Perhaps one of the most frustrating things we encounter when transitioning from primary school mathematics to secondary school mathematics is that textbooks replace the perfectly functional degree system with the radian system. That familiar 360^\circ for a full circle is inexplicably turned into an irrational number 2\pi, and a bunch of conversion formulas are added for good measure. This troubled me for quite a while back then. Why abandon a perfect 360^\circ in favor of an irrational number 2\pi? There are quite a few reasons involved here, and within these reasons, the consistency and simplicity of the mathematical system are once again reflected. Of course, the views in this article are merely my own opinions and are for reference only.
Radians: The Requirement of Simplicity
If the reader has already studied limit theory, then I can say directly that the introduction of the radian system is to satisfy the following condition under this specific measure of angles: \lim_{x\to 0} \frac{\sin x}{x}=1
It is not difficult to prove that only the radian system satisfies this limit. If we were to use degrees as the unit, we would have: \lim_{x\to 0} \frac{\sin x}{x}=\frac{\pi}{180}
This would appear far less elegant. What is the benefit of satisfying this limit? It is for the sake of further simplicity: only when this condition is met do we have the following concise formulas: (\sin x)'=\cos x, \quad (\cos x)'=-\sin x
Only with such concise formulas can the sine and cosine functions have simpler power series representations: \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots
And only under these conditions can we have the great Euler’s formula: e^{i\theta}=\cos\theta + i \sin\theta
From there, we arrive at the various magnificent mathematical achievements of complex analysis.
If we used degrees as the unit, then: (\sin x)'=\frac{\pi}{180}\cos x, \quad (\cos x)'=-\frac{\pi}{180}\sin x This would drive us to distraction. Therefore, we ultimately chose the radian system.
Radians: A Unified Description
In fact, there is another reason for using the radian system: it arises from our reflection on and generalization of the measurement of angles.
The angles we generally speak of are plane angles. In essence, an angle is the degree of opening between two rays originating from the same source. How do we measure this degree of opening? We first divide a full circle into 360 parts, each called a degree, then divide them further, and use this "protractor" to measure it. In primary school, we might have learned conversion rules like 1^\circ=60' and 1'=60''. This easily leads us to believe that a "degree" is a physical unit; in physical terms, that angles have dimensions. But in fact, angles are dimensionless; they do not have physical units like length or time. The degrees, minutes, and seconds we speak of are arbitrarily assigned by humans and do not reflect a physical reality.
In truth, measuring the degree of opening between two rays does not necessarily require the "protractor" mentioned above; it only requires calculating a ratio of lengths. For example, if we take an arbitrary point on one ray and drop a perpendicular to the other ray to form a right-angled triangle, the ratio of the opposite side to the hypotenuse—which is the value of \sin\theta—represents the magnitude of this opening. However, in this measurement method, \sin\theta does not change linearly with the uniform change of \theta, making it inconvenient to use. Later, mathematicians ingeniously conceived a definition for angles:
With the intersection of the angle as the center, draw a unit circle with a radius of unit length. The length of the arc intercepted by that angle is the magnitude of the angle.
This is the method mathematicians use to measure angles! Some readers might ask: doesn’t this give the angle a unit of length? Note that this is a unit circle; this is for convenience of description. It is actually the ratio of the arc length to the radius. Since it is a ratio, it naturally has no dimensions! In middle school, we learned that under the radian system, the arc length is l=R\theta, and we might even have been asked to prove it. Now you should understand that this is the definition of a radian! Asking us to prove it in middle school was actually putting the cart before the horse (though this is understandable). It is like asking, "Why does the first place belong to the top three?" Because "if you are not in the top three, you cannot be in first place!"
This definition of an angle allows us to easily generalize it to the concept of a "solid angle":
From a single point, extend three or more rays, and with this point as the center, draw a unit sphere. The area of the spherical polygon intercepted by these rays on the unit sphere is the magnitude of this solid angle.
Such a generalization is obvious and very useful. Of course, we will not further expand on its functions here. However, we can already see that the shift from degrees to radians reflects the fact that mathematicians pursue the harmony and simplicity of the mathematical whole!
Related Wikipedia Description:
http://zh.wikipedia.org/wiki/Trigonometric_functions
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