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Is units of angular displacement never a degree and is always a radian ?

  • $\begingroup$ It is always radians if there is any integration or differentiation to be done - I suspect this is why the textbook says so, to avoid confusion. In common use, degrees are more likely. $\endgroup$ Jan 29 at 13:17

I suspect this book is related to motion dynamics. The reason why it states that angular measurements are in radians only, is probably because it tries to avoid confusion and ambiguity.

The problem is that in most technical subjects degrees are used for angular measurements. However, when you start delving into physics and calculus- which are a prerequisite in your case since you are probably reading about dynamics- using radians make more sense. There are two main reasons (+1 which some times is as important) for that.

  1. There is very straight forward relationship between the arc of a circle $L$ and the angle $\theta$ when it is represented in radians. $$L = r\cdot \theta[rad]$$

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The same equation in degrees would be: $L = r\cdot \theta[deg] \cdot\frac{\pi}{180}$. Notice the extra factor $\frac{\pi}{180}$

  1. Similarly (and more importantly) angles in radians produce much simpler forms for angular displacement derivatives.

So for example:

$$\frac{d }{dx} \sin\theta[rad] = \cos\theta[rad]$$

The equivalent for degrees would be:

$$\frac{d }{dx} \sin\theta[deg] = \frac{\pi}{180} \cos\theta[deg]$$

It gets even worse for the double derivative with respect to time because then you have:

$$\frac{d^2 }{dx^2} \sin\theta[deg] = -\left(\frac{\pi}{180}\right)^2 \sin\theta[deg]$$

while when you are using radians its simply:

$$\frac{d^2 }{dx^2} \sin\theta[rad] = -\sin\theta[rad]$$

  1. Small angle approximation $\sin\theta\approx \theta$ works better with radians (because $1 [deg]\approx \frac{1}{56}rad$).

Bottom Line: You can use both degrees and radians for angular displacement, however when you need to take derivatives wrt to time, radians make it easier.


Often seen instructions to rotate something about the centre by 90 degrees or 180 degrees...

But for use in calculations radians can be easier for units.

  • $\begingroup$ It is not like it is never used right? Radians and degree both units can be used ? $\endgroup$
    – S.M.T
    Jan 29 at 10:44
  • $\begingroup$ @user15072279, right. Degrees can be and are used for angular displacements. $\endgroup$
    – NickB
    Jan 29 at 11:27

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