The sine values

The unit circle can be considered not only for individual trigonometric functions, but also under the microscope each function separately. Here are the sine values for different angles.

sin 0. Mathematics for blondes.
sin 0

sin 30. sin p/6, sine 1/6 pi. Mathematics for blondes.
sin 30
sin 45, sin p/4, sine pi/4. Mathematics for blondes.
sin 45
sin 60, sine p/3, sin pi/3. Mathematics for blondes.
sin 60
sin 90, sin p/2, sin pi/2, sine 1/2 pi. Mathematics for blondes.
sin 90
sin 120, sin 2p/3, sin 2pi/3, sine 2/3 pi. Mathematics for blondes.
sin 120
sin 135 degrees, sin 3p/4, sin 3pi/4, sine 3/4 pi radians. Mathematics for blondes.
sin 135
sin 150, sine 5/6 pi radians, sin 5p/6, sin 5 pi by 6. Mathematics for blondes.
sin 150
sin 180, sin pi. Mathematics for blondes.
sin 180
sin 210, sin 7p/6, sin 7/6 pi. Mathematics for blondes.
sin 210
sin 225, sin 5/4 pi, sin 5p/4. Mathematics for blondes.
sin 225
sin 240, sin 4p/3, sine 4 pi division by 3 radians, sin 4/3 pi. Mathematics for blondes.
sin 240
sin 270, sin 3pi/2, sine 3/2 pi. Mathematics for blondes.
sin 270
sin 300, sin 5p/3, sin 5/3 pi. Mathematics for blondes.
sin 300
sin 315, sin 7pi/4, sin 7/4 pi. Mathematics for blondes.
sin 315
sin 330, sin 11pi/6, sin 11/6 pi. Mathematics for blondes.
sin 330
sin 360, sin 2pi, sine 2 pi radians. Mathematics for blondes.
sin 360

The unit circle is like a movie that shows us mathematics. Here you see some pictures of the cult TV series.


Unit circle

Mathematicians consider themselves clever and all the rest. But not all of us are as smart as mathematics. The unit circle mathematics invented for themselves. For those who are just beginning to study mathematics, I suggest a simpler version - separately cosines and sines separately.

If we remove from the unit circle all that relates to the sine, we get the unit circle cosines.

Unit circle cosines. Cos 0, cos 30, cos 45, cos 60, cos 90. Mathematics for blondes.
Unit circle cosines

If we remove from the unit circle everything about cosines, we get a unit circle sinuses.

Unit circle sinuses. Sin 0, sin 30, sin 45, sin 60, sin 90. Mathematics for blondes.
Unit circle sinuses

Now, you will not confuse the values of sines and cosines.


Degrees, minutes, seconds

I was write: "Help me to understand. It is necessary to determine how many degrees, minutes, and seconds contains 4563 seconds corner. Only by actions, and I completely confused."

No harly-barly! We sit quietly and remember that one minute contains 60 seconds, in one degree contains 60 minutes. Now we pimenyaem our knowledge into practice. Transform seconds in a minutes.

4563 : 60 = 76.05 minutes

Determine how many seconds we leave intact:

4563 - 60 * 76 = 4563 - 4560 = 3 seconds

Now 76 minutes translate into degrees, still sadly dividing by 60:

76 : 60 = 1.26667 degrees

We calculate how many minutes do not fit into the clothes degrees:

76 - 60 * 1 = 76 - 60 = 16 minutes

Now our suffering ended and we in good conscience can write the final result:

4563 seconds = 1 degree 16 minutes 3 seconds

Checking? We translate everything back to the second:

1 * 60 * 60 + 16 * 60 + 3 = 3600 + 960 + 3 = 4563

As the math, the conversion is correct.


How to find the area of a rectangle?

Here's a problem about the area of a rectangle of a textbook on algebra in 7 class (Russian school program):

If the width of the rectangle by 2 dm enlarge and reduce a length of 0.5 m, then we obtain a square, an area 50 dm² less than the area of the rectangle. Find the area of a rectangle.

Interestingly, in 7 class school studying systems of linear equations with two unknowns? Judging by the fact that the task of the textbook on algebra, so it is necessary to solve this problem. Stupidly make up the system of equations, bluntly we decide - anguish. If I'm here just write a solution, and you simply write off, the smarter you from it will not. I propose first to solve this problem, and the system of equations with the decision at the end we write.

What is square? It's a rectangle whose sides are equal. What is rectangle? It is a square, whose side are different. You are their teachers of mathematics do not govoriteovetuyu - for them it sounds like a desecration. I like "definitely" use constantly. After all, the mathematical properties of geometric objects they pass very accurately.

To solve the problem, we denote the side of the rectangle: a - the length, b - the width. Now we start again to read the statement of the problem.

"If the width of the rectangle to increase by 2 dm ...". In the language of mathematics it can be written as: b + 2.

"... And to reduce the length of 0.5 m ..." Here please pay attention - only completely illiterate people in a problem using different units of length. For example, meters and decimeters. We are educated people, as opposed to the author of the textbook, and translate everything in decimeters. Why decimeters? Because our area is measured in square decimetres. How many decimeters in one meter? That's right, ten. A 0.5 meter - decimeter is how many? Yes, 0.5 * 10 = 5 decimeters. Now we can take our phrase in the language of mathematics: a-5.


