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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.
\begin{figure}
\begin{center}\mbox{\epsfig{file=PS/radian.ps,height=2.0in} }\end . . . 
 . . . gin{center}\mbox{\epsfig{file=PS/trig.ps,height=2.0in} }\end{center}\end{figure}

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 . . . .



Jess H. Brewer
1998-10-08