设f(x)=ln[(√3)*sin2x+cos2x](1)求定义域和值域(2)求x使得f(x)=0

Question

设f(x)=ln[(√3)*sin2x+cos2x]
(1)求定义域和值域
(2)求x使得f(x)=0

Answer 1

(1)
f(x)=In(√3sin2x+cos2x)
=In [2sin(2x+π/6)] ←补助角公式
2sin(2x+π/6)＞0
2kπ＜2x+π/6＜(2k+1)π,k∈Z
kπ - π/12＜x＜ kπ + 5π/12,k∈Z
∴定义域为(kπ - π/12,kπ + 5π/12) ,k∈Z
0＜2sin(2x+π/6)≤2
∴In [2sin(2x+π/6)]≤ln2
∴值域为(﹣∞,ln2]
(2)
f(x)=In [2sin(2x+π/6)]=0
2sin(2x+π/6)=1
sin(2x+π/6)=1/2
2x+π/6=π/6 + 2kπ或5π/6+2kπ,k∈Z
x=kπ或π/3+kπ,k∈Z
^.^