We'll use the auxiliary angle method to solve the
equation.
3 sin 5x - 3 cos 5x =
4
We'll divide entire equation by sqrt (3^2 + 3^2) =
sqrt18.
(3/3sqrt2)*sin 5x - (3/3sqrt2)*cos 5x =
4/3sqrt2
We'll consider sin a = (3/3sqrt2) and cos a =
(3/3sqrt2).
sin a*sin 5x - cos a*cos 5x =
4/3sqrt2
cos (a+5x) = -
4/3sqrt2
a + 5x = +/- arccos (- 4/3sqrt2) +
2k*pi
5x = +/- arccos (- 4/3sqrt2) + 2k*pi -
a
x = +/- arccos (- 4/3sqrt2)/5 + 2k*pi/5 -
a/5
The set of solutions of x is: {+/- arccos
(- 4/3sqrt2)/5 + 2k*pi/5 - a/5}.
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