Question

数学题求手写答案，谢谢。

Answer 1

(1)
a//b，则:ainx/cosx=-1/(3/2)=-2/3
===>
tanx=-2/3
===>
tan^2
x=4/9
===>
cos^2
x=1/sec^2
x=1/(1+tan^2
x)=1/[1+(4/9)]=9/13
sin2x=2tanx/(1+tan^2
x)=(-4/3)/[1+(4/9)]=-12/13
所以，cos^2
x-3sin2x=(9/13)-3*(-12/13)=45/13
(2)
f(x)=(a+b)·b=(sinx+cosx,1/2)·(cosx,3/2)
=(sinx+cosx)*cosx+(1/2)*(3/2)
=sinxcosx+cos^2
x+(3/4)
=(1/2)sin2x+[(cos2x+1)/2]+(3/4)
=(1/2)(sin2x+cos2x)+(5/4)
=(√2/2)sin[2x+(π/4)]+(5/4)
所以，f(x)的最小正周期为T=2π/2=π
单调递增区间为2x+(π/4)∈[2kπ-(π/2),2kπ+(π/2)]
===>
2x∈[2kπ-(3π/4),2kπ+(π/4)]
===>
x∈[kπ-(3π/8),kπ+(π/8)](k∈Z)