If-sin-4-cos-4-4sin-cos-0-pi-2-then-sin-cos-equal-to-

Question Number 138099 by liberty last updated on 10/Apr/21

I f \sin^{4} α + \cos^{4} β = 4 \sin α \cos β,

0 ⩽ α, β ⩽ \frac{π}{2}, t h e n \sin α + \cos β

e q u a l t o \dots

Commented by mr W last updated on 10/Apr/21

r e a s o n :

s a y p = \sin α, q = \cos β

f o r a g i v e n q, y o u c a n s o l v e f o r p :

p^{4} - 4 q p + q^{4} = 0

t h a t m e a n s p + q \neq c o n s t a n t

Commented by mr W last updated on 10/Apr/21

n o u n i q u e v a l u e!

0 ⩽ \sin α + \cos β ⩽ 1.25099

s o t h e q u e s t i o n s h o u l d h a v e b e e n :

f i n d t h e m a x i m u m v a l u e o f

\sin α + \cos β .

Commented by liberty last updated on 10/Apr/21

a n s w e r i s 2

Commented by mr W last updated on 10/Apr/21

w r o n g!

i f \sin α + \cos β = 2, t h e n

\sin α = 1, \cos β = 1 a n d

\sin^{4} α + \cos^{4} β = 2 \neq 4 \sin α \cos β = 4

Commented by mr W last updated on 10/Apr/21

y o u c a n a l s o s e e :

w i t h α = 0, β = \frac{π}{2} w h i c h f u l f u l l

\sin^{4} α + \cos^{4} β = 4 \sin α \cos β, b u t

\sin α + \cos β = 0 \neq 2 .

a n o t h e r e x a m p l e :

α = π / 3, β = 1.40749 w h i c h f u l f u l l

\sin^{4} α + \cos^{4} β = 4 \sin α \cos β, b u t

\sin α + \cos β = 1.0286 \neq 2 .

t h e s e e x a m p l e s s h o w a l s o t h a t

\sin α + \cos β m a y h a v e d i f f e r e n t v a l u e s .

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