a)

Korzystamy z twierdzenia cosinusów i mamy:

$c^2=a^2+b^2-2ab\cos \gamma$

$c^2=4^2+{\sqrt{3}}^2-2\cdot 4\cdot \sqrt{3}\cos 30^{\circ }$

$c^2=16+3-8\sqrt{3}\cdot \frac{\sqrt{3}}{2}$

$c^2=19-12$

$c^2=7|\sqrt{},c>0$

 $c=\sqrt{7}$

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b)

Korzystamy z twierdzenia cosinusów i mamy:

$a^2=b^2+c^2-2bc\cos \alpha$

$a^2=5^2+{\left(3\sqrt{2}\right)}^2-2\cdot 5\cdot 3\sqrt{2}\cdot \cos 45^{\circ }$

$a^2=25+9\cdot 2-30\sqrt{2}\cdot \frac{\sqrt{2}}{2}$

$a^2=25+18-30$

$a^2=13|\sqrt{},a>0$

$a=\sqrt{13}$

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c)

Korzystamy z twierdzenia cosinusów i mamy:

$c^2=a^2+b^2-2ab\cos \gamma$

$c^2=2^2+6^2-2\cdot 2\cdot 6\cdot \cos 120^{\circ }$

$c^2=4+36-24\cos \left(180^{\circ }-60^{\circ }\right)$

$c^2=40-24\cdot \left(-\cos 60^{\circ }\right)$

$c^2=40+24\cos 60^{\circ }$

$c^2=40+24\cdot \frac{1}{2}$

$c^2=40+12$

$c^2=52|\sqrt{},c>0$

$c=\sqrt{52}$

$c=\sqrt{4\cdot 13}$

$c=2\sqrt{13}$

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d)

Korzystamy z twierdzenia cosinusów i mamy:

$b^2=a^2+c^2-2ac\cos \beta$

$b^2={\left(2\sqrt{3}\right)}^2+4^2-2\cdot 2\sqrt{3}\cdot 4\cos 150^{\circ }$

$b^2=4\cdot 3+16-16\sqrt{3}\cdot \cos \left(180^{\circ }-30^{\circ }\right)$

$b^2=12+16-16\sqrt{3}\cdot \left(-\cos 30^{\circ }\right)$

$b^2=28+16\sqrt{3}\cdot \cos 30^{\circ }$

$b^2=28+16\sqrt{3}\cdot \frac{\sqrt{3}}{2}$

$b^2=28+24$

$b^2=52|\sqrt{},b>0$

$b=\sqrt{52}$

$b=\sqrt{4\cdot 13}$

$b=2\sqrt{13}$