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Show the trigonometric equation has no solutions

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,755
Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.
 

kaliprasad

Well-known member
Mar 31, 2013
1,322
Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.
if sin (x) = cos (y) then

$x+y = \pi/2 +2npi$ or $(2n+1)pi-\pi/2$

so lowest | cos a + sin a | = $\pi/2$

so |cos a + sin a| =$ \pi/2$ should have a solution

but |cos a + sin a | <= $\sqrt(2)$

so no solution