Why is radian so common in maths?

I have learned about the correspondence of radians and degrees so 360° degrees equals 2π radians. Now we mostly use radians (integrals and so on)

My question: Is it just mathematical convention that radians are much more used in higher maths than degrees or do radians have some intrinsic advantage over degrees?

For me personally it doesn’t matter if I write \cos(360°) or \cos(2\pi). Both equals 1, so why bother with two conventions?

Answer

The reasons are mostly the same as the fact that we usually use base e exponentiation and logarithm. Radians are simply the natural units for measuring angles.

  • The length of a circle segment is x\cdot r, where x is the measure and r is the radius, instead of x\cdot r\cdot \pi/180.
  • The power series for sine is simply \sin(x)=\sum_{i=0}^\infty(-1)^i{x}^{2i+1}/(2i+1)!, not \sin(x)=\sum_{i=0}^\infty(-1)^i(x\cdot \pi/180)^{2i+1}/(2i+1)!.
  • The differential equation \sin (and \cos) satisfies is f+f”=0, not f+f”\pi^2/(180)^2=0.
  • \sin’=\cos, not \cos\cdot 180/\pi.

You could add more and more to the list, but I think the point is clear.

Attribution
Source : Link , Question Author : Slater , Answer Author : tomasz

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