Question 1116853
<pre>
{{{10sin(3x+ pi/6) + 7sqrt(3) = 2sqrt(3)}}}

First subtract {{{7sqrt(3)}}} from both sides:

{{{10sin(3x+ pi/6) = -5sqrt(3)}}}

Divide both sides by 10

{{{sin(3x+ pi/6) = -5sqrt(3)/10}}}

{{{sin(3x+ pi/6) = -sqrt(3)/2}}}

The right side is negative.
The left side is a sine.
The sine is negative in QIII and QIV.
The 2 particular solutions in [0,2<font face="symbol">p</font>) are,
from the unit circle 4<font face="symbol">p</font>/3 and 5<font face="symbol">p</font>/3. 

{{{3x+pi/6 = 4pi/3}}},   {{{3x+pi/6=5pi/3}}}
{{{3x=4pi/3-pi/6}}},   {{{3x=5pi/3-pi/6}}}
{{{3x=8pi/6-pi/6}}},   {{{3x=10pi/6-pi/6}}}
{{{3x=7pi/6}}},      {{{3x=9pi/6}}}
{{{x=7pi/18}}},       {{{3x=3pi/2}}}
                {{{x=pi/2}}}

The exact general solutions are found by adding 2<font face="symbol">p</font>&#8729;n
to the exact particular solutions in [0,2<font face="symbol">p</font>),
where n is any integer positive, negative or 0.

So the general solutions are:

{{{matrix(1,3,

7pi/18+2pi*n, and,pi/2+2pi*n)}}}   

If you like you can do a little work on
those exact general solutions and get

{{{matrix(1,3,

((7+36n)/18)pi, and,((1+4n)/2)pi)}}}

Edwin</pre>