How do you solve $2cosx+2sinx=sqrt6$ in the interval [0,360]?

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Answer 1 · 2016-09-01T18:38:07

The solutions in the given interval are $x = 15˚$ and $x = 75˚$ .

Factor out a 2:

$2(cosx + sinx) = sqrt(6)$

$cosx + sinx = sqrt(6)/2$

Square both sides:

$(cosx + sinx)^2 = (sqrt(6)/2)^2$

$cos^2x + 2sinxcosx + sin^2x = 6/4$

Apply the identity $sin^2x + cos^2x = 1$ :

$1 + 2sinxcosx = 6/4$

Apply the identity $2sinxcosx = sin2x$ :

$sin2x = 1/2$

$2x = 30˚ and 2x = 150˚$

$x = 15˚ and 75˚$

Check both solutions in the original equation. Both work. Hence, our solution set is ${15˚, 75˚}$ .

Hopefully this helps!

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