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Caleb Proctor 2022-07-08 Answered
Prove that 0 1 + cos θ 2 + sin θ 4 3 for all real θ
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Elianna Wilkinson
Answered 2022-07-09 Author has 11 answers
Using half angle formula we convert it to tan function. (Hoping you know the formulae for sin,cos to convert it to tan ( x / 2 )). So we get it as
1 1 + tan 2 ( x 2 ) + tan ( x 2 )
The maximum of denominator is obvious which is infinity so minimum of the whole function is 0. Now we use calculus to find the minimum of the denominator. Differentiating only the denominator we have 2 tan ( x 2 ) = 1 knowing that we can cancel out sec 2 ( x 2 ) as it is never 0. Thus minimum is achieved at tan ( x 2 ) = 1 / 2 . Verify using the second derivative test. Thus min of denominator is maximum of the function . Substituting −1/2 back we get the max as 4 3 as desired.

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