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Math Help - Trig identities: problem involving tan and double angle forumlas

  1. #1
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    Trig identities: problem involving tan and double angle forumlas

    There is this problem:

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

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

    But what about the sign? For example, for the first period \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.
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    Re: Trig identities: problem involving tan and double angle forumlas

    Quote Originally Posted by Kurama View Post
    There is this problem:

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

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

    But what about the sign? For example, for the first period \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 0 \le \frac{x}{3} < \frac{\pi}{2} , \tan\left(\frac{x}{3}\right) \ge 0

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

    ... and so on for quads III and IV
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