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Determine the general solution of : `cos^2 theta + 3sin theta = - 3`     Thank you

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durbanville | High School Teacher | (Level 1) Educator Emeritus

Posted June 3, 2013 at 3:17 PM via web

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Determine the general solution of :

`cos^2 theta + 3sin theta = - 3`

 

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 3, 2013 at 3:24 PM (Answer #1)

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You need to use the basic trigonometric formula `1  - sin^2 theta = cos^2 theta` , to write the equation in terms of `sin theta` , only, such that:

`(1 - sin^2 theta) + 3sin theta = -3`

`1 - sin^2 theta + 3sin theta + 3 = 0`

`- sin^2 theta + 3sin theta +4 = 0`

Factoring out -1 yields:

`-1(sin^2 theta - 3sin theta - 4) = 0`

Replacing t for `sin theta` yields:

`t^2 - 3t - 4 = 0`

You may use quadratic formula to evaluate `t_(1,2)` such that:

`t_(1,2) = (3+-sqrt(9 + 16))/2 => t_(1,2) = (3+-5)/2`

`t_1 = 4, t_2 = -1`

Replacing back sin theta for t yields:

`sin theta = 4` invalid since `sin theta <= 1`

`sin theta = -1 => theta = (-1)^(n+1)(pi/2) + n*pi`

Hence, evaluating the general solution to the given equation, yields `theta = (-1)^(n+1)(pi/2) + n*pi.`

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