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Math Help - What Trig Identity are these?

  1. #1
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    What Trig Identity are these?

    I saw these trig identities at the beginning of this video on PatrickJMT Trigonometric Integrals Part 5 of 6 and I don't remember using these identities before. They weren't in the back of my textbook either. Does anyone know what they're called, and maybe a simple method for deriving them? And by simple I mean along the lines of deriving a half-angle formula using Pythagorean theorem and double-angle formulas. It doesn't have to be simple, I'm just hoping there's a simple way to memorize them!

    Here's the identities if the link doesn't work:

     sin(Ax)cos(Bx) = \frac{1}{2}[sin(Ax-Bx)+sin(Ax+Bx)]
     sin(Ax)sin(Bx) = \frac{1}{2}[cos(Ax-Bx)-cos(Ax+Bx)]
     cos(Ax)cos(Bx) = \frac{1}{2}[cos(Ax-Bx)+cos(Ax+Bx)]

    Edit: I just realize this might be better of in the Trig forum and not calculus. I posted here just out of habit! And fix't the typos
    Last edited by AZach; January 10th 2013 at 12:10 PM.
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  2. #2
    Super Member ebaines's Avatar
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    Re: What Trig Identity are these?

    I believe you have mis-written these identities; they should be:

     \sin(Ax)\cos(Bx) = \frac 1 2 [ \sin(Ax -Bx) + \sin(Ax + Bx)]

     \sin(Ax) \sin(Bx) = \frac 1 2[ \cos(Ax-Bx) - \cos(Ax + Bx) ]


     \cos(Ax) \cos(Bx) = \frac 1 2[ \cos(Ax-Bx) + \cos(Ax + Bx) ]


    These can all be derived from the basic identities:

     \sin (Ax + Bx) = \sin (Ax) \cos(Bx) + \cos(Ax)\sin(Bx),
     \cos(Ax + Bx) = \cos(Ax) \cos(Bx) - \sin(Ax) \sin (Bx)

    For example:

    \frac 1 2 [\sin(Ax -Bx) + \sin (Ax + Bx)] = \frac 1 2 [\sin(Ax) \cos(Bx) - \cos(Ax) \sin(Bx) + \sin(Ax)\cos(Bx) + \cos(Ax)\sin(Bx) ] = \sin(Ax)\cos(Bx)

    The other two can be derived in similar fashion.
    Last edited by ebaines; January 10th 2013 at 07:53 AM.
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  3. #3
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    Re: What Trig Identity are these?

    Hello, AZach!

    Careful! . . . Your identities have terrible typos.


    Product-to-Sum Indentities

    . . \begin{array}{ccc}\sin A\sin B &=& \frac{1}{2}\left[\cos(A-B) - \cos(A+B)\right] \\ \\[-3mm] \cos A\cos B &=& \frac{1}{2}\left[\cos(A-B) + \cos(A+B)\right] \\ \\[-3mm] \sin A\cos B &=& \frac{1}{2}\big[\sin(A-B) + \sin(A+B)\big] \end{array}


    Sum-to-Product Identities

    . . \begin{array}{ccc}\sin A + \sin B &=& 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2}) \\ \\[-3mm] \sin A - \sin B &=& 2\cos(\frac{A+B}{2})\sin(\frac{A-B}{2}) \\ \\[-3mm] \cos A + \cos B &=& 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2}) \\ \\[-3mm] \cos A - \cos B &=& \text{-}2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2}) \end{array}


    These are lesser-known identities,
    . . but become more important later on.
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