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 September 28th, 2016, 12:56 AM #1 Senior Member   Joined: Jul 2011 Posts: 405 Thanks: 16 real values of a if $\sin x+\cos x= y^2-y+a$ the equation has no real values of $x$ for any real values of $y,$Then values of $a$
 September 28th, 2016, 07:39 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,552 Thanks: 1402 show some work Thanks from panky
 September 28th, 2016, 02:42 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond $\displaystyle a\gt\sqrt2+\frac14$ Thanks from panky and topsquark
 September 29th, 2016, 01:05 AM #4 Senior Member   Joined: Jul 2011 Posts: 405 Thanks: 16 Using greg 1313 hint $-\sqrt{2} \leq \sin x+\cos x \leq \sqrt{2}\;\forall x\in \mathbb{R}$ So $\displaystyle \underbrace{\sin x+\cos x}_{\bf{\leq \sqrt{2}}} = y^2-y+\frac{1}{4}+a-\frac{1}{4} = \left(y-\frac{1}{2}\right)^2+a-\frac{1}{4}$ So $\displaystyle a-\frac{1}{4}>\sqrt{2}$ for no real solution. So we get $\displaystyle a>\sqrt{2}+\frac{1}{4}$

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