Mathematical trigonometric function

1.(cosx-sinx)^2= 1-2sinx*cosx=2/3

π/4 & lt； x & ltπ/2

= = & gtcosx-sinx & gt； 0

= = & gtcosx-sin=√6/3

2. Sina ≤√3/2

= = & gt0≤A≤π/3，2π/3≤A≤π

cosA≥√3/2

= = & gt0≤A≤π/6

3. Sina+COSA+3 > 0 is a constant.

(tanA-3)(sinA+cosA+3)=0

= => Tana -3=0

= = & gttanA=3

= = & gt3/2(sinA)^2+ 1/4(cosA)^2

=[3/2(sina)^2+ 1/4(cosa)^2]/[(sina)^2+(cosa)^2]

=[3/2(tana)^2+ 1/4]/[(tana)^2+ 1]

= 1 1/8

4.[sina+cosa]^2= 1/25= 1+2sinacosa

= = & gtsinAcosA=- 12/25

= = & gt(sinAcosA)^2= 144/625

[a+b]^3=a^3+b^3+3a^2b+3ab^2

= = & gt 1={(sina)^2+(cosa)^2}^3=(sina)^6+(cosa)^6+3(sina)^4(cosa)^2+3(sina)^2(cosa)^4

=(sina)^6+(cosa)^6+3*(sina)^2* 144/625+3*(cosa)^2* 144/625

=(sinA)^6+(cosA)^6+432/625

= = & gt(sinA)^6+(cosA)^6= 193/625

5.0<A< pi?

==.& gt0 & ltA & ltπ/2

= = & gt0 & ltsin(A/2)，cos(A/2)& lt； 1

= = & gtsin(A/2)+cos(A/2)>0

√[ 2 sin(A/2)* cos(A/2)]+√[ 1+2 sin(A/2)* cos(A/2)]

=√(2sina)+√[sin(a/2)+cos(a/2)]^2

=√(2sinA)+sin(A/2)+cos(A/2)

6 .π/2 & lt； A< pi?

= = & gt- 1 & lt； cosA & lt0，0 & lt； Sina & lt 1

= = & gtsin(cosA)& lt； 0, cos (Sina)>0

= =>p is in the second quadrant