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How can I solve this trigonometric equation? Sin(x)+ `sqrt(3) Cos (x)=0` Please help...

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princess90000 | eNotes Newbie

Posted May 14, 2013 at 2:01 AM via web

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How can I solve this trigonometric equation?

Sin(x)+ `sqrt(3) Cos (x)=0`

Please help me

I need process

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

Posted May 14, 2013 at 2:43 AM (Answer #1)

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Solve `sinx+sqrt(3)cosx=0` :

Wih some algebraic manipulation we can rewrite as an equation involving just tanx:

`sinx+sqrt(3)cosx=0`

`sinx=-sqrt(3)cosx` Now divide by cosx*

`sinx/cosx=-sqrt(3)`

`tanx=-sqrt(3)`

`x=tan^(-1)(-sqrt(3))`

`x=-pi/3+npi` for integer n.

(The inverse tangent returns an angle in the 4th quadrant if the argument is negative. The tangent is negative in the 2nd and 4th quadrants, so the complete solution includes adding multiples of `pi` )

* Since we divided by cosx, we must check that cosx=0 is not a solution. cosx=0 ==> x=`pi/2"or" (3pi)/2` . In each case `sinx != 0` so cosx=0 is not a solution.

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The solutions are `x=-pi/3+npi`

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The graph:

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