Square Trapped in Pentagon

$$ See the picture below.

Square Trapped in Pentagon

Apply Sine rule in $\Delta\text{ASP}$ and $\Delta\text{PBQ}$ .

$\bullet\;\;\Delta\text{ASP}$

$\begin{aligned} \frac{y}{\sin 36^\circ}&=\frac{b}{\sin 108^\circ}\\ y&=\frac{\sin 36^\circ}{\sin 108^\circ}b \end{aligned}$

$\bullet\;\;\Delta\text{PBQ}$

$\begin{aligned} \frac{a-y}{\sin 18^\circ}&=\frac{b}{\sin 108^\circ}\\ a-y&=\frac{\sin 18^\circ}{\sin 108^\circ}b\\ a-\frac{\sin 36^\circ}{\sin 108^\circ}b&=\frac{\sin 18^\circ}{\sin 108^\circ}b\\ a&=\frac{\sin 36^\circ}{\sin 108^\circ}b+\frac{\sin 18^\circ}{\sin 108^\circ}b\\ \frac{a}{b}&=\frac{\sin 36^\circ+\sin 18^\circ}{\sin 108^\circ}\\ &=\frac{2\sin 18^\circ\cos 18^\circ+\sin 18^\circ}{\sin (90^\circ+18^\circ)}\\ &=\frac{2\sin 18^\circ\cos 18^\circ+\sin 18^\circ}{\cos 18^\circ}\\ &=2\sin 18^\circ+\tan 18^\circ. \end{aligned}$ $$ Thus, $\Psi+\Theta+\Phi=2+18+18=\boxed{38}$ . $$

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$\text{\# }\mathbb{Q.E.D.}\text{ \#}$