题目

复平面内,矩形OABC的顶点O、B所对应的复数分别为0、√2(1+i)
且│OA│=√3,│OC│=1,则A、C所对应的复数为______

提问时间：2020-12-29

答案

在坐标系中,OB与x轴角度为π/4,又│OC│=1,│OA│=√3,│OB│=2,故COB为直角三角形,∠COB=π/6,故OC与x轴角度为π/4+π/6=5π/12或π/4+π/3=7π/12,OA与OC角度为π/4-π/3=-π/12或π/4-π/6=π/12,
对于OC为5π/12,OA为-π/12时,A的复数为√3（cos5π/12+isin5π/12）,C的复数为cosπ/12-isinπ/12；
对于OC为7π/12,OA为π/12时√3（cos7π/12+isin7π/12）,OC为cosπ/12+isinπ/12