Trig identity in complex multiplication

  • Thread starter Shaybay92
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  • #1
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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?
 

Answers and Replies

  • #2
737
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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)
 
  • #3
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Thanks I hadn't seen these identities before
 
  • #4
mathman
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Thanks I hadn't seen these identities before
These are basic identities, which are taught in the first course of trigonometry.
 
  • #5
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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!
 
  • #6
mathman
Science Advisor
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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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