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Radians

  

Figure: [top] Definition of the angle $\theta \equiv \ell/r$. [bottom] Illustration of the trigonometric functions $\cos(\theta) \equiv x/r$, $\sin(\theta) \equiv y/r$, $\tan(\theta) \equiv x/y$ etc. describing the position of a point B in circular motion about the centre at O.

In Physics, angles are measured in radians. There is no such thing as a ``degree,'' although Physicists will sometimes grudgingly admit that $\pi$ is equivalent to $180^\circ$. The angle $\theta$ shown in Fig. 10.1 is defined as the dimensionless ratio of the distance $\ell$ travelled along the circular arc to the radius r of the circle. There is a good reason for this. The trigonometric functions $\cos(\theta) \equiv x/r$, $\sin(\theta) \equiv y/r$, $\tan(\theta) \equiv x/y$ etc. are themselves defined as dimensionless ratios and their argument ($\theta$) ought to be a dimensionless ratio (a ``pure number'') too, so that these functions can be expressed as power series in $\theta$:
 
$\begin{array}[c]{rrrrrrrr} \cos(\theta) =& 1 & &- {\displaystyle {\theta^2 . . . . . . \over 3!} } & &+ {\displaystyle {\theta^5 \over 5!} } & \cdots \end{array}$
 
 
Why would anyone want to do this? You'll see, heh, heh . . . .