Verify the inequality : sinx<1/2cosx.

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to verify sin x < 1 / 2 cos x.

sin x < 1 / 2 cos x

=> 2 sin x * cos x < 1

=> sin 2x < 1

But sin 2x can also be equal to 1.

Note: Therefore the inequality should be sin x < = 1/2 cos x.

We therefore prove that sin x < = 1/ 2 cos x.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll multiply by 2 cos x both sides:

2 sin x * cos x<1

Instead of the value 1, we'll put the fundamental relation of trigonometry:

1= (sin x)^2 + (cos x)^2

The inequality will become:

2 sin x * cos x < (sin x)^2 + (cos x)^2

We'll subtract 2sin x*cos x both sides:

(sin x)^2 -2 sin x * cos x + (cos x)^2>0

The expression from the left side is a perfect square:

(sin x  - cos x)^2 > 0, true, for any real value of x.

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