Question 756874
One solution for {{{tan(3x)=1}}} comes from {{{3x=45^o}}} or {{{3x=pi/4}}}.
The period of the function tangent is {{{180^o}}} or {{{pi}}}, so we can express all solutions, using {{{k}}}= any integer, as
{{{3x=45^o+k*180^o}}} --> {{{x=(45^o+k*180^o)/3}}} --> {{{highlight(x=15^o+k*60^o)}}} or
{{{3x=pi/4+k*pi}}} --> {{{x=((pi/4+k*pi))/3}}} -->  {{{x=pi/12+k*pi/3}}} --> {{{x=pi/12+4k*pi/12}}} --> {{{highlight(x=(4k+1)pi/12)}}}
 
EXTRA:
If you are looking for solutions between {{{0^o}}} and {{{360^O}}},
{{{k=0}}} --> {{{highlight(x=15^o)}}},
{{{k=1}}} --> {{{x=15^o+1*60^o}}} --> {{{x=15^o+60^o}}} --> {{{highlight(x=75^o)}}},
{{{k=2}}} --> {{{x=15^o+2*60^o}}} --> {{{x=15^o+120^o}}} --> {{{highlight(x=135^o)}}},
{{{k=3}}} --> {{{x=15^o+3*60^o}}} --> {{{x=15^o+180^o}}} --> {{{highlight(x=195^o)}}},
{{{k=4}}} --> {{{x=15^o+4*60^o}}} --> {{{x=15^o+240^o}}} --> {{{highlight(x=255^o)}}},
{{{k=5}}} --> {{{x=15^o+5*60^o}}} --> {{{x=15^o+300^o}}} --> {{{highlight(x=315^o)}}}
We stop at {{{k=5}}} because
{{{k=6}}} --> {{{x=15^o+6*60^o}}} --> {{{x=15^o+360^o=375>360^o}}}
 
Measuring angles in radians instead of degrees, the solutions for {{{0<=x<2pi}}}
can be calculated from {{{x=(4k+1)pi/12}}} making {{{k}}}= 0, 1, 2, 3, 4, and 5.
{{{k=0}}} --> {{{highlight(x=pi/12)}}},
{{{k=1}}} --> {{{x=(4*1+1)pi/12}}} --> {{{highlight(5pi/12)}}}
{{{k=2}}} --> {{{x=(4*2+1)pi/12}}} --> {{{highlight(9pi/12)}}}
{{{k=3}}} --> {{{x=(4*3+1)pi/12}}} --> {{{highlight(13pi/12)}}}
{{{k=4}}} --> {{{x=(4*4+1)pi/12}}} --> {{{highlight(17pi/12)}}}
{{{k=5}}} --> {{{x=(4*5+1)pi/12}}} --> {{{highlight(21pi/12)}}}