How to memorize trigonometric functions?

How to memorize trigonometric functions? This question is set to itself very many at the study of trigonometric functions. On own experience I know that this business is not simple. Through many years I succeeded to carry two moments only, that behave to the trigonometric functions. First, that well burned into my memory, this determination of trigonometric functions generally: a trigonometric function is this relation something to something. Thus in memory at once there is a shot, at that I do not remember that stands in a numerator forgot that was in a denominator. From all trigonometric relations I remember only a tangent: a tangent is a sine on a cosine. A sine costs on the first place, he means in the numerator of shot. A cosine stands on the second place, he means in the denominator of shot. Sines with cosines I generally on life always mixed up. To the last time...

I succeeded to find a that failing element that so was not enough in the days of my educating. Today even I can faultlessly dab with a finger in a cosine to tell that on what in him divided.

The portrait of tangent appeared this magic stick. Yes, yes, not surprised - exactly portrait, at a look to that faultlessly know a tangent. This portrait of tangent is only work by oil of genius artist of beginning twenty the first century, that on auction of Sothebys in 2137 was sold for one hundred million dollars. Oddly enough, but a picture is written was on reverse side of canvas. Where do I it know from? I am a mathematician and I am able to see the future. By the way, yesterday I spoke with this artist, he asserts that such picture did not yet write, but an idea very pleased him. Specially for you I publish reproduction of this masterpiece here.

Masterpiece. Tangent portrait.

Now we have all for creation of our own trigonometric Bible that will need to be learned by heart. Do not be frightened, in our Bible there will be only one line:

Tangent - is sine by cosine

One-only picture from this Bible I already showed you. Now, as the most real preacher, I will teach you all of it it is correct to read. I think, the best method of educating will be a gape-seed of trigonometric comics. Let in the near time by it and we will engage: trigonometric functions.

Why are sines and cosines needed?

Why are sines and cosines needed? Really, interesting question. In comments to the trigonometric circle of sines and cosines such question appeared:

where will sin and cos be useful in life?
p.s why are they needed sines cosines?

Let us will call a spade a spade. To swingeing majority from you they will be never useful. Unless, when will your children go out into school and will begin to study trigonometric functions, they too will put question you "Why are sines and cosines needed?" and, in addition, will ask to explain, what is it.

Money we use every day already not alone thousand years and perfectly we do without every sines, cosines and other elegant mathematical pieces. I assure you, and through millions of years in the count of money nothing will change. Not because we are such dull, and because such are mathematical properties of money : it is impossible to increase roubles on roubles and with money in the second degree to hurry in a motor show to buy "Lamborghini".

On a kitchen, in culinary recipes, you will meet neither sines nor cosines too. If to give a glance soberly on our everyday life, then all our everyday mathematics remains somewhere at the level of knowledge of Ancient Greece. We are enough with a head.

So why are sines and cosines needed? As compared to Ancient Greece, we have very much different pieces about that ancient Greeks could not dream even today. Even their Gods did not ride on machines, did not use mobile communication, did not communicate on the Internet. But we have all of it and we use this constantly. Did all this extraordinary riches undertake from where? He was created by us. At first scientists did the scientific opening. Then engineers, on the basis of done by the scientists of opening, created every useful things. We use these things today, not having not the least concept about that is into these things and what scientific laws are fixed in basis of their work. So, if there were not sines and cosines, there would not be all these useful things.

Sines and cosines are used most effectively scientists and engineers. I will not say that they continuously trigonometric functions are used only. No, they use them rarely, but well-aimed. Sines and cosines often are in the formulas of different calculations an engineer or scientific.

Often with sines and cosines it is necessary to clash to the geodesists. They have the special instruments for an exact goniometry. Through sines and cosines corners can be converted into lengths or coordinates of points on an earth surface.

The teachers of mathematics on the sort of the duties constantly deal with trigonometric functions. This year they told about sines and cosines to you, the next year the teachers of mathematics will tell the same to other students. Such for them work - to teach.

Schoolchildren and students study trigonometric functions on the lessons of mathematics. Personally I got through tortures sines and cosines at school, техникуме, institute.

Adults sometimes engage in sines and cosines then, when their toschoolchildren need a help at preparation of homeworks.

All! Sines and cosines do not need other generally! In everyday life most people they are not used hardly ever. If I wrong, remedy me.

So why then generally to teach these sines and cosines? Well, firstly, such is the school program. Secondly, if you in life may need apply a sine or cosine, you know already, what is it and where it is needed to search information about them. The knowledge gained at school will fully have you, what is independent in everything to understand.

So what such the sines, cosines and other trigonometric functions? It is a mathematical instrument it is needed that to be able to use. That we this instrument we do not use hardly ever, talks not that studying them is not necessary, and that efficiency of application of the knowledge gained by us is practically equal to the zero. But it is quite another theme already.

How to calculate the area of surface of direct three-cornered prism

How to calculate the area of surface of direct three-cornered prism? For this purpose the best of all to imagine a three-cornered prism in all her beauty. How does a three-cornered prism look? Look at a picture

Three-cornered prism

Now next question: do you have a tube of lipstick in form three-cornered prism? If you answered this question (and or no) though as, means now you already know about what three-cornered prism and as she looks. Here those two triangles, from above and from below, are named the grounds of prism. Three rectangles on each side are named the verges of prism. As a triangle lies in founding of prism, verges for us exactly three. If there will be a pentagon in founding, verges it will be five. And if is there 1234-cornered in founding of prism? Correctly, verges there will be 1234 things. With the construction of direct prism we understood, after it it is possible to take up the mathematical calculations of area of surface of three-cornered prism.

As a prism is a geometrical body, her structure can be investigated on the example of body of blonde. That is included in the area of surface of prism? If you will come a heel on something sharp or knocked by the обо-что top of the blond head, it will be badly you. That is why both heels and top of head enter in the complement of surface of your body. Just overhead and lower triangles are included in the area of surface of three-cornered prism. If to dab with you a finger in a side, you will say "Ouch"! because a side belongs to your body. Here those three rectangles, that are from the sides of three-cornered prism, make the area of her side.

By the result of our scientific researches of bodies of blonde and three-cornered prism, we came to the conclusion, that the area of surface of three-cornered prism consists of areas overhead and lower grounds (they are equal) and area of side. The area of triangle needs to be found on one of formulas. I remind that the area of triangle found on a formula at the calculation of area of surface of prism needs to be taken two times, id est to increase her on two.

The area of side of prism is determined as a sum of areas of rectangular verges. It is needed to increase length of every side of triangle (that triangle that is in founding) on the height of prism and lay down three got areas together. It is possible to act simpler: to increase the perimeter of triangle (a perimeter of triangle is a sum of lengths of all his parties) on the height of prism. But adding up not to avoid you in any case: either to the increase (we fold long parties) or after an increase (we fold the areas of rectangular verges). Another trouble that you can lie in wait on a way to the area of surface of three-cornered prism is this absence of values of all lengths of parties. It is not mortal for the calculation of area of triangle. And for the area of side of prism you will have preliminary to find lengths of all three parties of triangle, applying the mathematical knowledge.