What does this leave us? Juggle this expression, we can express the length of one side through the length of the other side. In the future, it is useful to us. Personally, I do not like the "minus" sign. Now we get rid of him.


Something we digress from the conditions of the problem. Turn rewind and read the phrase: "... then we get a square with an area of 50 dm² is less than the area of a rectangle".

How to find the area of a rectangle? Area, perimetr of a rectangle. Mathematics for blondes.
How to find the area of a rectangle?


What's next? Now we can party instead of the original a substitute the result of juggling a=b+7.


The width of the rectangle we already know - 18 decimeters. We are looking for long.


Now we can easily determine the area of a rectangle 25 * 18 = 450 dm². The notebook can write all this as a system of two equations with two unknowns. I will give just two equations, choose any.

How to find the area of a rectangle. Decision. Mathematics for blondes.
How to find the area of a rectangle. Decision.

On the left side, we expressed a square area over the length of the rectangle on the right side - in width. In the course of solving the problem, we have considered a third option - the area of the square is represented as the product of the length by the width. All three options provide the same result. Here on the use of mathematics equations to solve problems.


The problem about the juice

The problem about the juice. Mathematics For Blondes
The problem about the juice
 On the Internet there are a lot of things. Yesterday I saw an interesting problem about the juice. I understand that the person made a mistake copying the text of the problem from the book. But it turned out very interesting. I used to see all over the math, just as I am translating the language of mathematics to the surrounding reality. That's how I read this problem and this is the decision I did.

A liter of grape juice is worth 6 manat. It was mixed with a liter of mulberry juice manat. A liter of juice sold for 10 manats. What benefits can be obtained from the sale of 10 liters of the mixed juice?

For those other than dollars knows nothing, to announce that manat - is the monetary unit of Azerbaijan (there is such a country). By the way, Muslims do not eat pork, mathematicians so do not use math-degree measure of angles in Calculus. As they say, find ten differences. It was the information for the overall development, but will return to the problem about the juice.

The number in the value of mulberry juice available. The man forgot to write. But mulberry juice can be stolen. Then it really does not cost anything. Such "schemes" are thriving in our lives. The lack of numbers in mathematics to denote the numeral zero. If we substitute the value of zero in mulberry juice, then the problem is easily solved.

For a start, we determine the number of cocktail, which is obtained by mixing two different juices. Each juice we take in the amount of one liter. If you do not like liters, take one gallon.

1 + 1 = 2 liters

Now we consider the cost of the resulting cocktail

6 + 0 = 6 manats

Calculate the cost per liter of cocktail

6: 2 = 3 manats

Who is the most interesting point - we determine the profit from the sale of a liter of cocktail

10 - 3 = 7 manats

At the end of the general view of the profits from this scam

10 * 7 = 70 manats


1. With revenues of these scams is not to compare, but enough to start.
2. The juice can be diluted with water, then stealing nothing.


1. For the theft could be imprisoned.
2. For juice dilution with water can beat face.

If in the problem still listed price for mulberry juice, instead of the zero substitute that number. The solution of the problem will not change. Incidentally, the legal business from the business of the criminal also a little different.


The problem about the garden area

The problem about the garden area. The garden occupies 80 hectares. Apple trees occupy 5/8 of the area, and 31% of cherries. Mathematics for blondes.
The problem about the garden area

Why such a strange image to task about the area of ​​the garden? Because the problem itself, to put it mildly, very strange. Here's how it sounds:

The garden occupies 80 hectares. Apple trees occupy 5/8 of the area, and 31% of cherry trees. How many hectares of area under apple trees larger than the area under the cherry trees?

Let's first solve this problem for those students who want to write off the stupid decision, and only then talk about the strangeness of this problem.

The first action we define the area that is occupied by apple trees in the garden. To do this, the total area of ​​the garden should be multiplied by the fractional expression of the area under apple trees.

80 * 5/8 = 50 ha

The second area of ​​action is determined that the garden occupied cherry trees. Take the common garden area, multiplied by the number of cherry trees per cent and 100 per cent share. Interest on interest reduced and as a result we get an area of ​​hectares.

80 * 31%: 100% = 24.8 ha

The area under apple trees we really get more than the area under the cherry trees. Takes away from smaller and larger area of ​​the results.

50 - 24.8 = 25.2 ha

Answer: The area under apple trees on 25.2 hectares more than the area under the cherry trees.

Without checking any decision can be considered incorrect. How to check the solution to this problem? It is necessary to put together the area under apple trees and the area under the cherry trees. This result should be compared with the total area of ​​the garden. If the sum is greater than the total area, so we decided not to challenge properly. If the sum is equal to or less than the total area, so our decision is correct.

50 + 24.8 = 74.8 hectares of less than 80 hectares

In most curious students immediately arises a natural question: what else is growing in this delicious daze, about what we were afraid to tell?

