2siny 8cosy 4 2 find the value of y

f(x)=(sinx)^2+2sinx(4siny+4)+(4siny+4)^2+(cosx)^2-10cosxcosy+25(cosy)^2=1+8sinx(siny+1)-10cosxcosy+16(siny+1)^2+25(cosy)^2=1+sin(x-α)64(siny+1)^2+100(cosy)^2)+16(siny+1)^2+25(cosy)^2

where cos = 8sinx (64(siny+1) 2+100(cosy) 2).

When sin(x- )=-1, f(x) is the smallest, i.e., g(y)=1- (64(siny+1) 2+100(cosy) 2)+16(siny+1) 2+25(cosy) 2=( (16(siny+1) 2+25(cosy) 2)-1) 2, so that cosy=x, -1"x", when x=1, take the maximum value of 7

I can only work out an approximation, y=63° or, probably not right.

Let t=cosxcosy-sinxsiny=cos(x+y), then.

sin(2x+2y)=2sin(x+y)cos(x+y)=-3 staring at the dust and regretting 8, square brother Xunde 4t 2*(1-t 2)=9 64,256t 4-256t 2+9=0,t 2=(128 soil 16 55) Kaizheng 256

8 soil 55) 16, t = soil (8 soil 55) 4 (with 4 values), is what is sought.

Add the two formulas. y+1/2=sin(x+y)

1≤y+1/2≤1

3 2 2 y 1 2

Subtract the two formulas. y-1/2=sin(y-x)

1≤y-1/2≤1.

1/2≤y≤3/2

value range [-1 or hui2,1 2].

Let cosx+2siny=m, square both sides cos x+4sin y+4cosxsiny=m (table squared, the same below).(1) From the known sinx+2cosy=2, the square of the two sides of the sedan sin x+4cos y+4sinxcosy=4(2) (1)+(2):

1+4+4sin(x+y)=m +4 1+4sin(x+y)=m 0 m 5 -5 m 5 i.e. Dan Fan Jane - Die pants 5....

Suppose (2xcosy+y 2*cosx)dx+(2ysinx-x 2*siny)dy

The full differentiation of the pose width u(x,y) of a function.

du/dx=2xcosy+y^2*cosx...1)du/dy=2ysinx-x^2*siny...2) x points for the family orange (1).

u=x^2*cos(y)

y^2*sin(x)..3)

Y integral megalowatt for (2).

u=x^2*cos(y)

y^2*sin(x)..4)

Formula 3 is equal to Formula 4.

u(x,y)=x^2*cos(y)

y^2*sin(x)

Let dz=(2siny)dx+(2xcosy+1)dy, then z x=2siny So: z=2xsiny +g(y) z y=2xcosy +g'(y), and it is known: z y = 2xcosy+1

Hence g'(y)=1, so: g(y)=y+c

Original function: z=2xsiny+y+c

Let the comic only sensitive cosx+2siny=m, the square of both sides cos x+4sin y+4cosxsiny=m (table squared, the same below).1) From the known sinx+2cosy=2, the square of the branches of the two mountains sin x+4cos y+4sinxcosy=4....2) (1)+(2):

1+4+4sin(x+y)=m +4 1+4sin(x+y)=m 0 m 5 -5 m 5 i.e. - 5 cosx+2siny 5

y''=1/2sin2y

y'=-1/4cos26+c

y=-1/8sin2y+c

Hope it helps, hope. Good luck with your studies.