What is a sine?

Who among us did not shout at school that mathematics to himnever useful. It seemed to all of us that all these abstruse formulas, cumbersome equations and complex names have nothing to do with real life. But sooner or later all the knowledge we got at school finds its application. And knowing what a sine, cosine or tangent can save your reputation.

A bit of school geometry

So, the sine is the aspect ratio in a right-angled triangle. Let's recall what the rectangular triangle consists of.

Angles. The sum of the angles in the triangle is 180about. The forward angle is 90about. Hence, the other two in the sum should also give 90about. That is, we have one right angle and two sharp ones.

Parties. The rectangular triangle consists of a hypotenuse and two legs. Two legs form a right angle, and the hypotenuse lies opposite it.

What is the sine of an angle? As already mentioned, this aspect ratio. But which ones? The sinus of an acute angle is the ratio of the leg, which lies opposite this angle, to the hypotenuse. Consider the example:

The sine of angle A is the ratio of side a (opposite leg) to side b (hypotenuse).

The sine of angle C is the side relation with (the cathete lies opposite side C) to side b (hypotenuse).

That is, if the sides are equal to a = 3, c = 4, b = 5, then the sine of angle A will be 3/5, and the sine of angle C will be 4/5.

What does this give us? So far, nothing, but let's look at another example. Let's increase the triangle by extending the sides. Now we have done this:

As can be seen from the figure, the sides' lengths increased, but the corners did not. But what is most interesting - the ratio also did not change!

Suppose, d = 6, k = 8, m = 10. Then the sine of the angle A is d / m = 6/10. We cut it by two sides of the equation and get the same 3/5, as in the first case! And no matter how you change, extend or shorten the parties, the attitude of the parties will still be the same.

Therefore it is clear that the sine is a constant value.

And now - trigonometry

The ancient Greeks noticed this for a long time. They calculated the sinuses of the main corners and recorded them, in order to continue to use the already ready quantities, and not to invent new ones.

In addition to the sine, the angle also has a cosine(the relation of the adjacent leg to the hypotenuse), the tangent (the ratio of the opposing leg to the adjacent) and cotangent (the ratio of the adjacent leg to the opposite). All these quantities are called trigonometric functions of the angle, and are used for calculations and problem solving.

Each time you do not need to calculate the sine. There are specially compiled Bradis tables, in which all the sines, cosines, tangents and cotangents are already recorded. From here we receive information. For example, if we know the angle, we know its sine and cosine. Or vice versa - if a sine or cosine is known - we can easily find which angle is given.

Naturally, these trigonometric functions are huge. Remember them all is simply impossible, but actually not necessary. They use, in the main, only some of them.

A little bit about the corners

But trigonometric functions are not onlysharp and right angles, they are also for the stupid, but here for their finding, a circle and a graph of the coordinate axes will already be needed. And this is a completely different story.

Sinewave

Now let's see what a sinusoid is. It looks like a sinusoid like this:

And is a graph of the sine change independing on the angle change. As mentioned above, the sides can change, and the angle remains the same - then the sine will be unchanged. But if the angle changes, then the aspect ratio changes, and, consequently, the sine value.

The sinusoid displays numerical changes in the sineangle and is a graph of the function y = sin (x). There is nothing complicated here, especially since the values ​​of the sines of all angles are written in the Bradys tables. But we will remember only the most basic.

A little more about the designation of angles

Everyone knows that angles are measured in degrees orradians. Degrees we measure with a protractor, which looks like a semicircle. One degree is 1/360 of a circle. Why so? Because any angle can be "opened" or "closed". You can even open it for the entire turn and get a circle.

Full turnover, as is known, 360about. A straight line is an angle of 180about. That is, the diameter of the circle is 180about.Or the number Pi. Therefore, it turns out that the angle can be as 90about(in degrees), and Pi / 2 (in radians).

Now try to remember the most basic sinuses.