Question

1.已知向量a=（cosa,sina）,向量b(cosa,sina),且向量a⊥向量b,则向量b=?2.设p=sinas

Answer 1

1.取內積得 cos(a)cos^2(a)+sin(a)sin^2(a)=0,所以 [cos(a)+sin(a)][cos^2(a)-cos(a)sin(a)+sin^2(a)]=0 不過cos^2(a)-cos(a)sin(a)+sin^2(a)=1-(1/2)sin(2a)>0 所以cos(a)+sin(a)=0 即cos(a)=-sin(a) 所以cos(a)=-sin(a)=±1/√2 所以向量b=(1/2,1/2) 2.p+q=sin(a)sin(b)+cos^2[(a+b)/2] =(1/2)[cos(a-b)-cos(a+b)]+(1/2)[1+cos(a+b)] 【積化和差、半角公式】 =(1/2)[cos(a-b)+1] 所以0≤p+q≤1 3.sin(7π/3)*cos(-11π/6)+tan(-15π/4)*1/tan(13π/6) =sin(π/3)*cos(π/6)+tan(π/4)*1/tan(π/6) =(√3)/2 * (√3)/2 + 1*1/(1/√3) =2√3 4.每個三棱錐的體積是 (1/6)*(1/2)*(1/2)*(1/2)=1/48 所以截去了體積1/6 所以剩下了體積5/6的截半立方體