Question

In triunghiul dreptunghic ABC, masura lui A=90 grade, AD _I_ BC, tg BAD=3/4 si DC=48 cm. Determinati lungimile laturilor triunghiului si valorile functiilor trigonometrice ale unghiurilor B si C.

Answer 1

tgBAD=DB/AD=3/4⇒AD=4DB/3
conform th.inaltimii avem:AD²=DB·DC⇒(4DB/3)²=DB·48
16DB²/9=DB·48⇒DB=48·9/16⇒DB=27cm
BC=DB+DC=27+48⇒BC=75cm;AD=4DB/3=4·27/3⇒AD=36cm
inΔADB aplic th.pit.:AB²=AD²+DB²=36²+27²=1296+729=2025
AB=√2025=45cm
inΔADC aplic th.pit.:AC²=AD²+DC²=36²+48²=1296+2304=3600
AC=√3600=60cm
sinB=AC/BC=60/75=4/5;cosB=AB/BC=45/75=3/5
tgB=AC/AB=60/45=4/3;ctgB=AB/AC=45/60=3/4
sinC=AB/BC=45/75=3/5;cosC=AC/BC=60/75=4/5
tgC=AB/AC=45/60=3/4;ctgC=AC/AB=60/45=4/3