[tex]m\sphericalangle BAC=90\°\\m\sphericalangle ABC=30\°\\AB=12\text{ cm}[/tex]
[tex]A_{\Delta ABC}=\text{ ?}[/tex]
Răspuns:
[tex]A_{\Delta ABC}=24\sqrt{3}\text{ cm}[/tex]
Explicație pas cu pas:
Vom afla aria [tex]\Delta ABC[/tex] folosind formula [tex]A_{\Delta\text{dr.}}=\dfrac{cateta_1\cdot cateta_2}{2}[/tex].
[tex]m\sphericalangle ACB=180\°-90\°-30\°\\m\sphericalangle ACB=60\°\\[/tex]
Pentru a afla lungimea celelalte catete [tex]AC[/tex], folosim tangenta în [tex]\sphericalangle ABC[/tex]:
[tex]\Delta ABC\text{ dr.}\Rightarrow \tan{\sphericalangle ABC}=\dfrac{\text{cat. op.}}{\text{cat. al.}}\\m\sphericalangle ABC=30\°\Rightarrow \dfrac{1}{\sqrt3}=\dfrac{AC}{12}\\\Rightarrow AC = \dfrac{12}{\sqrt3}\\\Rightarrow AC = \dfrac{12\sqrt{3}}{3}\Rightarrow\\\Rightarrow AC=4\sqrt{3}[/tex]
[tex]A_{\Delta ABC}=\dfrac{AB\cdot AC}{2}\\\Rightarrow A_{\Delta ABC}=\dfrac{12\cdot4\sqrt{3}}{2}\Rightarrow\\\Rightarrow A_{\Delta ABC}=24\sqrt{3}\text{ cm}[/tex]
