a)

$a=16\left[\text{cm}\right],\alpha ={45}^{\circ }$

Korzystając z twierdzenia sinusów dostajemy:

$\frac{a}{\sin \alpha }=2R$

$\frac{16}{\sin {45}^{\circ }}=2R$

$\frac{16}{\frac{\sqrt{2}}{2}}=2R$

$16\cdot \frac{2}{\sqrt{2}}=2R\mid \cdot \frac{1}{2}$

$\frac{16}{\sqrt{2}}=R$

$R=\frac{16}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{16\sqrt{2}}{2}=8\sqrt{2}\left[\text{cm}\right]$       

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b) 

$c=9\left[\text{cm}\right],\gamma ={60}^{\circ }$  

Korzystając z twierdzenia sinusów dostajemy:

$\frac{c}{\sin \gamma }=2R$

$\frac{9}{\sin {60}^{\circ }}=2R$

$\frac{9}{\frac{\sqrt{3}}{2}}=2R$

$9\cdot \frac{2}{\sqrt{3}}=2R\mid \cdot \frac{1}{2}$

$\frac{9}{\sqrt{3}}=R$

$R=\frac{9}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{9\sqrt{3}}{3}=3\sqrt{3}\left[\text{cm}\right]$  

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c)

$b=24\left[\text{cm}\right],\beta ={150}^{\circ }$  

Korzystając z twierdzenia sinusów dostajemy:

$\frac{b}{\sin \beta }=2R$

$\frac{24}{\sin {150}^{\circ }}=2R$

$\frac{24}{\sin \left({180}^{\circ }-{30}^{\circ }\right)}=2R$

$\frac{24}{\sin {30}^{\circ }}=2R$  

$\frac{24}{\frac{1}{2}}=2R$  

$24\cdot 2=2R\mid \cdot \frac{1}{2}$

$R=24\left[\text{cm}\right]$

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d)

$c=28\left[\text{cm}\right],\gamma ={48}^{\circ }$  

Korzystając z twierdzenia sinusów dostajemy:

$\frac{c}{\sin \gamma }=2R$

$\frac{28}{\sin {48}^{\circ }}=2R\mid \cdot \frac{1}{2}$

$R=\frac{14}{0,7431}\approx 18,84\left[\text{cm}\right]$  

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e)

$a=10\left[\text{cm}\right],\alpha ={128}^{\circ }$  

Korzystając z twierdzenia sinusów dostajemy:

$\frac{a}{\sin \alpha }=2R$

$\frac{10}{\sin {128}^{\circ }}=2R\mid \cdot \frac{1}{2}$

$R=\frac{5}{\sin \left({180}^{\circ }-{52}^{\circ }\right)}=\frac{5}{\sin {52}^{\circ }}\approx \frac{5}{0,7880}\approx 6,35\left[\text{cm}\right]$    

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f)

$b=120\left[\text{cm}\right],\beta ={144}^{\circ }$  

Korzystając z twierdzenia sinusów dostajemy:

$\frac{b}{\sin \beta }=2R$

$\frac{120}{\sin {144}^{\circ }}=2R\mid \cdot \frac{1}{2}$

$R=\frac{60}{\sin \left({180}^{\circ }-{36}^{\circ }\right)}=\frac{60}{\sin {36}^{\circ }}\approx \frac{60}{0,5878}\approx 102,08\left[\text{cm}\right]$