Solving 2tan(x) - 3sin(x) = 0 for -pi ≤ x < pi

The first step to answering this question is to get the equation in a simpler form. Aim to have it solely in terms of sin(x) and cos(x) with no fractional parts.                                                                                                                                                                                                       To begin doing this, remember that tan(x) = sin(x)/cos(x). Finally, to make the equation easier to deal with, times both sides of the equation by cos(x). This will leave the equation with no fractions and only in terms of sin(x) and cos(x).                                                                                                                                                       2tan(x) - 3sin(x) = 0 ----> using tan(x)=sin(x) / cos(x) ----> 2sin(x)/cos(x) - 3sin(x) = 0                                                                                                                                                                                                                     2sin(x)/cos(x) - 3sin(x) = 0 ----> times both sides by cos(x) ----> 2sin(x) - 3sin(x)cos(x) = 0                                                                                                                                              Now we can factorise out the sin(x). Do not simply divide by it, to remove it, as it provides more solutions to the equation as will be clear later. This means that the new equation is:                                                                                                                                                                                sin(x)(2 - 3cos(x)) = 0.                                                                                                                                                                                                    To find all the values of x where this equation is true you need to find the values of x where sin(x) = 0 and where 2 - 3cos(x) = 0. This is because any value times 0 = 0 and so if:                                                                                                                                                                                      sin(x) = 0 or (2 - 3cos(x)) = 0                                                                                                                                                                                            Then when these two parts are times together the answer will also be 0 and the equation will be true for that value of x. It is now clear why you could not simply divide by sin(x), as there are some solutions when sin(x) = 0 that would have been lost.                                                                                                                           So the answers to these two separate parts are:                                                                                                                                                                                sin(x) = 0 ----> x = sin-1(0) = 0 and -pi                                                                            2- 3cos(x) = 0 ---- rearranging ----> cos(x) = 2/3 -----> x = cos-1(2/3) = 0.841 and -0.841.                                                                                                                                              These are all the solutions to this question.                                                                                                                                                                                    The second solution for each part shown above is found by drawing the sin(x) and cos(x) curves in the range -pi ≤ x < pi and seeing the other value of x for which the equation holds. For example the cos(x) graph, if you draw it and look at it, has an obvious line of symmetry down the y axis. This means for any positive answer, there will be an answer exactly the same except negative.