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Math Help - Mnemonic Device for the Addition of Angles Formulas

  1. #1
    A Plied Mathematician
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    Mnemonic Device for the Addition of Angles Formulas

    If you're like me, you have a hard time remembering the addition of angles formulas for sin and cos:

    \cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B), and
    \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B).

    The problem is, those formulas are extremely important ones to remember. So, how to do it?

    Here's an alternative that I, for one, find easier to remember: rotation matrices.

    Let's suppose you have a unit vector \hat{n}. Well then, it's at some angle B with respect to the positive x-axis, so we can represent it as

    \hat{n}=\begin{bmatrix}\cos(B)\\ \sin(B)\end{bmatrix}.

    This is no different from the conversion from cartesian coordinates to polar coordinates: x=r\cos(\theta) and y=r\sin(\theta). Just set r=1 and \theta=B in those equations to get my unit vector.

    Now then, suppose we rotate this unit vector through an angle A? That is equivalent to multiplying by the rotation matrix

    R_{A}=\begin{bmatrix}\cos(A) &-\sin(A)\\ \sin(A) &\cos(A)\end{bmatrix}.

    What is the result going to be? Another unit vector, \hat{n}_{\text{rot}}. And what will \hat{n}_{\text{rot}} look like? Well, the matrix multiplication will look like the following:

    \begin{bmatrix}\cos(A) &-\sin(A)\\ \sin(A) &\cos(A)\end{bmatrix}\begin{bmatrix}\cos(B)\\ \sin(B)\end{bmatrix}=\begin{bmatrix}\cos(A)\cos(B)-\sin(A)\sin(B)\\ \sin(A)\cos(B)+\cos(A)\sin(B)\end{bmatrix}.

    But rotating a unit vector at angle B through an angle A is going to result in a unit vector at angle A+B. The representation of that vector is going to be

    \hat{n}_{\text{rot}}=\begin{bmatrix}\cos(A+B)\\ \sin(A+B)\end{bmatrix}.

    Hence,

    \begin{bmatrix}\cos(A+B)\\ \sin(A+B)\end{bmatrix}=\begin{bmatrix}\cos(A)\cos(  B)-\sin(A)\sin(B)\\ \sin(A)\cos(B)+\cos(A)\sin(B)\end{bmatrix},

    which is equivalent to the addition of angles formulas.

    Since matrix multiplication is in my blood to stay, and the conversion from cartesian coordinates to polar coordinates is in my blood, I've reduced remembering the addition of angles formula down to remembering the rotation matrix form, which I think is easier to remember.

    To sum up, remember this:

    \begin{bmatrix}\cos(A+B)\\ \sin(A+B)\end{bmatrix}=\begin{bmatrix}\cos(A) &-\sin(A)\\ \sin(A) &\cos(A)\end{bmatrix}\begin{bmatrix}\cos(B)\\ \sin(B)\end{bmatrix},

    or "rotated vector equals rotation matrix times original vector."

    As always with mnemonic devices, take this or leave it. If it's useful, great. If something else works better for you, then ignore this!

    Regards.
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  2. #2
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    This works well if you're in a situation where you need to have rotation matrices memorized. If you're not (and many people first learning trigonometry are not), then remembering the standard double angle formulas (i.e. cos( 2x )=cos^2( x )-sin^2( x ), etc.) , combined with the symmetry of A and B is useful. As far as remembering the double angle formulae goes, I don't think I know a good method.
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  3. #3
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    I usually derive the double-angle formulas from the addition of angles formulas.

    You're right about rotation matrices usually being taught after trig. Perhaps, though, they could be moved up earlier in the curriculum with advantage.
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  4. #4
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    Quote Originally Posted by Ackbeet View Post
    I usually derive the double-angle formulas from the addition of angles formulas.
    I didn't mean to imply that I'd seen them taught any other way. But usually the double-angle formulas come right after, and they're slightly simpler to remember.
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  5. #5
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    I know this is trig section but I couldn't resist. You can use calculus. I find it easier to remember the \sin(A+B) formula

    Here is a trick I found:

    \displaystyle \frac{d}{dA}\sin(A+B) = \frac{d}{dA}\sin(A)\cos(B)+\frac{d}{dA}\cos(A)\sin  (B)  = \cos(A)\cos(B) - \sin(A)\sin(B)

    Or for double angle identity

    \displaystyle \frac{d}{dA}\sin(2A) = 2\cos(2A)

    and

    \displaystyle \frac{d}{dA}\sin(2A) = 2\frac{d}{dA}\sin(A)\cos(A) = 2 \cos(A)\cos(A) -2 \sin(A)\sin(A)

    You basically only need to remember one of the identities and you get the others for free (well, almost).

    Also, using complex numbers it's quite awesome:

    e^{i (A+B)} = \cos(A+B) + i \sin(A+B)

    But also

    e^{i (A+B)} = e^{iA}e^{iB} = (\cos(A) + i \sin(A))(\cos(B)+i \sin(B))

    You multiply these out and equate real and imaginary parts to get both identities at the same time. Of course this is essentially the same as Ackbeet's method with the matrices (modulo notation). If you insist on using one of these, it's either multiplication with rotation matrices or complex numbers. I find it easier to remember Euler's identity and derive the identities from it if I need. It'll take up less space than the matrices method, i think. As far as teaching goes... it's tough.
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  6. #6
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    Data Recovery Software

    You're right about rotation matrices usually being taught after trig. It’s working well. Many thanks to you…




    Regards,
    Data Recovery Software
    http://www.datadoctor.biz
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