Find solution: `cos^3x+sinx=1`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to re-write the equation such that:

`cos^3 x = 1 - sin x => cos^2 x*cos x = 1 - sin x`

Using the trigonometric identity `cos^2 x = 1 - sin^2 x` yields:

`(1 - sin^2 x)*cos x = 1 - sin x`

You need to convert the difference of squares `1 - sin^2 x` into a product, such that:

`(1 - sin x)(1 + sin x)*cos x = 1 - sin x`

Moving all terms to one side, yields:

`(1 - sin x)(1 + sin x)*cos x - (1 - sin x) = 0`

Factoring out `(1 - sin x)` yields:

`(1 - sin x)(cos x*(1 + sin x) - 1) = 0 => {(1 - sin x = 0),(cos x*(1 + sin x) - 1 = 0):} => {(sin x = 1),(cos x*(1 + sin x) = 1):}`

`{(x = (-1)^n*(pi/2) + n*pi),(cos x = 1 => x = 2npi):}`

Hence, evaluating the solutions to the given equation yields `x = (-1)^n*(pi/2) + n*pi` and` x = 2npi.`

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oldnick | (Level 1) Valedictorian

Posted on

Not complete, how do ya solve cubic equation?

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