Homework Help: Problem in finding the General Solution of a Trigonometric Equation v2

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1. May 19, 2017

Wrichik Basu

1. The problem statement, all variables and given/known data:

Find the general solution of the equation: $$\tan {x}+\tan {2x}+\tan {3x}=0$$

Answer given: $x=$ $\frac {n\pi}{3}$, $n\pi \pm \alpha$ where $\tan {\alpha} = \frac {1}{\sqrt {2}}$.

2. Relevant equations:

These equations may be used:

3. The attempt at a solution:

The answer from the "EITHER" is correct, but how do I simplify the second part?

Last edited: May 19, 2017
2. May 19, 2017

BvU

I find it hard to read and it is badly focused, so I am reluctant to type it out (after all you are too lazy to do so too ...), but I can distinguish you go from $$\left [ 1\ + \ {1\over 1-\tan\alpha\tan 2\alpha} \right ] = 0$$ to something I don't understand and -- If I read it right -- don't believe...

Could you post the steps (typed) ?

3. May 19, 2017

Wrichik Basu

I've posted a better picture. Please see. And the $\alpha$ in your post will be $x$.

4. May 19, 2017

BvU

Better focused, yes. Understand or believe ?

5. May 19, 2017

haruspex

Looks good as far as you went, but why did you stop?
You can throw away the denominator, and convert the cos 3x back to trig terms in x and 2x again.

6. May 19, 2017

BvU

Ah, the last $\cos 3x$ is more like $\cos 3x$ plus .....

7. May 19, 2017

Wrichik Basu

I could, but I would've also got a sin term, which I can take to other side, and then cross multiply to get tan terms in x and 2x. Then?

8. May 19, 2017

haruspex

You should get an equation involving tan x, tan 2x and a constant.

9. May 19, 2017

Wrichik Basu

Yes, $2\tan {x}\tan {2x}=0$.

Then....I understood. Thanks a lot.

10. May 19, 2017

haruspex

did you mean, $2-\tan {x}\tan {2x}=0$?

11. May 19, 2017

Wrichik Basu

No, I did a wrong calculation. Then how will I proceed after that?

12. May 19, 2017

BvU

I second haru: $2-\tan {x}\tan {2x}=0$ to be solved . Repeat: see post #2.

13. May 19, 2017

haruspex

If you now have that equation, expand the tan 2x. If not, please post your working.