Question

Triunghiul ABC este dreptunghic in A. Daca tgB=3 supra4 si BC=15cm, calculate perimetru triunghiului si functiie trignmetrice ae unghiuui B

Answer 1

ΔABC
AB=c
AC=b
BC= 15 = a
tg B= [tex] \frac{b}{c} = \frac{3}{4} \\ b= \frac{3c}{4} \\ a^{2}= b^{2} +c^{2} \\ 15^{2} = \frac{9c^{2} }{16} + c^{2} \\ 225= \frac{9 c^{2}+16c^{2}}{16} \\ 225= \frac{25 c^{2} }{16} \\ c^{2} =144 \\ c=12 \\ b=9 \\ P=15+12+9=36 \\ sinB= \frac{b}{a} = \frac{9}{15} = \frac{3}{5} \\ cosB= \frac{c}{a} = \frac{12}{15} = \frac{4}{5} \\ ctgB = \frac{c}{b} = \frac{12}{9} = \frac{4}{3} [/tex]

Answer 2

tg B=AC/AB=3/4 ⇒ AC=3x, AB=4x 
aplic teorema lui Pitagora in trg ABC ⇒ BC^2=AB^2+AC^2=(4x)^2+(3x)^2=16x^2+9x^2=25x^2
15^2=225=25x^2 ⇒ x^2=225:25=9 ⇒ x=3 ⇒ AC=3*3 cm=9 cm si AB= 4*3 cm=12 cm
sin B=AC/BC=9/15=3/5
cos B=AB/BC=12/15=4/5
ctg B=1/tg B = 4/3