2016-II-39

考慮 $\mathrm{\Delta }ABC$，

$\begin{array}{rcl}A{C}^2& =& A{B}^2+B{C}^2\\ AC& =& \sqrt{{16}^2+{12}^2}\\ AC& =& 20\text{ cm}\end{array}$

考慮 $\mathrm{\Delta }QPC$，

$\begin{array}{rcl}P{Q}^2& =& Q{C}^2+P{C}^2\\ PQ& =& \sqrt{9^2+{10}^2}\\ PQ& =& \sqrt{181}\text{ cm}\end{array}$

考慮 $\mathrm{\Delta }APF$，

$\begin{array}{rcl}F{P}^2& =& A{P}^2+A{F}^2\\ FP& =& \sqrt{{10}^2+{24}^2}\\ FP& =& 26\text{ cm}\end{array}$

考慮 $\mathrm{\Delta }FHQ$，

$\begin{array}{rcl}F{Q}^2& =& F{H}^2+H{Q}^2\\ FQ& =& \sqrt{{20}^2+{15}^2}\\ FQ& =& 25\text{ cm}\end{array}$

考慮 $\mathrm{\Delta }FPQ$，

$\begin{array}{rcl}\cos \mathrm{\angle }PFQ& =& {\displaystyle \frac{F{Q}^2+F{P}^2–P{Q}^2}{2(FQ)(FP)}}\\ \cos \mathrm{\angle }PFQ& =& {\displaystyle \frac{{25}^2+{26}^2–(\sqrt{181}{)}^2}{2(25)(26)}}\\ \cos \mathrm{\angle }PFQ& =& {\displaystyle \frac{56}{65}}\\ \mathrm{\angle }PFQ& =& 30.510237{41}^{\circ }\\ \sin \mathrm{\angle }PFQ& =& {\displaystyle \frac{33}{65}}\end{array}$