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prove this quantity (1-x^2/2)<=cosx<= (1-x^2/2)+(x^4/4*3*2)  

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samihaalhabsi | Student, Undergraduate | (Level 2) eNoter

Posted November 14, 2011 at 4:15 PM via web

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prove this quantity

(1-x^2/2)<=cosx<= (1-x^2/2)+(x^4/4*3*2)


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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 25, 2011 at 11:20 PM (Answer #1)

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Remember that the values of cosine function are included in the interval [-1;1].

You must show that the inequality is true.

`1- (x^2/2) =lt cos x = lt 1- (x^2/2) + x^4/24`

Add the quantity `x^2/2`  to the left and to the right of inequality, to keep it stable.

`1- (x^2/2)+ (x^2/2)=lt cos x = lt 1- (x^2/2) + (x^2/2) +x^4/24`

Yo  may reduce the opposite terms:

`1=lt cos x = lt 1+x^4/24`

Notice that the left side is a false statement because the values of the cosine function do not overpass 1.

ANSWER: The inequality is not true.

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