The Meaning of $u=c\sin\theta$ Equation in Special Relativity In the Introductory Special Relativity book, by W G V Rosser, page 19, the author is formulating the relativistic kinetic energy equation. In his derivation, he writes the following equation:
$$u=c\sin\theta$$
I have two questions regarding Equation (2.7), the above equation, in the book:

*

*How did the author prove this equation?


*Could anyone please explain the physical interpretation of this equation?
Any help is much appreciated. Thank you so much.

 A: In this passage by Rosser, $u=c\sin\theta$ is the spatial velocity, where $\theta$ is a parameter used in a "Loedel diagram" (see Wikipedia entry
and Paul Gruner's Elementary geometric representation of the formulas of the special theory of relativity), which is a limited variant of the Minkowski spacetime diagram that tries to use Euclidean trigonometry instead of the more appropriate hyperbolic trigonometry (which uses the rapidity parameter $w$, geometrically interpreted as the Minkowski-arc-length of the unit hyperbola).
So,
$$\text{spatial velocity}=u=c\sin\theta=c\tanh w.$$
[Incidentally, $\theta=\text{gd}(w)$ where $\text{gd()}$ is the Gudermannian function which could be defined as $\text{gd}(w)=\arcsin(\tanh w)$.]
With the above definitions, the time-dilation function is
$$\gamma=(\text{Rosser's}\ \alpha)=\frac{1}{\sqrt{1-u^2/c^2}}=\sec\theta=\cosh w,$$
and the spatial-component of the 4-momentum [the relativistic momentum] is
$$\begin{align*}p 
&=\alpha mu =\sec\theta\ m (c \sin\theta)=mc\tan\theta\\
&=\gamma mu
=\cosh w(mc\tanh w)=mc\sinh w\end{align*}.$$
As noted by one of the commenters, the $\theta$-parameter is a convenient choice of variables that make the equations of special relativity easier to analyze with Euclidean trigonometry.
There is a limited geometric interpretation of $\theta$ using the Loedel diagram.
A fuller explanation of this aspect would require a detailed treatment of the Loedel diagram.
In my opinion, the Minkowski diagram and its hyperbolic trigonometry using rapidity $w$ is superior (and does not suffer from the limitations of the Loedel diagram). But I can't elaborate on this point here.
