# 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:

Please see the pic below:

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.

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