a)

$\alpha +\beta +\gamma ={180}^o$  

${45}^o+{60}^o+\gamma ={180}^o$  

${105}^o+\gamma ={180}^o$  

$\gamma ={75}^o$  

$\sin \alpha =\sin {45}^o\approx 0,7071$  

$\sin \beta =\sin {60}^o\approx 0,8660$  

$\sin \gamma =\sin {75}^o\approx 0,9659$  

 

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

$a=\frac{c\cdot \sin \alpha }{\sin \gamma }\approx \frac{6\cdot 0,7071}{0,9659}\approx \frac{4,2426}{0,9659}\approx 4,39$  

 

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

$b=\frac{c\cdot \sin \beta }{\sin \gamma }\approx \frac{6\cdot 0,8660}{0,9659}\approx \frac{5,196}{0,9659}\approx 5,38$  

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b)

$\alpha +\beta +\gamma ={180}^o$  

${45}^o+{105}^o+\gamma ={180}^o$  

${150}^o+\gamma ={180}^o$  

$\gamma ={30}^o$  

$\sin \alpha =\sin {45}^o\approx 0,7071$  

$\sin \beta =\sin {105}^o=\sin \left({90}^o+{15}^o\right)=\cos {15}^o\approx 0,9659$  

$\sin \gamma =\sin {30}^o=0,5$  

 

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

$a=\frac{c\cdot \sin \alpha }{\sin \gamma }\approx \frac{8\cdot 0,7071}{0,5}\approx 11,31$  

 

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

$b=\frac{c\cdot \sin \beta }{\sin \gamma }\approx \frac{8\cdot 0,9659}{0,5}\approx 15,45$  

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c)

$\alpha +\beta +\gamma ={180}^o$   

${30}^o+{30}^o+\gamma ={180}^o$  

${60}^o+\gamma ={180}^o$  

$\gamma ={120}^o$  

$\sin \alpha =\sin \beta =\sin {30}^o=0,5$  

$\sin \gamma =\sin {120}^o=\sin \left({90}^o+{30}^o\right)=\cos {30}^o\approx 0,8660$  

 

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

$a=\frac{c\cdot \sin \alpha }{\sin \gamma }\approx \frac{10\cdot 0,5}{0,8660}\approx 5,77$  

 

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

$b=\frac{c\cdot \sin \beta }{\sin \gamma }\approx \frac{10\cdot 0,5}{0,8660}\approx 5,77$  

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d)

$\alpha +\beta +\gamma ={180}^o$  

$\alpha +{60}^o+{45}^o={180}^o$  

$\alpha +{105}^o={180}^o$  

$\alpha ={75}^o$   

$\sin \alpha =\sin {75}^o\approx 0,9659$  

$\sin \beta =\sin {60}^o\approx 0,8660$  

$\sin \gamma =\sin {45}^o\approx 0,7071$  

 

$\frac{b}{\sin \beta }=\frac{a}{\sin \alpha }\mid \cdot \sin \beta$  

$b=\frac{a\cdot \sin \beta }{\sin \alpha }\approx \frac{5\cdot 0,8660}{0,9659}\approx 4,48$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }\approx \frac{5\cdot 0,7071}{0,9659}\approx 3,66$