Question

Catetele unui triunghi dreptunghic sunt AB=6 cm si AC=8 cm. Calculând
sin unghi B + sin unghi C se obtine ....
AJUTOR,VĂ ROG⚠️⚠️​

Answer 1

Teorema sinusurilor: a/sinA=b/sinB=c/sinC=2R=abc/2S

S=AB*AC/2=6*8/2=6*4=24 cm²

In ΔABC,m(∡A)=90°=>(T.Pitagora):BC²=AB²+AC²=6²+8²=36+64=100=>BC=√100=10 cm

AC/sinB=AB*BC*AC/2S=>sinB=2S/AB*BC=2*24/6*10=48/60=4/5

AB/sinC=AB*BC*AC/2S=>sinC=2S/AC*BC=2*24/8*10=24/40=3/5

sinB+sinC=4/5+3/5=7/5

Answer 2

Răspuns: $\frac{7}{5}$

Explicație pas cu pas:

$AB \: şi \: AC \: sunt \: catetele \: în \: \triangle \: ABC \: dreptunghic = > \\ = > m( \widehat{A}) = 90 \degree = > BC \: ipotenuz \breve{a} \\ \stackrel {T.Pitagora}{ \longrightarrow} {BC}^{2} = {AB}^{2} + {AC}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {BC}^{2} = {6 \: cm}^{2} + {8 \: cm}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {BC}^{2} = 36 \: cm + 64 \: cm \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {BC}^{2} = 100 \: cm \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: BC = \sqrt{100 \: cm} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{BC = 10 \: cm} \\ sinus = \frac{cateta \: opus \breve{a}}{ipotenuz \breve{a}} \\ sinus( \widehat{B}) = \frac{AC}{BC} = \boxed{ \frac{8}{10} } \\ sinus( \widehat{C}) = \frac{AB}{BC} = \boxed{\frac{6}{10} } \\ \frac{8}{10} + \frac{6}{10} = \frac{8 + 6}{10} = { \frac{{14}^{(\div 2}{10} }=\boxed{\frac {7}{5}}$