Prove that xy = rt(cos(u+v) + i sin(u+v))

Mathematics
Tutor: None Selected Time limit: 1 Day

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) + 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.

 

Apr 15th, 2015

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



Apr 15th, 2015

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