Prove-that-the-equation-sin-x-1-x-is-impossible-if-x-be-real-

Question Number 37733 by kunal1234523 last updated on 17/Jun/18

Prove that the equation \sin θ = x + \frac{1}{x}

is impossible if x be real .

Answered by MrW3 last updated on 17/Jun/18

i f x > 0 :

\sin θ = x + \frac{1}{x} = {(\sqrt{x} - \frac{1}{\sqrt{x}})}^{2} + 2 ⩾ 2

b u t \sin θ ⩽ 1 \Rightarrow n o s o l u t i o n

i f x < 0 :

\sin θ = x + \frac{1}{x} = - {(\sqrt{- x} - \frac{1}{\sqrt{- x}})}^{2} - 2 ⩽ - 2

b u t \sin θ ⩾ - 1 \Rightarrow n o s o l u t i o n

Commented by kunal1234523 last updated on 17/Jun/18

thank you very much MrW 3