you can always determine that:
The tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent
The cotangent of an acute angle in a right-angled triangle is the ratio of the adjacent leg to the opposite one.
WORD-LOGICAL METHOD
About tangent. Remember the bunch:
That is, if you need to remember the definition of the tangent, using this logical connection, you can easily remember that it is
"... the relation of the opposite leg to the adjacent one"
If it comes to cotangent, then remembering the definition of tangent, you can easily voice the definition of cotangent -
"... the relationship of the adjacent leg to the opposite"
There is an interesting technique for memorizing tangent and cotangent on the site " Math tandem " , take a look.
UNIVERSAL METHOD
You can just memorize. But as practice shows, thanks to verbal and logical connections, a person remembers information for a long time, and not only mathematical.
I hope the material was useful to you.
Best regards, Alexander Krutitskikh
P.S: I would be grateful if you could tell us about the site on social networks.
