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1.函数f(x)=sin²(x+π/4)-sin²(x-π/4)是()A.周期为π的偶函数B.周期为π的奇函数C.周期为2π的偶函数C.周期为2π的奇...
1.函数f(x)=sin²(x+π/4)-sin²(x-π/4)是 ( )
A.周期为π的偶函数 B.周期为π的奇函数
C.周期为2π的偶函数 C.周期为2π的奇函数
2.当-π/2≤x≤π/2时,函数f(x)=sinx+√3 cosx有 ( )
A.最大值1,最小值-1 B.最大值1,最小值-1/2
C.最大值2,最小值-2 D.最大值2,最小值-1
3.函数y=1/2+sinx+cosx的最大值是 ( )
A.√2 /2-1 B.√2 /2+1 C.1-√2 /2 D.-1-√2 /2
4.f(x)=sinx+cosx,若π/4<x1<x2<π/2,则f(x1)f(x2)的大小关系是( )
5.函数f(x)=cos2x-2√3 sinxcosx的最小正周期是( )
6.已知sina+sinβ+sinr=0,cosa+cosβ+cosr=0,求值cos(β-r)的值.
7.求值:(1+cos20°/2sin20° )-sin10°(1/tan50°-tan5°) 展开
A.周期为π的偶函数 B.周期为π的奇函数
C.周期为2π的偶函数 C.周期为2π的奇函数
2.当-π/2≤x≤π/2时,函数f(x)=sinx+√3 cosx有 ( )
A.最大值1,最小值-1 B.最大值1,最小值-1/2
C.最大值2,最小值-2 D.最大值2,最小值-1
3.函数y=1/2+sinx+cosx的最大值是 ( )
A.√2 /2-1 B.√2 /2+1 C.1-√2 /2 D.-1-√2 /2
4.f(x)=sinx+cosx,若π/4<x1<x2<π/2,则f(x1)f(x2)的大小关系是( )
5.函数f(x)=cos2x-2√3 sinxcosx的最小正周期是( )
6.已知sina+sinβ+sinr=0,cosa+cosβ+cosr=0,求值cos(β-r)的值.
7.求值:(1+cos20°/2sin20° )-sin10°(1/tan50°-tan5°) 展开
1个回答
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1/A
f(x)=sin²(x+π/4)-sin²(x-π/4)
=1/2(1-cos(2x+π/2)+1/2(cos(2x-π/2)-1)
=1/2[sin2x+sin2x)
=sin2x
sin2x是周期为π的奇函数
2/D
sinx+√3cosx
=2sin(x+π/3)
因为-π/2≤x≤π/2
所以-π/6≤x+π/3≤5π/6
所以-1≤2sin(x+π/3)≤2
3、B
sinx+cosx=√2sin(x+π/4)
所以-√2<=sinx+cosx<=√2
2-√2<=2+inx+cosx<=2+√2
1/(2+√2)<=1/(2+inx+cosx)<=1/(2-√2)
分母有理化
最大值=(2+√2)/2 =1+√2/2
4、f(x)=sinx+cosx
=√2(√2/2sinx+cosx√2/2)
=√2sin(x+π/4)
π/4<x<π/2
π/2<x+π/4<3/4π
x1<x2
∴f(x1)>f(x2)
5、cos2x-2√3 sinxcosx
=cos2x-√3sin2x
=2(cos2x*1/2-√3/2*sin2x)
=2cos(2x+π/3)
所以周期T=2π/2=π
6、sin²a+cos²a=1
(sinb+sinr) ²+(cosb+cosr) ²=1
sin²b+2sinbsinr+sin²r+cos²b+2cosbcosr+cos²r=1
1+2(sinbsinr+cosbcosr)+1=1
cos(b-r)=-1/2
:((1+cos20°)/2sin20°)-sin10°(cot5°-tan5°) =[(1+cos20°)/4sin10°cos10°]-sin10°(cot5°-tan5°)
= [(1+cos²10°-sin²10)/4sin10°cos10°]-sin10°(cot5°-tan5°)
=2cos²10°/4sin10°cos10°-sin10°(cot5°-tan5°)
=(2cos10°/4sin10°)-2sin5°cos5°(cot5°-tan5°)
=(cos10°/2sin10°)-2(cosp²5°-sin²5°)
=(cos10°/2sin10°)-2cos10°
=(cos10°-4sin10°cos10°)/2sin10°
=(sin80°-2sin20°)/2sin10°
=[(sin80°-sin20°)-sin20°]/2sin10°
=(2cos50°sin30°-sin20°)/2sin10°
=(sin40°-sin20°)/2sin10°
=2cos30°sin10°/2sin10°
=cos30°
=√3/2
f(x)=sin²(x+π/4)-sin²(x-π/4)
=1/2(1-cos(2x+π/2)+1/2(cos(2x-π/2)-1)
=1/2[sin2x+sin2x)
=sin2x
sin2x是周期为π的奇函数
2/D
sinx+√3cosx
=2sin(x+π/3)
因为-π/2≤x≤π/2
所以-π/6≤x+π/3≤5π/6
所以-1≤2sin(x+π/3)≤2
3、B
sinx+cosx=√2sin(x+π/4)
所以-√2<=sinx+cosx<=√2
2-√2<=2+inx+cosx<=2+√2
1/(2+√2)<=1/(2+inx+cosx)<=1/(2-√2)
分母有理化
最大值=(2+√2)/2 =1+√2/2
4、f(x)=sinx+cosx
=√2(√2/2sinx+cosx√2/2)
=√2sin(x+π/4)
π/4<x<π/2
π/2<x+π/4<3/4π
x1<x2
∴f(x1)>f(x2)
5、cos2x-2√3 sinxcosx
=cos2x-√3sin2x
=2(cos2x*1/2-√3/2*sin2x)
=2cos(2x+π/3)
所以周期T=2π/2=π
6、sin²a+cos²a=1
(sinb+sinr) ²+(cosb+cosr) ²=1
sin²b+2sinbsinr+sin²r+cos²b+2cosbcosr+cos²r=1
1+2(sinbsinr+cosbcosr)+1=1
cos(b-r)=-1/2
:((1+cos20°)/2sin20°)-sin10°(cot5°-tan5°) =[(1+cos20°)/4sin10°cos10°]-sin10°(cot5°-tan5°)
= [(1+cos²10°-sin²10)/4sin10°cos10°]-sin10°(cot5°-tan5°)
=2cos²10°/4sin10°cos10°-sin10°(cot5°-tan5°)
=(2cos10°/4sin10°)-2sin5°cos5°(cot5°-tan5°)
=(cos10°/2sin10°)-2(cosp²5°-sin²5°)
=(cos10°/2sin10°)-2cos10°
=(cos10°-4sin10°cos10°)/2sin10°
=(sin80°-2sin20°)/2sin10°
=[(sin80°-sin20°)-sin20°]/2sin10°
=(2cos50°sin30°-sin20°)/2sin10°
=(sin40°-sin20°)/2sin10°
=2cos30°sin10°/2sin10°
=cos30°
=√3/2
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