It was a children's nursery problem solution. Now the conversation for adults. This is the task of the textbook, which approved the Ministry of Education as an educational tool. The condition of this problem and at the same time used the percentage of fractional parts of a whole. Normal literate people'd never allow it. They used a ratio or percentage. Only an idiot is able to dump everything into one pile. The author of this task is illiterate idiot who either do not understand what he was doing, or for approval of a textbook command is ready to do anything. Stati, the quality of textbooks is very well characterizes the quality of all education. Our education is built on the principle of "one fools compose tutorials, others say they are fools".

For example, I'll write down the number on the same principle that is used by the author about the problem area of ​​the garden. I simultaneously use two forms of records: number and letter. That's what I got:

2 thousand three hundred 45

As you can see, only idiots can do so. Competent people will write down this number as follows:

2345 or two thousand three hundred forty-five

Why am I so vehemently opposed to such problems? Children - they are like a sponge that absorbs everything. If it is written in the book, then you can do so. As a result, we get the next portion of the idiots who write stupid textbooks that stupid leadership approves a flock of idiots. Just because they are so used to be taught.

What would you say there is not mathematics, but knowledge of mathematics lies not in the ability to accurately repeat all the things the teacher taught. Knowledge of mathematics - is the ability to competently and simply express their thoughts in the language of mathematics.


The six faces of a parallelepiped

Here is such a question: "Parallelepiped. Explain where he has six faces?". If mathematics is not able to clearly explain to you the parallelepiped design, then I'll try to do it. Will speak only about the faces of the parallelepiped, without delving into other structural details of the mathematical model, we are not in the showroom, and I'm not a manager trying to sell you an outdated model of the parallelepiped.

So, imagine that you, as if nothing had happened, went to sleep in his rectangular (this clarification is important) room. And at night you wake up in the current model of the box in full size! No need to panic. Easy start counting the brink of mathematical miracle. The wall with the window - this is the first bound. The wall opposite the window - this is the second face. The walls of the left and right of the window - this is the third and fourth edges. The floor - this is the fifth face. Ceiling - this is the sixth and final face. Great mathematical revelation: the number of faces does not depend on the order of their conversion, the main thing - do not miss it.

If you are up to this point has not yet fallen asleep, the next question is what to do next? Mentally, we develop a mathematical papyrus entitled "Set Theory", looking for the head of "Infinite math scores sheep" and start counting. People say this is very good mathematical procedure helps insomnia.

Just I want to honestly admit that I have lied a little. Not a rectangular room is a working model of a rectangular parallelepiped, but quite the contrary - a cuboid is a mathematical model of the room. This is particularly evident during the repair. The area of ​​the walls will be the surface area of ​​the side faces of the cuboid. The area of ​​floor or ceiling is determined as well as in the area of ​​the base of the box. Of course, the builders made their nuances in the rules for determining the areas of math, but we will not specify now.

Incidentally, the rectangular room is entirely dependent on the quality of construction. It is only in ancient Greece mathematics was so developed that famous building of the Parthenon in Athens was built with almost no right angles and straight lines. There's a base architecture of the building was laid not a mathematical perfection, and optical illusions. I'm afraid of modern mathematics such a task is not on the shoulder - too high in the clouds, they soar. But we digress from faces of a parallelepiped.

If you count the face of the parallelepiped you wanted during the day and not at night, then pull out from the wardrobe of a rectangular box with shoes. The bottom of the box - it is one face, it is also the lower base of the box. The lid of the box - a second side, she's upper base. The four walls of a shoebox - it faces the third to sixth.

We have considered six faces of a rectangular parallelepiped. If angles are not straight and curves? In this case we are dealing with a conventional parallelepiped, not rectangular. On the number of faces is not affected. Well, think, a little crushed a parallelepiped. By the way, mathematics bend rectangular parallelepipeds or align usual? I'm on the algebra of interesting to see. However, the mathematician's simple: uttered the sacred mantra "Let there be given a parallelepiped" and that he had gleaming white chalk on a blackboard. Life is complicated. There are many ways to straighten the curvature and parallelepipeds - by a heavy sledgehammer, to flirty "Oh, please!". About algebra of these methods can not even ask.

Seriously speaking, the algebra and rectangular, and the usual parallelepipeds is exactly the same. Bends and leveled a parallelepiped using sines of the angles between the ribs. In all corners of the rectangular parallelepipeds straight sinuses and their unity. Lazy math just does not write these sinuses in formulas. In conventional Parallelepipeds sines of the angles is less than one, so willy-nilly have to mathematicians in their formulas to write.

In conclusion, as they say teachers, fix the passed material. As a fixative use a simple children's coloring book, which paint the six faces of a parallelepiped.

The six faces of a parallelepiped. What is a parallelepiped. Mathematics for blondes.
The six faces of a parallelepiped

I recall that the parallelepiped, unlike the quadrangular prism opposing faces are parallel, and the base is a quadrilateral with parallel sides: square, rectangle, diamond or parallelogram. However, mathematicians believe that a parallelepiped is a kind of prism. So they have written in the definition. With the same success it can be argued that the prism is a kind of parallelepiped. Simply rewrite the definition, as a textbook on mathematics - it is not the Bible.