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Math Help - Expand tan(x+pi/4)

  1. #1
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    Expand tan(x+pi/4)

    What is the quickest method to do this Taylor expansion for the first three terms? Somebody previously suggested doing the expansion for tan(x) and then substituting in (x + pi/4) but if I do that I get the following working and run into problems:



    Thanks in advance.
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  2. #2
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    First of all, the series for tan is incorrect.

    Substitution will give you the Taylor series about -\pi/4. Is it the Maclaurin series that you're looking for? If so, you need to differentiate and evaluation at 0.
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  3. #3
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    f(x) = f(c) + f'(c)(x - c) + \dfrac{f''(c)(x-c)^2}{2!} + ...

    f(x) = \tan{x} centered at c = \frac{\pi}{4} ...

    f(x) = f\left(\frac{\pi}{4}\right) + f'\left(\frac{\pi}{4}\right)\left(x - \frac{\pi}{4}\right) + \dfrac{f''\left(\frac{\pi}{4}\right)\left(x - \frac{\pi}{4}\right)^2}{2!} + ...

    \tan{x} = 1 + 2\left(x - \frac{\pi}{4}\right) + 2\left(x - \frac{\pi}{4}\right)^2 + ...

    \tan\left(x + \frac{\pi}{4}\right) = 1 + 2x + 2x^2 + ...
    Attached Thumbnails Attached Thumbnails Expand tan(x+pi/4)-tanseries.jpg  
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  4. #4
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    Looks correct.
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