tga+ctga=2
已知a为锐角,求证tga+ctga≥2,并说明不等式取等号的条件.解:根椐基本不等式a+b≥2√(ab),其中:a>0,b>0当a=b取等号.∵a为锐角,∴tga>0,ctga>0,→1/tga>0∴tga+ctga=tga+(1/tga)≥2√tga*(1/tga)=2√1=2即tga+ctga≥2,当tga=ctga不等式取等号即tga=1/tga,(tga)^2=1,tga=±1,但a为锐角,∴tga=1,a=π/4
a为锐角,tga>0、ctga>0(根号tga-根号ctga)^2>=0(根号tga-根号1/tga)^2>=0tga+ctga>=2tga=ctga时,tga+ctga=2a=45度
答:详情>>