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Math Help - Understanding 2t , 1-t^2 etc

  1. #1
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    Understanding 2t , 1-t^2 etc

    Hi
    Can someone explain the following diagram I have drawn.
    For example where the t and 2 came from and their use in identities as I cannot make the relationship between them in trig identities involving tan
    Thanks
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  2. #2
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    Using pythagorean theorem you can see the relationship between the sides.

    A^2 = B^2 + C^2
    Where A is the hypotenuse (across from the 90 degree angle) and B and C are the vertical and horizontal sides.
    So,
    A^2 = (2t)^2 + (1-t^2)^2
    A^2 = 4t^2 + (1-2t^2+t^4) = t^4 +2t^2 + 1
    The right side is a perfect square and thus simplifies to:
    A^2 = (t^2 + 1)^2
    Rooting both side we see A, the hypotenuse, is t^2+1
    Does this clarify things for you?
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  3. #3
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    Hi
    Thanks, I get that bit but Im not sure on its use in double angle idnetities and when and where to use it
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  4. #4
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    Hi

    \sin 2t = 2 \sin t \cos t

    \cos 2t = \cos^2 t - \sin^2 t

    Therefore \tan 2t = \frac{2 \sin t \cos t}{\cos^2 t - \sin^2 t}

    Dividing numerator and denominator by \cos^2 t

    \tan 2t = \frac{2 \tan t}{1 - \tan^2 t}

    Let T = tan(t)

    \tan 2t = \frac{2T}{1 - T^2}
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  5. #5
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    beautiful
    thanks very much
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