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Explain how solve this hard equation sin^3 x+sin^2 3x+sin x=3?

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cane25 | Student, Undergraduate | eNoter

Posted May 8, 2013 at 5:56 PM via web

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Explain how solve this hard equation sin^3 x+sin^2 3x+sin x=3?

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oldnick | Valedictorian

Posted May 9, 2013 at 2:46 AM (Answer #2)

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 yields:

Hence, evaluating the solution to the given equation yields

                    IS'NT ENOUGH!

HOW DID YOU GET HERE? MAY YOU EXPLAIN?

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 8, 2013 at 6:09 PM (Answer #1)

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You need to remember that the sine function take on y values in interval [-1,1], hence, the following inequality holds for sine function, such that:

`-1 =< sin x <= 1 => -1 =< sin 3x <= 1`

Raising to square `sin 3x <= 1` yields:

`sin^2 3x <= 1`

Raising to cube `sin x <= 1` yields:

`sin^3 x <= 1`

Hence, evaluating the given relation yields:

`sin^3 x + sin^2 3x + sin x <= 1 + 1 + 1 = 3 => sin x = 1 => x = sin^(-1) 1 + n*pi => x = (-1)^k*pi/2 + n*pi` Hence, evaluating the solution to the given equation yields `x = (-1)^k*pi/2 + n*pi.`

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