a)

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

$\sin \beta =\frac{b\cdot \sin \alpha }{a}=\frac{5\cdot \sin {30}^o}{3}=\frac{5\cdot \frac{1}{2}}{3}\approx 0,833$  

Z tablic matematycznych możemy odczytać, że:

$\sin 56^{\circ }\approx 0,8290$

oraz

$\sin 57^{\circ }\approx 0,8387$

Zauważmy, że odległość liczby $0,833$ jest mniejsza od liczby $0,8290$ niż od liczby $0,8387$.

Zatem $\beta \approx 56^{\circ }$

$\beta \approx {56}^o\text{lub}\beta \approx {180}^o-{56}^o={124}^o$  

 

Przypadek I.

$\beta \approx {56}^o$  

$\gamma \approx {180}^o-\left({30}^o+{56}^o\right)$  

$\gamma \approx {94}^o$  

$\sin {94}^o=\sin \left({90}^o+4^o\right)=\cos 4^o\approx 0,9976$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }=\frac{3\cdot \sin {94}^o}{\sin {30}^o}\approx \frac{3\cdot 0,9976}{0,5}\approx 5,986$  

 

Przypadek II.

$\beta \approx {124}^o$  

$\gamma \approx {180}^o-\left({30}^o+{124}^o\right)$  

$\gamma \approx {26}^o$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }=\frac{3\cdot \sin {26}^o}{\sin {30}^o}\approx \frac{3\cdot 0,4384}{0,5}\approx 2,609$  

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b)

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

$\sin \beta =\frac{b\cdot \sin \alpha }{a}=\frac{10\cdot \sin {45}^o}{8}\approx \frac{10\cdot 0,7071}{8}\approx 0,884$  

$\beta \approx {62}^o\text{lub}\beta \approx {180}^o-{62}^o={118}^o$  

 

Przypadek I.

$\beta \approx {62}^o$  

$\gamma \approx {180}^o-\left({45}^o+{62}^o\right)$  

$\gamma \approx {73}^o$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }=\frac{8\cdot \sin {73}^o}{\sin {45}^o}\approx \frac{8\cdot 0,9563}{0,7071}\approx 10,819$  

 

Przypadek II.

$\beta \approx {118}^o$  

$\gamma \approx {180}^o-\left({45}^o+{118}^o\right)$  

$\gamma \approx {17}^o$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }=\frac{8\cdot \sin {17}^o}{\sin {45}^o}\approx \frac{8\cdot 0,2924}{0,7071}\approx 3,308$  

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c)

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

$\sin \beta =\frac{b\cdot \sin \alpha }{a}=\frac{10\cdot \sin {60}^o}{9}\approx \frac{10\cdot 0,8660}{9}\approx 0,962$  

$\beta \approx {74}^o\text{lub}\beta \approx {180}^o-{74}^o={106}^o$  

 

Przypadek I.

$\beta \approx {74}^o$  

$\gamma \approx {180}^o-\left({60}^o+{74}^o\right)$  

$\gamma \approx {46}^o$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }=\frac{9\cdot \sin {46}^o}{\sin {60}^o}\approx \frac{9\cdot 0,7193}{0,8660}\approx 7,479$  

 

Przypadek II.

$\beta \approx {106}^o$  

$\gamma \approx {180}^o-\left({60}^o+{106}^o\right)$  

$\gamma \approx {14}^o$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }=\frac{9\cdot \sin {14}^o}{\sin {60}^o}\approx \frac{9\cdot 0,2419}{0,8660}\approx 2,514$  

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d)

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

$\sin \alpha =\frac{a\cdot \sin \beta }{b}=\frac{1\cdot \sin {45}^o}{2}\approx \frac{0,7071}{2}\approx 0,354$  

$\alpha \approx {21}^o\text{lub}\alpha \approx {180}^o-{21}^o={159}^o$  

 

Przypadek I.

$\alpha \approx {21}^o$  

$\gamma \approx {180}^o-\left({21}^o+{45}^o\right)$  

$\gamma \approx {114}^o$  

$\sin {114}^o=\sin \left({90}^o+{24}^o\right)=\cos {24}^o\approx 0,9135$  

 

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

$c=\frac{a\cdot \sin \gamma }{\sin \alpha }=\frac{1\cdot \sin {114}^o}{\sin {21}^o}\approx \frac{0,9135}{0,3584}\approx 2,549$  

 

Przypadek II.

$\alpha \approx {159}^o$  

$\gamma \approx {180}^o-\left({159}^o+{45}^o\right)<0^o$  

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