= = = & gtsinx-cosx>0

= = = & gt√2sin[x-(π/4)]>0

Therefore, x-(π/4)∈(2kπ, 2kπ+π)

Therefore, x∈(2kπ+π/4, 2kπ+5π/4)(k∈Z)

15、

f(x)=cos^2 x+sinx=( 1-sin^2 x)+sinx

=-sin^2 x+sinx+ 1

Let sinx=t

So f (t) =-t2+t+1=-[t2-t+(1/4)]+(5/4) =-[t-(1/2)] 2+(.

It is known that |x|≤π/4, so t=sinx∈[-√2/2, √2/2]

Then when t=-√2/2, there is a minimum value =-[-√ 2/2-1/2] 2+(5/4) = (2-2 √ 2)/4 = (1-2)/2.

17、f(x)= sin(2x/3)+cos[(2x/3)-(π/6)]

= sin(2x/3)+[cos(2x/3)cos(π/6)+sin(2x/3)sin(π/6)]

= sin(2x/3)+(√3/2)cos(2x/3)+( 1/2)sin(2x/3)

=(3/2)(sin2x/3)+(√3/2)cos)(2x/3)

=√3*sin[(2x/3)+(π/3)]

Therefore, the period of f(x) is T=2π/(2/3)=3π.

Then, the distance between two adjacent symmetry axes =T/2=3π/2.

18、

According to sine theorem, s △ ABC = (1/2) ab * BC * cosb = (1/2) ab *1* (√ 3/2) = √ 3.

So, AB=4.

The cosine theorem is as follows: AC 2 = Ab2+BC 2-2ab * BC * COSB.

= 16+ 1-2*4* 1*( 1/2)

= 13

Therefore, AC=√ 13.

Then, AC/sinB=AB/sinC.

= = = & gt(√ 13)/(√3/2)=4/sinC

= = = & gtsinC=2√39/ 13

Therefore, cosC=-√ 13/ 13.

Then, tanC=sinC/cosC=-2√3.