cos3x + sin(2x+5pi/6)= -2, then x =(pi/3)(6k+1)?

Question

xplain???????

Answer 1

If cos3x + sin(2x+5pi/6) == -2,
then both cos3x == -1 and sin(2x+5pi/6) == -1.

Thus 3x is of the form pi*(2m+1),
and 2x+5pi/6 is of the form pi*(2n-1/2).

So from these, x is of the form (pi/3)*(2m+1),
and also 2x is of the form pi*(2n-1/2-5/6).

This means 2x is of the form pi*(2n-4/3)
or pi*(2n+2/3) if we shift n by replacing n with n+1.

.So x is of the form (pi/3)*(2m+1),
and also of the form pi*(n+1/3) or (pi/3)*(3n+1).

So x is of the form (pi/3)*r
where r is one more than a multiple of 2
but also one more than a multiple of 3.

So r is one more than a multiple of 2*3 = 6, i.e. r = 6k+1.

So x = (pi/3)*(6k+1).

Answer 2

substitute give value of x and slove. if you get -2, then you know that x = (pi/3)(6k+1)