How to beat angles and jutes

The math of the sine

The sine in a right triangle


Fig. 7567 A right triangle (SVG)
Do you remember your geometry class? The calculation of triangles was certainly a very big deal at the time. We want to build on that and look at the triangle in Figure 7567. Do you still know the names for the three different sides of this triangle? We have


Fig. 7571 Four triangles (SVG)
Determine the values ​​by measuring the sides of the triangle in Fig. 7571 and using the sine formula:


Based on your calculations, you may already see a slight tendency: Even if you can't imagine a triangle, that's an angle of 90 ° or even 0 °, it can still make sense to expand the value range of the sine for these limit values. The exercise shows us that the following must probably apply:


The sine in the unit circle
It is quite obvious that there must be more behind the sine than the simple right triangle suggests. To determine the range of values ​​for any angle, we have to introduce the so-called unit circle. This will give us the possibility to get the sine for any angle without a lot of computational effort. Let's look at figure 7621. We are dealing with three circles in which different right triangles are drawn. We want to work out in four steps why the sine in the unit circle is so pleasant to treat.


Fig. 7621 Three circles (SVG)


Fig. 7597 Three unit circles (SVG)


Fig. 7622 The sine in the unit circle (SVG)
Is the sine of an angle are searched, then as in Figure 7622 this angle is drawn in a unit circle and the perpendicular leads from point A to the horizontal axis. The length from A to B is then the same . This length is also called the vertical component.
You may already recognize the greater benefit that can be gained from looking at the unit circle compared to right triangles: the angle can now assume any values. So in Figure 4854 are angles to recognize. The length from A to B is of course still the same as the sine of .

Fig. 4854 The unit circle with alpha> 90 °
Determine the values ​​for the sine from Figure 4854.


We immediately notice that the sine yields negative values ​​for angles greater than 180 °. In addition, it only covers a range of values ​​for all possible angles to , it never gets bigger or smaller. We hang on to:
For angles the sine is positive and always less than or equal to 1.
For angles the sine is negative and always greater than or equal to .
We are now able to assign a sine value to every angle from 0 ° to 360 °:


We should visualize this in a coordinate system.
Can you imagine how one can define the sine for angles even greater than 360 °? Look at figure 4856.

Fig. 4856 The sine for angles> 360 ° and their tracing back to the first period

Fig. 4857 The sine for negative alpha's

Fig. 4858 The entire sine function
Up to this point we have always given the angles in degrees (°). In physics, on the other hand, a different approach is chosen: the radian measure. Both sizes are completely equivalent, but the radian measure often has the advantage of clarity and naturally forms the literal arc to circular motion (more on this later).
Preview and definition

Fig. 4859 The radian measure
Against this, however, is for a fixed angle The relationship



Fig. 4860 The theorem of rays
constant, because


(see also Figure 4860) and has no unit (it cancels out). With this size you can clearly assign the swept distance on a circle to the angle. That should be reason enough for us to give this size a name:
With the designations in the adjacent figure, the radian measure of an angle (Unit: radian, rad) defined as the ratio:



Fig. 4861 4
Conversion from degrees to radians
Since the angle can be fully described by both degrees and radians, there should be a way to convert both sizes into one another. Let us first consider how the radian measure behaves when the angle changes. To do this, we take a look at Figure 4862.

Fig. 4862 Behavior of the radian measure when changing the angle
We see that doubling the angle is accompanied by a doubling of the radian measure. The following applies in general:
Radians and degrees are proportional to each other.
But what is the quantitative relationship between the two quantities? We can very easily think about this ourselves for an angle of 360 ° (Figure 4863):

Fig. 4863 An angle of 360 °
  1. The length of an arc that exceeds an angle of 360 ° is equal to the circumference of the circle. So:


  2. For the radian one obtains:


Since the radian measure is proportional to the degree measure, we can immediately calculate the respective radians for the three cases in Figure 4864:



Fig. 4864 The radian measure for three different angles
So that we can do this conversion for any angle, we need a general conversion formula. We hang on to:


The rest is a simple rule of three. Correspond to a degree


Radians. The conversion formula between the two angle sizes is thus:
If an angle is given in degrees, the radians can be calculated using the following formula:


or


The radians and the sine
Of course, the laws of the sine function do not change with the transition to radians. But a lot becomes clearer. Or if you at first glance that 1080 ° corresponds to three complete revolutions? is much more accessible to the eye. In addition, the sine does not necessarily have to have something to do with angles. For these cases we need abstract quantities, such as radians are (because is nothing else than ). The fundamental problem with the differentiation between the two perspectives is of a more technical nature: When you study the operating instructions for your pocket calculator, you will surely find that it can handle these two degrees, among other things. The pocket calculator calculates in degrees in "DEG" mode, but in radians in "RAD" mode. If you want to calculate the values ​​of the sine function with the help of the pocket calculator, you must first tell the pocket calculator in which degree the calculations should take place. If it is set to DEG, it will be for provide a different value than for , although both values ​​should give the same result depending on the perspective The following applies (use this to practice with your calculator!):


Also:


So that you can also admire the sine function in radians, Figure 4865 is recommended to you at this point.

Fig. 4865 The sine as a function of radians

Fig. 4867 The cosine in the unit circle
The following applies to the most important points:



The course of both functions is very similar, only that the cosine precedes the sine. This fact is even reflected in a separate term. It is also said that both functions are 90 ° resp. out of phase. This concept is relatively important, so we will devote an entire section to it later (section "The phase concept"). Figure 4868 shows the cosine function for two and a half periods.

Fig. 4868 The cosine function
However, there is another very important relationship between the sine and the cosine. In the accompanying text "Graphical Differentiation" it was shown that the derivative of the sine results in the cosine. If you derive the cosine, then you get If this function is derived again, one obtains etc. This cyclical behavior can be easily remembered in the following form:


Fig. 7572 Derivatives of the sine and cosine (SVG)
As a reminder: the sign is a mathematical notation for the derivative.
We have now dealt enough with the dry math of the sine. It is a good idea to summarize all the important facts of this chapter again: