My textbook, Fundamentals of Photonics, Third Edition, by Saleh and Teich, gives the following:
This seems to be mathematically incorrect to me?
Firstly, the author stated that $\phi = \psi - \theta_1 \approx \dfrac{y}{-R} - \theta_1$, and then substitutes this into $\theta_2 = 2\phi + \theta_1$ to get $\theta_2 = 2\phi + \theta_1 = 2\left[ \dfrac{y}{-R} - \theta_1 \right] + \theta_1$. But shouldn't this be $\theta_2 = 2\phi + \theta_1 \approx 2\left[ \dfrac{y}{-R} - \theta_1 \right] + \theta_1$?
And lastly, the author stated that $y \approx y_1 + \theta_1 z_1$, and then substitutes this into $\dfrac{2y}{-R} - \theta_1$ to get $\dfrac{2y}{-R} - \theta_1 = \dfrac{2(y_1 + \theta_1 z_1)}{-R} - \theta_1$. But shouldn't this be $\dfrac{2y}{-R} - \theta_1 \approx \dfrac{2(y_1 + \theta_1 z_1)}{-R} - \theta_1$?
Taking all of this into account, the result would be
$$\theta_2 = 2\phi + \theta_1 \approx 2\left[ \dfrac{y}{-R} - \theta_1 \right] + \theta_1 = \dfrac{2y}{-R} - \theta_1 \approx \dfrac{2(y_1 + \theta_1 z_1)}{-R} - \theta_1,$$
which, if my understanding is correct, is a very different mathematical result, in terms of the mathematical conclusions we can draw from this, than what the author has, due to the differences between $=$ and $\approx$ and how we treat them in mathematical calculations.
I'm not familiar with how physicists go about their calculations, but, if my understanding of the mathematics is correct, $\approx$ is not necessarily transitive, so if we have that $A \approx B$ and $B \approx C$, it is not necessarily true that we therefore have $A \approx C$?
The author illustrates what I mean here in their next conclusion, which is a consequence of the, what I believe to be, erroneous mathematics of the last result:
If my understanding of the mathematics is correct, then due to the differences between $=$ and $\approx$, we cannot simply treat $\approx$ as $=$ and draw conclusions in a "chain" of equations and approximations such as $\theta_2 = 2\phi + \theta_1 \approx 2\left[ \dfrac{y}{-R} - \theta_1 \right] + \theta_1 = \dfrac{2y}{-R} - \theta_1 \approx \dfrac{2(y_1 + \theta_1 z_1)}{-R} - \theta_1$. Here, we have a case of $A = \theta_2 \approx B = 2\left[ \dfrac{y}{-R} - \theta_1 \right] + \theta_1$ and $B \approx C = \dfrac{2(y_1 + \theta_1 z_1)}{-R} - \theta_1$, and the author assumes that $\approx$ is transitive so that, logically, we have $(A \approx B) \land (B \approx C) \Rightarrow (A \approx C)$. I do not think this is correct?
I think my understanding of the mathematics is correct here, but I wonder if there is something about physics conventions that I am unaware of that makes this actually acceptable in physical calculations?
I would greatly appreciate it if people could please take the time to clarify this.