For-R-cos-cosx-siny-sinx-x-R-then-find-the-sum-of-the-possible-values-of-sin-siny-

Question Number 27958 by bmind4860 last updated on 17/Jan/18

F o r α \in R, c o s α c o s x + s i n y ⩾ s i n x, \forall x \in R,

t h e n f i n d t h e s u m o f t h e p o s s i b l e v a l u e s

o f s i n α + s i n y .

Answered by ajfour last updated on 17/Jan/18

\sin y ⩾ \sin x - \cos α \cos x

\Rightarrow ⩾ \sqrt{1 + \cos^{2} α} [\frac{1}{\sqrt{1 + \cos^{2} α}} \sin x - \frac{\cos α}{\sqrt{1 + \cos^{2} α}} \cos x]

\sin y ⩾ \sqrt{1 + \cos^{2} α} \sin [x - \tan^{- 1} (\cos α)]

s i n c e x c a n b e a n y v a l u e,

a n d \sin y ⩽ 1 \Rightarrow \cos α = 0

\Rightarrow \sin α = \pm 1

t h e n t h i s i m p l i e s

\sin y ⩾ \sin x

a n d a g a i n f o r t h i s t o b e a l w a y s

t r u e \sin y = 1

h e n c e \sin y + \sin α = 0 o r 2 .

Commented by bmind4860 last updated on 17/Jan/18

t h a n k y o u! w o u l d y o u s u g g e s t w h a t d o

y o u t h i n k a b o u t t h e f o l l o w i n g q u e s t i o n,

w h e t h e r i t i s r i g h t o r w r o n g

f i n d g e n e r a l s o l u t i o n, s i n x + s i n 3 x + s i n \sqrt{x} = 0 .

Commented by ajfour last updated on 17/Jan/18

c a n t s a y, i {c o u l d n}^{'} t f i n d a w a y .

Commented by bmind4860 last updated on 17/Jan/18

m e t o o {t h a t}^{'} s w h y i t h i n k i t i s w r o n g

a n d t h a n k y o u f o r s u g g e s t i o n