B. ถ้า $ \sin A + \sin B +\sin C = \cos A+\cos B+ \cos C = 0$
หาค่า $ \cos^2A+ \cos^2B+ \cos^2C$
$$ sin A + sin B +sin C=0$$
$$ sin A + sin B =-sinC$$
$$sin^2A+sin^2B+2sinAsinB=sin^2C$$
$$2-(cos^2A+cos^2B)+2sinAsinB=1-cos^2C$$
$$-1+(cos^2A+cos^2B)-2sinAsinB=cos^2C...[1]$$
$$ cos A + cosB +cosC=0$$
$$ \cos A + \cos B =-cosC$$
$$cos^2A+cos^2B+2cosAcosB=cos^2C...[2]$$
$[1]=[2];$
$$2cosAcosB=-1-2sinAsinB$$
$$cosAcosB+sinAsinB=-\frac{1}{2} $$
$$cos(A-B)=-\frac{1}{2} $$
$$cos \frac{A-B}{2}=\frac{1}{2}$$
โจทย์
$cos^2A+cos^2B+cos^2C$
$=2[cos^2A+cos^2B]+2cosAcosB$
$=2[(cosA+cosB)^2-2cosAcosB]+2cosAcosB$
$=2[(2cos\frac{A+B}{2}cos\frac{A-B}{2})^2-2cosAcosB]+2cosAcosB$
$=2cos^2\frac{A-B}{2}-2cosAcosB$
$=2cos^2\frac{A-B}{2}-cos(A+B)-cos\frac{2\pi}{3}$
$=1-cos\frac{2\pi}{3}=1.5$
|