Solve 5x tan (8x ) =3x .

Solution : Firstly, before we even begin solving we have to make one thing clear.  DO NOT CANCEL AN x FROM BOTH SIDES!!! Whereas this may appear like a natural thing to do it WILL cause us to lose solution here.

Hence, to solve this equation first we'll get all the terms on one side of the equation & then factor out an x of the equation.  If we can cancel out an x from all terms then it can be factored out.  Doing this gives,

                                     5x tan (8x ) - 3x = x (5 tan(8x ) - 3) = 0

Upon factoring we can illustrates that we ought to have either,

                                   x= 0      OR      tan (8x ) = 3/5

Note as well that if we'd canceled out the x we would have missed the first solution.  Now, we solved out an equation along with a tangent in it in previous section hence we'll not go into the details of this solution here.  Following is the solution to the trig equation.

x= 0.0676 + ? n/4                                                 n= 0, ±1, ±2,..........

x= 0.4603 + ? n/4

The complete set of solutions then to the original equation is following,

x= 0

x= 0.0676 + ? n/4

x= 0.4603 + ? n/4

n= 0, ±1, ±2,..........