Solve the trigonometric equation sin 6x=-cos (3x), using double angle identity.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll use the double angle identity to re-write the term sin 6x:

sin 6x = sin 2*(3x) = 2 sin 3x*cos 3x

We'll re-write the equation, moving all terms to one side:

2 sin 3x*cos 3x + cos 3x = 0

We'll factorize by cos 3x:

cos 3x(2 sin 3x + 1) = 0

We'll set each factor as zero:

cos 3x = 0

3x = +/-arccos 0 + 2kpi

3x = +/-(pi/2) + 2kpi

We'll divide by 3:

x = +/-(pi/6) + 2kpi/3

We'll set the next factor as 0:

2 sin 3x + 1 = 0

sin 3x = -1/2

3x = (-1)^k*arcsin(-1/2) + kpi

x = (-1)^(k+1)*(pi/18) + kpi/3

The solutions of trigonometric equation are: {+/-(pi/6) + 2kpi/3 ; k integer}U{(-1)^(k+1)*(pi/18) + kpi/3 ; k integer}.

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