Show that if 0 &lt; x &lt; 1 2 </mfrac> &#x03C0;<!-- π --> , then

Dania Mueller

Dania Mueller

Answered question

2022-06-22

Show that if 0 < x < 1 2 π, then 0 < cos x < sin x x < 1 cos x

Answer & Explanation

America Barrera

America Barrera

Beginner2022-06-23Added 23 answers

By the mean value theorem,
sin x x = sin x sin 0 x = cos c x
for some c x ( 0 , x ) . Because cos is strictly decreasing on [ 0 , π / 2 ] ,, we have cos x < cos c x , giving the first inequality. The second inequality is the same as
sin x cos x x < 1.
That equals ( sin 2 x ) / ( 2 x ) , and because sin u < u for all u>0, we have the second inequality.

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