Będziemy korzystać z twierdzenia dotyczącego związku sinusów kątów α, β, γ z długościami odpowiednich boków trójkąta a, b, c (jak w zadaniu)

$\frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }$

 

$\text{a) }\frac{a}{\sin \alpha }=\frac{b}{\sin \beta }$  

$\frac{20}{\sin {45}^{\circ }}=\frac{24}{\sin \beta }\mid \cdot \sin \beta$

$\frac{20}{\frac{\sqrt{2}}{2}}\cdot \sin \beta =24$   

$20\cdot \frac{2}{\sqrt{2}}\cdot \sin \beta =24$  

$\frac{40}{\sqrt{2}}\cdot \sin \beta =24\mid \cdot \frac{\sqrt{2}}{40}$  

$\sin \beta ={\cancel{24}}^3\cdot \frac{\sqrt{2}}{{\cancel{40}}^5}$  

$\sin \beta =\frac{3\sqrt{2}}{5}$  

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$\text{b) }\gamma ={180}^{\circ }-\left({75}^{\circ }+{45}^{\circ }\right)={180}^{\circ }-{120}^{\circ }={60}^{\circ }$  

$\frac{c}{\sin \gamma }=\frac{b}{\sin \beta }$

  $\frac{36}{\sin {60}^{\circ }}=\frac{b}{\sin {45}^{\circ }}$

$\frac{36}{\frac{\sqrt{3}}{2}}=\frac{b}{\frac{\sqrt{2}}{2}}\mid \cdot \frac{\sqrt{2}}{2}$  

$b=\frac{36}{\frac{\sqrt{3}}{2}}\cdot \frac{\sqrt{2}}{2}$  

$b=36\cdot \frac{{\cancel{2}}^1}{\sqrt{3}}\cdot \frac{\sqrt{2}}{{\cancel{2}}^1}$  

$b=36\cdot \frac{\sqrt{2}}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}$  

$b=\frac{{\cancel{36}}^{12}\sqrt{6}}{{\cancel{3}}^1}=12\sqrt{6}$  

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$\text{c) }\frac{a}{\sin \alpha }=\frac{c}{\sin \gamma }$  

$\frac{11}{\sin \alpha }=\frac{17}{\sin {71}^{\circ }}\mid \cdot \sin {71}^{\circ }$  

$\frac{11}{\sin \alpha }\cdot \sin {71}^{\circ }=17\mid \cdot \sin \alpha$  

$11\cdot \sin {71}^{\circ }=17\sin \alpha \mid :17$

$\frac{11\cdot 0,9455}{17}=\sin \alpha$  

$\sin \alpha =\frac{10,4005}{17}\approx 0,612$   

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$\text{d) }\alpha ={180}^{\circ }-\left({48}^{\circ }+{26}^{\circ }\right)={180}^{\circ }-{74}^{\circ }={106}^{\circ }$  

Zanim przejdziemy dalej ze wzoru redukcyjnego zapiszemy sinus jako

$\sin {106}^{\circ }=\sin \left({180}^{\circ }-{74}^{\circ }\right)=\sin {74}^{\circ }$  

$\frac{a}{\sin \alpha }=\frac{c}{\sin \gamma }$

$\frac{16}{\sin {74}^{\circ }}=\frac{c}{\sin {48}^{\circ }}\mid \cdot \sin {48}^{\circ }$  

$c=\frac{16}{\sin {74}^{\circ }}\cdot \sin {48}^{\circ }$

$c=\frac{16}{0,9613}\cdot 0,7431$

$c\approx 16,6441\cdot 0,7431=12,3682\approx 12,37$