Determine all solutions of the equation sin14x=sin(-7x).

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First, we'll recall the fact that the sine function is odd, such as:

sin(-x) = -sin x

We'll re-write the equation, with respect to all the above:

sin 14x = - sin 7x

We'll add sin 7x both sides:

sin 14 x + sin 7x = 0

We can use two methods to solve this problem. One of them is to transform the sum of matching functions into a product. The aother method is to re-write the first term, using the double angle identity, into:

sin 14x = sin 2*(7x) = 2 sin 7x*cos 7x

We'll re-write the equation:

2 sin 7x*cos 7x + sin 7x = 0

We'll factorize by sin 7x:

sin 7x(2 cos 7x + 1) = 0

We'll set each factor as zero:

sin 7x = 0

7x = (-1)^k*arcsin 0 + 2kpi

7x = 0 + 2kpi

We'll divide by 7:

x =  2kpi/7

2 cos 7x + 1 = 0

cos 7x = -1/2

7x = arccos(-1/2) + kpi

x = +/-(pi/21) + kpi/7

The solutions of trigonometric equation are: {2kpi/7 ; k integer}U{+/-(pi/21) + kpi/7 ; k integer}.

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