Trigonometry Geometry

[Guest]

$(c\times\sin B)^2+(b\times\cos C)^2=b^2$

$(c\times\sin A)^2+(a\times\cos C)^2=a^2$

$(c\times\sin B)^2+(c\times\cos B)^2=c^2$

$(c\times\sin A)^2+(c\times\cos A)^2=c^2$

$(b\times\cos C)+(c\times\cos B)=a$

$(a\times\cos C)+(c\times\cos A)=b$

$(a\times\cos B)+(b \times\cos A)=c$

$(c\times\cos B)+(b \times\cos C)=a$

$(c\times\cos A)+(a\times\cos C)=b$

and to find one of the altitude:

$h_a=\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2a}$
$h_b=\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2b}$
$h_c=\frac{\sqrt{2(a^2 b^2+a^2 c^2+b^2 c^2)-a^4-b^4-c^4}}{2c}$
I have three altitudes and all three sides and and I able to find all three angles with the help of SohCahToa, but I unable to find a general equation with the help of pythagorean identities for side of c without using one of the altitudes or height.

[Baltuilhe]

Good night

Sorry, my friend! I didn't catch what is your doubt

Can you give me an example?

Thanks