第14章 14.6

(1)

$A = 180 - 75 - 45 = 60$

$c = a\frac{\sin A}{\sin C} = 6\frac{\sin 60}{\sin 45} = 6\frac{\dfrac{\sqrt3}{2}}{\dfrac{1}{\sqrt2}} = 3\sqrt6$

$S = \frac{1}{2}ac\cdot\sin B = \frac{1}{2}\cdot 6\cdot 3\sqrt6\cdot\sin(45+30) = 9\sqrt6\cdot\frac{\sqrt3+1}{2\sqrt2} = \frac{9}{2}\left(3+\sqrt3\right)$

(2)

$C = 180 - 30 - 105 = 45$

$a = 2R\sin A = 6\sin 30 = 3$

$c = 2R\sin 45 = 6\sin 45 = 3\sqrt2$

$S = \frac{1}{2}ac\sin B = \frac{9\sqrt2}{2}\sin(45+60) = \frac{9\sqrt2}{2}\cdot\frac{\sqrt3+1}{2\sqrt2} = \frac{9\sqrt3+9}{4}$

(3)

$\frac{1}{2}(a+b+c) = 9$

$S = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{216} = 6\sqrt6$