For all θ in [ 0 , π / 2 ] I need to show...

Mara Cook

Mara Cook

Answered

2022-06-26

For all θ in [ 0 , π / 2 ] I need to show that cos ( sin θ ) > sin ( cos θ )

Answer & Explanation

pheniankang

pheniankang

Expert

2022-06-27Added 22 answers

Over I = [ 0 , π 2 ] we have:
sin ( θ ) + cos ( θ ) = 2 sin ( θ + π 4 ) 2 < π 2
hence:
(1) θ I , cos θ < π 2 sin θ .
The LHS of (1) belongs to [0,1], the RHS belongs to [ π 2 1 , π 2 ] ; the sine function is increasing over [ 0 , π 2 ] , hence:
(2) θ I , sin ( cos θ ) < cos ( sin θ ) .

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