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tan(3x) = tan(x)
By Anonymous on Saturday, June 17, 2000 - 05:39 pm:

What's the best way to get tan(3x) = tan(x) into a single trigonometric ratio (or two)?

By Patrick Aouad (P2687) on Tuesday, July 18, 2000 - 10:25 am:

I have found a way to make this trig equation into a single term. From the graphs, the answer is obviously 0°, 180°, 360° etc. However this is not your question.

To get this equation into a single solvable term I used the equation for a sum of angles and the special case of above; a double angle ratio, these are both below:


tan(x+y)
=
 tanx + tany
1-tanxtany
(1)
tan(2x)
=
 2tanx
1-tan2x
(2)


here we go, please bear with me:


tan(3x)
=
tanx
tan(2x+x)
=
tanx
 tan(2x) + tanx
1 - tan(2x)tanx
=
tanx
(from 1)
æ
è		  2tanx
1-tan2x
 + tanx ö
ø
æ
è		 1 -  2tanx
1 - tan2x
 tanx ö
ø
=
tanx
(from 2)
 2tanx + tanx(1-tan2x)
1 - tan2x - 2tan2x
=
tanx
(multiplying through by (1-tan2x)
 3tanx - tan3x
1 - tan2x - 2tan2x
=
tanx
3tanx - tan3x
=
tanx(1 - 3tan2x)
tanx + tan3x
=
0
tanx(1 + tan2x)
=
0
tanx
=
0 or tan2x=-1

So the only ratio that applies is tan(x)=0, so we get 0 degrees and therefore the general solution to this trig equation is:

x=180 degrees × n , where n is any integer.

These can be substituted and found correct as i mentioned earlier, I don't know whether this is the -best- way to get a nice single ratio, but it works, I hope I helped a little. If there is any problem please let me know.

Patrick.