Stim catetele triunghiului dreptunghic, calculam ipotenuza folosind teorema lui Pitagora
[tex]BC^{2}=AB^{2}+AC^{2}=9^{2}+12^{2}=81+144=225=15^{2}\Rightarrow BC=15[/tex]
b) Functia sinus e definita ca
[tex]sinus=\frac{cateta opusa}{ipotenuza}[/tex] Atunci avem
[tex]\sin{B}=\frac{AC}{BC}=\frac{12}{15}=\frac{4}{5}[/tex]
[tex]\sin{C}=\frac{AB}{BC}=\frac{9}{15}=\frac{3}{5}[/tex]
Functia cosinus e definita ca
[tex]cosinus=\frac{cateta alaturata}{ipotenuza}[/tex]
Atunci avem
[tex]\cos{B}=\frac{AB}{BC}=\frac{9}{15}=\frac{3}{5}[/tex]
[tex]\cos{C}=\frac{AC}{BC}=\frac{12}{15}=\frac{4}{5}[/tex]
Functia tangenta e definita ca
[tex]tg=\frac{sinus}{cosinus}[/tex]
Atunci avem
[tex]tg B=\frac{\sin{B}}{\cos{B}}=\frac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}[/tex]
[tex]tg C=\frac{\sin{C}}{\cos{C}}=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4}[/tex]
Pentru cotangenta avem
[tex]ctg=\frac{1}{tg}[/tex] Atunci avem
[tex]ctg B=\frac{1}{tg B}=\frac{1}{\frac{4}{3}}=\frac{3}{4}[/tex]
[tex]ctg C=\frac{1}{tg C}=\frac{1}{\frac{3}{4}}=\frac{4}{3}[/tex]
c) stim ca unghiul A=90 si [tex]\sin{A}=\sin{90}=1[/tex]
Atunci avem [tex](sin{A})^{2}+(sin{B})^{2}+(sin{C})^{2}=1^{2}+(\frac{4}{5})^{2}+(\frac{3}{5})^{2}=1+\frac{16}{25}+\frac{9}{25}=1+\frac{16+9}{25}=1+\frac{25}{25}=1+1=2[/tex]
d) Aria triunghiului dreptunghic este produsul catetelor/2
[tex]A_{ABC}=\frac{AB*AC}{2}=\frac{9*12}{2}=9*6=54[/tex]
e) Aria triunghiului mai poate fi scrisa si ca produsul inaltimii cu ipotenuza. De aici putem calcula inaltimea
[tex]A_{ABC}=\frac{AD*BC}{2}\Rightarrow AD=\frac{2*A_{ABC}}{BC}=\frac{2*54}{15}=\frac{36}{4}=7.2[/tex]
