lượng giác

Question

Nếu $\text{sin}x+\text{sin}y+\text{sin}z=0$ và $\text{cos}x+\text{cos}y+\text{cos}z=0$, chứng minh :

1. $\text{cos}(x-y)=\frac{1}{2}$

2. $\text{sin}^2(x)+\text{sin}^2(y)+\text{sin}^2(z)=\frac{3}{2}$

Trần Nhật Tân · Answer 1 · 10/13/2012 10:37:37 PM

2. Kí hiệu $\left\{\begin{array}{c}
\cos x+i\cdot\sin x=u\\\
\cos y+i\cdot\sin y=v\\\
\cos z+i\cdot\sin z=w\end{array}\right|\implies u+v+w=0\ \wedge\ uv+vw+wu=0\ \implies\ u^2+v^2+w^2=0$
$\ \implies\sum\cos 2x=0\ \wedge\ \sum\sin 2x=0$ .
Do đó, $\sum\cos 2x=0\implies \sum\left(1-2\sin^2x\right)=0\implies$ $\sum\sin^2x=\frac 32$

Trả lời 13-10-12 10:37 PM

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