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How to solve the equation square root 3*sin x-cosx=0?

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How to solve the equation square root 3*sin x-cosx=0?

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giorgiana1976's profile pic

Posted (Answer #1)

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First, we'll shift cos x to the right side:

sqrt3*sin x = cos x

We'll divide by cos x both sides:

sqrt3*sin x/cos x = 1

But the fraction sin x/cos x can be replaced by the tangent function tan x.

sqrt3*tan x = 1

tan x = 1/sqrt3

Since it is not allowed to keep the square root to denominator, we'll multiply both, numerator and denominator, by sqrt3.

tan x = sqrt3/3

x = arctan (sqrt3/3) + k*pi

x = pi/6 + k*pi

The set of solutions of the equation is: {pi/6 + k*pi}.

chrisbond437's profile pic

Posted (Answer #2)

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sin x/cos x=1/sqrt(3)

tan x=1/sqrt 3

we know tan 30=1/sqrt 3


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Posted (Answer #3)

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The question is not very clear, i.e is it sqrt(3) sinx-cosx =0 or sqrt(3sinx)-cos x=0

I will do it for the questions:

1) sqrt(3) sinx - cos x =0

or sqrt (3) sinx = cos x 

or, sinx/cosx = tanx = 1/sqrt(3) 

i.e. x = pi/6 (or 30 degrees)

2) sqrt (3 sinx)-cosx= 0

or 3sinx = cos^2 x = 1- sin^2 x

(using sin^2 x + cos^2 x = 1)

or, sin^2x + 3 sinx -1 = 0

solving this quadratic equation, we get sin x = (1/2) x (-3+ sqrt (13))

or, x = 17.62 degrees.

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