已知函数f（x）=sin²x＋2sinxconx＋3con²x.1求函数f（x）的单调曾区间,2.在△ABC中,角A,B,C所对的边分别为a,b,c若f（C）=3,且c=√2.a=2.求b的值,

已知函数f（x）=sin²x＋2sinxconx＋3con²x.1求函数f（x）的单调曾区间,2.在△ABC中,角A,B,C所对的边分别为a,b,c若f（C）=3,且c=√2.a=2.求b的值,
已知函数f（x）=sin²x＋2sinxconx＋3con²x.1求函数f（x）的单调曾区间,2.在△ABC中,角A,B,C所对的边分别为a,b,c若f（C）=3,且c=√2.a=2.求b的值,

已知函数f（x）=sin²x＋2sinxconx＋3con²x.1求函数f（x）的单调曾区间,2.在△ABC中,角A,B,C所对的边分别为a,b,c若f（C）=3,且c=√2.a=2.求b的值,
f(x)=sin2x+cos2x+2
=√2sin(2x+π/4)+2
（1） 2kπ-π/2

1-cos^2x+3cos^2x+2sinxcosx
=cos2x+2+sin2x
=根2（sin（2x+派/4））+2
-派/2+2k派＜2x+派/4＜派/2+2k派
[-3派/4+k派，派/4+k派]1-cos^2x+3cos^2x+2sinxcosx
=cos2x+2+sin2x
=根2（sin（2x+派/4））+2
-派/2+2k派＜...

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1-cos^2x+3cos^2x+2sinxcosx
=cos2x+2+sin2x
=根2（sin（2x+派/4））+2
-派/2+2k派＜2x+派/4＜派/2+2k派
[-3派/4+k派，派/4+k派]

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f(x)=sin²x＋2sinxcosx＋3cos²x
=2sinxcosx＋2cos²x+1
=sin2x＋cos2x+2
=√2sin(2x+π/4)+2
1. 2kπ-π/2≤2x+π/4≤2kπ+π/2
kπ-3π/8≤x≤kπ+π/8
2kπ+π/2≤2x+π/4≤2kπ+3π/2

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f(x)=sin²x＋2sinxcosx＋3cos²x
=2sinxcosx＋2cos²x+1
=sin2x＋cos2x+2
=√2sin(2x+π/4)+2
1. 2kπ-π/2≤2x+π/4≤2kπ+π/2
kπ-3π/8≤x≤kπ+π/8
2kπ+π/2≤2x+π/4≤2kπ+3π/2
kπ+π/8≤x≤kπ+5π/8
单增区间:[kπ-3π/8,kπ+π/8];单减区间:[kπ+π/8,kπ+5π/8]
2. ∵c∴0f(C)=√2sin(2C+π/4)+2=3
2C+π/4=3π/4
C=π/4
c²=b²+a²-2abcosC
b²-2√2b+2=0
(b-√2)²=0
b=√2

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