Question

求帮助， 已知函数f(x)=2sinxcosx+cos2x(x∈r) 1、当x取什么值时，函数f(x

求帮助， 已知函数f(x)=2sinxcosx+cos2x(x∈r)
1、当x取什么值时，函数f(x)取得最大值，并求其最大值；
2、若a为锐角，且f(a+兀/8)=√2/3，求tana的值。

Answer 1

f(x)=sin2x+cos2x
=√2sin(2x+π/4)
所以T=2π/2=π
当2x+Pai/4=2kPai+Pai/2,即X=KPai+Pai/8时有最大值=√2

f(θ+π/8)=√2sin(2θ+π/4+π/4)
=√2cos2θ
=√2/3
cos2θ=1/3
θ锐角则sin2θ>0
sin²2θ+cos²2θ=1
sin2θ=2√2/3
tan2θ=2√2
tan2a=2tana/(1-tan^2a)=2根号2
tana=根号2-根号2tan^2a

根号2tan^2a+tana-根号2=0
tana=(-1+3)/2根号2=根号2/2

Answer 2

(1)
f(x)=2sinxcosx+cos2x

= sin2x+cos2x
= √2(sin(2x+π/4)
max f(x) = √2
2x+π/4 = 2kπ+π/2
x= kπ+π/8
(2)
f(a+π/8) = √2/3
1/3 = sin( 2a+π/2)
= cos2a
2(cosa)^2 - 1 = 1/3
cosa = 2/3
tana = √5/2

Answer 3

1，f(x)=2sinxcosx+cos2x=sin2x+cos2x=根号2 [sin(2x+π/4)],最大值等于根号2，当x=π/8 +kπ

2，代入f(x)=sin(2a+π/2)=1/3 ，得cos2a=1/3,求得tana=√2/2