Key words: Geometry, Unusual geometry, Math, Physics, Chemistry, High school, Geometry Olympiads, Free Geometry, Euclidean Geometry, Calculus, Geometric Construction. Oh yes, going-nowhere discussions, haikus, and poems too.
I'd think that the maximum value occurs when α=β=θ or Max[(sin(α)*sin(β)*sin(θ)] =3√3/8Vihaan
Correct!
But why?
Henkie, see six' solution below:six said... So we have,x + y + z = 360 and sinx*siny*sinz-> f(x,y) = sin(x)*sin(y)*sin(360-x-y)-> f(x,y) = sin(x)*sin(y)*-sin(x+y)Now to find the critical points of f(x,y), we just need to find the partial derivative with respect to x and y and solve for 0.f_x(x,y) = sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)solve for 0.sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)=0-> tan(x) = -tan(x+y)Since the function is symmetric, we should get the same partial derivative for y.-> tan(y) = -tan(y+x)-> tan(x)=tan(y)-> x = y or they are opposites. However, if they are opposites, the original function just becomes 0. Thus, x = y. Now substitute in x for y in the original equation and find its critical points.Eventually, you will get sin(3x)=0x = 120 degreesy = 120 degreesz = 120 degreesAnswer (D)
I'd think that the maximum value occurs when α=β=θ or Max[(sin(α)*sin(β)*sin(θ)] =3√3/8
ReplyDeleteVihaan
Correct!
ReplyDeleteBut why?
ReplyDeleteHenkie, see six' solution below:
ReplyDeletesix said...
So we have,
x + y + z = 360 and sinx*siny*sinz
-> f(x,y) = sin(x)*sin(y)*sin(360-x-y)
-> f(x,y) = sin(x)*sin(y)*-sin(x+y)
Now to find the critical points of f(x,y), we just need to find the partial derivative with respect to x and y and solve for 0.
f_x(x,y) =
sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)
solve for 0.
sin(x)sin(y)(-cos(x+y)-cos(x)sin(y)sin(x+y)=0
-> tan(x) = -tan(x+y)
Since the function is symmetric, we should get the same partial derivative for y.
-> tan(y) = -tan(y+x)
-> tan(x)=tan(y)
-> x = y or they are opposites. However, if they are opposites, the original function just becomes 0. Thus, x = y.
Now substitute in x for y in the original equation and find its critical points.
Eventually, you will get sin(3x)=0
x = 120 degrees
y = 120 degrees
z = 120 degrees
Answer (D)