第14章 14.8

(1)

$S = \frac{1}{2}bc \cdot \sin A = \frac{1}{2} \cdot 5 \cdot 4 \cdot \sin 60 = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}$

(2)

$s = \frac{1}{2}(a+b+c) = \frac{1}{2}(13+14+15) = 21$

$S = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84$

(3)

$A = 180 - 60 - 45 = 75$

$\sin 75 = \sin(30+45) = \sin 30 \cos 45 + \cos 30 \sin 45 = \frac{1+\sqrt{3}}{2\sqrt{2}}$

$b = \frac{\sin B}{\sin A}a = \frac{\sin 60}{\sin 75}3 = \frac{\dfrac{\sqrt{3}}{2}}{\dfrac{1+\sqrt{3}}{2\sqrt{2}}}3 = \frac{3\sqrt{6}}{1+\sqrt{3}}$

$S = \frac{1}{2}ab \cdot \sin C = \frac{1}{2}\frac{9\sqrt{6}}{1+\sqrt{3}}\frac{1}{\sqrt{2}} = \frac{9\sqrt{3}}{2(1+\sqrt{3})} = \frac{9}{4}\sqrt{3}(-1+\sqrt{3}) = \frac{9}{4}(3-\sqrt{3})$

(4)

$s = \frac{1}{2}(a+b+c) = 27$

$S = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{27 \cdot 14 \cdot 7 \cdot 6} = \sqrt{15876} = 126$