Explicație pas cu pas:
ABC triunghi dreptunghic, ∢A = 90°
AD ⊥ BC, DC = 36 cm
[tex]\tan(C) = \frac{2}{3}; \: \: \tan(C) = \frac{AD}{DC} \\ \frac{AD}{36} = \frac{2}{3} = > AD = 24 \: cm[/tex]
teorema înălțimii:
AD² = DC×DB
24² = 36×DB => DB = 16 cm
BC = DC + DB = 36 + 16 = 52 cm
teorema catetei:
AB² = DB×BC = 16×52 = 832
[tex]AB = \sqrt{832} = 8 \sqrt{13} \: cm [/tex]
AC² = DC×BC = 36×52 = 1872
[tex]AC = \sqrt{1872} = 12 \sqrt{13} \: cm[/tex]
[tex]\sin(B) = \frac{AD}{AB} = \frac{24}{8 \sqrt{13}} = \frac{3 \sqrt{13} }{13} \\ [/tex]
[tex]\cos(B) = \frac{DB}{AB} = \frac{16}{8 \sqrt{13} } = \frac{2 \sqrt{13} }{13} \\ [/tex]
[tex]\sin(C) = \frac{AD}{AC} = \frac{24}{12 \sqrt{13} } = \frac{2 \sqrt{13} }{13} \\ [/tex]
[tex]\cos(C) = \frac{DC}{AC} = \frac{36}{12 \sqrt{13} } = \frac{3 \sqrt{13} }{13} \\ [/tex]
