Description
Use the following to complete part B:
• Let x = r(cos u + i sin u)
• Let y = t(cos v + i sin v)
Prove that xy = rt(cos(u+v) + i sin(u+v))
Provide a correct proof that includes written justification for each step showing the following:
1. The angle (or argument) of the product xy is (u + v).
2. The radius (or modulus) of the product xy is rt.
Explanation & Answer
x = r(cos u + i sin u)
y = t(cos v + i sin v)
xy = r(cos u + i sin u) . t(cos v + i sin v)
xy = rt ( cosu.cosv + i cosu sinv + i sinu cos v + i^2 sinu sinv )
xy = rt ( cos u cos v + i cos u sin v + i sin u cos v - sin u sin v )..............................i^2 = -1
xy = rt ( cos u cos v - sin u sin v + i ( cos u sin v + sin u cos v ) )
xy = rt ( cos ( u - v ) + i sin ( u + v) )
Clearly the argument is = u + v
radius = rt
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