Here we have only the first reflection.

What we need here is the thickness of the thinnest soap film, $n=1.33$, considering the appearance of a bright stripe. The light that is used in this problem has a wavelength $lambda=5.75times10^{-7}text{ m}$

The definition of thickness when black stripe appears is

$$
begin{align}
d=frac{1}{2}cdotleft(m+frac{1}{2} right)cdotfrac{lambda}{n_f}
end{align}
$$

For a thinnest film $m=0$, but for the second thinnest width of a film $m=1$. So, we have

$$
begin{align}
d&=frac{1}{2}cdotleft(1+frac{1}{2} right)cdotfrac{5.75times10^{-7}text{ m}}{1.33} \
&boxed{d=3.24times 10^{-7}m}
end{align}
$$

$$
d=3.24times 10^{-7}m
$$

$a)$ The light ray goes through the air, $n=1$, and hits the plastic film, $n_f=1.83$. Here the first reflection and phase inversion happens because $n<n_f$.

Another situation is when ray goes through a film and hits the air. Since $n<n_f$, on the second reflection there is no phase inversion.

We need to find the thinnest film that reflects yellow-green light.

Let’s start with known equation

$$
begin{align}
d=frac{1}{2}cdotleft(m+frac{1}{2} right)cdotfrac{lambda}{n_f}
end{align}
$$

For a thinnest film $m=0$. So, we have

$$
begin{align}
d&=frac{1}{2}cdot 1cdotfrac{5.55times10^{-7}text{ m}}{1.83} \
&boxed{d=7.58times 10^{-8}m}
end{align}
$$

$b)$ For the second thinnest width of a film $m=1$. So, we have

$$
begin{align}
d&=frac{1}{2}cdotleft(1+frac{1}{2} right)cdotfrac{5.55times10^{-7}text{ m}}{1.83} \
&boxed{d=2.27times 10^{-7}m}
end{align}
$$

$a)$ $d=7.58times 10^{-8}m$

$b)$ $d=2.27times 10^{-7}m$

Small-angle approximation for $sin$ and $tan$

$$
begin{align}
sintheta&approxtheta \
tantheta&approxtheta \
sintheta&approxtantheta
end{align}
$$

is defined up to $9.91^{circ}$.

If we increase precision in our approximation, we need to add more elements in it so it becomes

$$
begin{align}
sintheta&approxtheta-frac{theta^3}{3!}+frac{theta^5}{5!}-… \
tantheta&approxtheta-frac{theta^2}{2!}+frac{theta^4}{4!}-… \
end{align}
$$

So the angle changes to $2.99^{circ}$