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Trig identity in complex multiplication

  1. Jun 8, 2010 #1
    Just wondering how this is simplified to the third line:

    If w, z are complex numbers

    wz = rs( cos[tex]\alpha[/tex] + isin [tex]\alpha[/tex] ) (cos [tex]\varphi[/tex] + isin [tex]\varphi[/tex])

    wz = rs(cos[tex]\alpha[/tex] cos [tex]\varphi[/tex] - sin [tex]\alpha[/tex]sin[tex]\varphi[/tex]) + i(sin [tex]\alpha[/tex]cos[tex]\varphi[/tex] + cos [tex]\alpha[/tex] sin [tex]\varphi[/tex]))

    wz = rs(cos ([tex]\alpha[/tex] +[tex]\varphi[/tex]) + i sin([tex]\alpha[/tex] +[tex]\varphi[/tex]))

    What sort of trigonometric identity is used here between the 2nd and 3rd lines?
  2. jcsd
  3. Jun 8, 2010 #2
    Exactly those as written:

    sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
    cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
  4. Jun 8, 2010 #3
    Thanks I hadn't seen these identities before
  5. Jun 9, 2010 #4


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    Science Advisor

    These are basic identities, which are taught in the first course of trigonometry.
  6. Jun 9, 2010 #5
    You would think so but apparently my school doesnt see the importance in teaching this stuff. The only identity we were taught was

    sin^2(x) + cos^2(x) = 1

    not even all the half angle ones which I'm finding out about now... How helpful for me!
  7. Jun 10, 2010 #6


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    It looks like your school must teach trigonometry for a couple of weeks within a broader math course. When I was in high school, we had a one semester course for trig.
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