Question

Laturile unui triunghi ABC verifica relatiile 3a=7c si 3b=8c.Sa se determine tg$\frac{A}{2}$ si m(A).
Multumesc mult!

Answer 1

3a=7c    ⇒a=7c/3
3b=8c    ⇒b=8c/3
teorema cosinusului
cos A=(b²+c²-a²)/2bc=[(8c/3)²+c²-(7c/3)²]/(2×8c/3×c)=
=(64c²/9+c²-49c²/9)/(16c²/3)= 
=(15c²/9+c²)/(16c²/3)=
=24c²/9 ×3/16c²=
=8c²/3 ×3/16c²=1/2

sin A=√(1-cos²A)=√[1-(1/2)²]=√(1-1/4)=√3/4=√3/2

tg A/2=(1-cosA)/sinA=(1-1/2)/(√3/2)=1/2×2/√3=1/√3=√3/3⇒
m(∡A/2)=30°  ⇒ m(∡A)=60°