How to find all solutions of equation (2*cosx-square root3)(11sinx-9)=0?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the solutions of the equation: (2*cos x - sqrt 3) (11*sin x - 9 ) = 0.

If 2*cos x - sqrt 3 =0

=> cos x = sqrt 3/2

=> x  arc cos (sqrt 3 / 2)

=> x = pi/6 + 2*n*pi

If 11*sin x - 9 = 0

=> sin x = 9/11

=> x = arc sin ( 9/11) + 2*n* pi

The required solutions are x = pi/6 + 2*n*pi and x = arc sin ( 9/11) + 2*n* pi

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll start from the fact that a product is zero if one of it's factors is zero.

We'll set the first factor as zero.

2*cosx-sqrt3 = 0

We'll add sqrt3 both sides:

2cos x = sqrt3

cos x = sqrt3/2

x = +/-arccos(sqrt3/2) + 2kpi, k is an integer number

x = +/-(pi/6) + 2kpi

Let's put the next factor equal to zero.

11sinx-9 = 0

We'll add 9 both sides:

11sin x = 9

We'll divide by 11:

sin x = 9/11

x = (-1)^k*arcsin(9/11) + k*pi

The solutions of the equation are: {+/-(pi/6) + 2kpi} U {(-1)^k*arcsin(9/11) + k*pi}.

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