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Thread: Theory behind compound and double angle formulas

  1. #1
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    Question Theory behind compound and double angle formulas

    Hi!

    I hope someone can help. I'm just trying to have a solid understanding of trig compound and double formulas. Let me provide an example:

    Sin2x = 2CosxSinx

    So in the example above, is the left-side and the right-side it's own function? It appears that they are two separate functions that are equivalent to each other and hence why there is an equal sign. I say this because I can solve for either side, I get the same answer. I would appreciate if someone clarified this thinking for me.

    If what I said above is correct, then I have two other statements:
    1. The only reason we have developed the double-angle formulas is only make it easier for us humans to make calculations, right? I'm suggesting this since the calculator can solve cos(5pi/11) automatically, but humans can't do that. We instead need to split cos(5pi/11) into smaller pieces, and that's why we developed the compound and the double-angle formulas.

    The only reason we can solve cos(pi/4) in our heads is because it is a common angle on the unit circle that is derived from the 45-45-90 triangle... once we face angles that are not part of the 16 common angles on the unit circle, then we need to create compound angle formulas to help us find the value in these. Am I right about this thinking?

    2. Is it possible to create compound and double-angle like formulas for non-trig functions? I understand that non-trig functions don't have angles, but I have a feeling that we can find functions that are equivalent to each other...and even if they did exist, I wonder if they would have much utility considering that its not common to come across such a thing in the math curriculum. I would really appreciate some clarification on this.

    Sincerely,
    Olivia
    Last edited by otownsend; Jun 4th 2017 at 07:01 AM.
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  2. #2
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    Re: Theory behind compound and double angle formulas

    1. Yes, more or less. There are other ways to find closed forms for specific angles
    2.
    $2x=x+x$
    $a^{2x}=a^xa^x$
    $\log (2x)=\log 2+\log x $
    Last edited by SlipEternal; Jun 4th 2017 at 08:02 AM.
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  3. #3
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    Re: Theory behind compound and double angle formulas

    Cool, thanks for your help! I'm going to think about this for a bit longer and then get back to you if I have any more questions.
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    Re: Theory behind compound and double angle formulas

    Quote Originally Posted by otownsend View Post
    Cool, thanks for your help! I'm going to think about this for a bit longer and then get back to you if I have any more questions.
    There is another way to look at this, a bit more advanced, which requires an understanding of complex numbers.

    It's a bit outside the level of the pre-calculus forum but I'll put it here anyway for your enjoyment.

    There's a famous formula, Euler's Identity, that relates the exponential function of a complex variable with the trig functions of real variables.

    $e^{i \theta} = \cos(\theta) + i \sin(\theta)$

    where $i^2 = -1$

    what this allows us to do is things like the following

    $\sin(\theta) = Im(e^{i \theta})$

    $\begin{align*}
    &\sin(2\theta) = \\ \\
    &Im(e^{i 2\theta})= \\ \\
    &Im((e^{i \theta})^2)= \\ \\
    &Im((\cos(\theta) + i \sin(\theta))^2) =\\ \\
    &Im(\cos^2(\theta) + 2 i \sin(\theta)\cos(\theta) + \sin^2(\theta))= \\ \\
    &2 \sin(\theta)\cos(\theta)
    \end{align*}$

    Using the binomial formula to expand $(\cos(\theta)+i\sin(\theta))^n$ you can use Euler's Identity for the trig function of any integer factor $n$ of an original angle.
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