Trig identities: problem involving tan and double angle forumlas

There is this problem:

If $\displaystyle tan{x}={a}{\tan{\frac{x}{3}}}$, express $\displaystyle \tan{\frac{x}{3}}$ in terms of *a*.

I expressed tan in terms of $\displaystyle \tan{\frac{x}{3}}$ and isolated it. I obtained $\displaystyle \tan{\frac{x}{3}}=\pm{\sqrt{{\frac{3-a}{1-3a}}}}$.

But what about the sign? For example, for the first period $\displaystyle \tan{\frac{x}{3}}$ should be positive for values of x greater than 0 and negative for values of x less than 0, but there does not seem to be anything in the expression I obtained that can account for the sign. Could somebody tell me what I have done wrong?

Thanks a lot.

Re: Trig identities: problem involving tan and double angle forumlas

Quote:

Originally Posted by

**Kurama** There is this problem:

If $\displaystyle tan{x}={a}{\tan{\frac{x}{3}}}$, express $\displaystyle \tan{\frac{x}{3}}$ in terms of *a*.

I expressed tan in terms of $\displaystyle \tan{\frac{x}{3}}$ and isolated it. I obtained $\displaystyle \tan{\frac{x}{3}}=\pm{\sqrt{{\frac{3-a}{1-3a}}}}$.

But what about the sign? For example, for the first period $\displaystyle \tan{\frac{x}{3}}$ should be positive for values of x greater than 0 and negative for values of x less than 0, but there does not seem to be anything in the expression I obtained that can account for the sign. Could somebody tell me what I have done wrong?

Thanks a lot.

the tangent function is positive in quad I , negative in quad II , positive in quad III and negative in quad IV ...

for $\displaystyle 0 \le \frac{x}{3} < \frac{\pi}{2}$ , $\displaystyle \tan\left(\frac{x}{3}\right) \ge 0$

for $\displaystyle \frac{\pi}{2} < \frac{x}{3} \le \pi$ , $\displaystyle \tan\left(\frac{x}{3}\right) \le 0$

... and so on for quads III and IV