Square Trapped in Pentagon

Geometry Level 4

The square \(\color{blue}{\text{PQRS}}\) is inscribed in the regular pentagon \(\color{red}{\text{ABCDE}}\) as shown.

If the side length of the pentagon is \(\,a\) and the side length of the square is \(\,b\), then the ratio of \(\;\dfrac{a}{b}\) can be expressed as \(\,\Psi\sin\Theta+\tan\Phi\), where \(\Theta\) and \(\Phi\) are in degrees. Find the value of \(\;\Psi+\Theta+\Phi\).

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