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Math Help - Trig identity, tan(2A)

  1. #1
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    Trig identity, tan(2A)

    Have a question i'm having some trouble with,

    Verify the identity;

    (sin A + sin 3A) / (cos A + cos 3A) = tan 2A

    not sure best way to go about this one,
    tried using sin3A = sin(2A + A) and expanding out but didn't seem to give me anything i wanted.

    Any help would be greatly appreciated.

    Cheers.
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  2. #2
    Member Black's Avatar
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    Using the identities

    \sin(x \pm y) = \sin x \cos y \pm \sin y \sin x

    \cos(x \pm y) = \cos x \cos y \mp \sin x \sin y,

    you can get

    \sin x + \sin y = 2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)

    \cos x + \cos y = 2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right).

    Set x=3A and y=A.
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  3. #3
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    Ok, i can see how to finish the problem and this may seem like a stupid question but how did you get from the sin(x + y) formulae to the sinx + siny ones
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  4. #4
    Member Black's Avatar
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    So you have

    sin (a + b) = sin a cos b + sin b cos a

    sin (a - b) = sin a cos b - sin b cos a

    Adding the two equations gives you

    sin (a + b) + sin (a - b) = 2 sin a cos b.

    Now set x = a + b and y = a - b, find a and b in terms of x and y, and you'll get the identity for sin x + sin y
    (similar for cos x + cos y).
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