Question

If $A+B+C=\pi$ and $\cos A+\cos B+\cos C$$=0=\sin A+\sin B+\sin C$ then $\cos 3A+\cos 3B+\cos 3C=1$

• A. True
• B. False

Answer 1

The correct option is B False

Given

$\sin A+\sin B+\sin C=0$

$\cos A+\cos B+\cos C=0$

Let $Z_1=e^{iA},Z_2=e^{iB},Z_3=e^{iC}$

Consider

$Z_1+Z_2+Z_3$

$⟹e^{iA}+e^{iB}+e^{iC}$

$⟹\cos A+i\sin A+\cos B+i\sin B+\cos C+i\sin C$

$⟹(\cos A+\cos B+\cos C)+i(\sin A+\sin B+\sin C)$

$⟹0$

As $Z_1+Z_2+Z_3=0$

$⟹Z_1^3+Z_2^3+Z_3^3-3Z_1{Z}_2{Z}_3=0$

$⟹e^{i3A}+e^{i3B}+e^{i3C}=3e^{i(A+B+C)}$

$⟹\cos 3A+\cos 3B+\cos 3C=3\cos (A+B+C)=3\cos \pi =-3$

So the given relation is $\text{False}$