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

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}.