problem 1 - umd physics€¦ · problem 1: algebraic manipulation of imaginary numbers the attached...
TRANSCRIPT
Problem 1: Algebraic manipulation of imaginary numbers The attached appendix B is from “Principles of Physics,” J.B. Marion and W.F. Hornyak, Saunders College Publishing, New York 1984 (see also Appendix D in Tipler, Ch. 3 in Hirose and Langren, class notes) Simplify expressions a) - d) and write the answer in the Cartesian (x+iy) form: a)
!
2 " i( ) " i 1" i 2( ) b)
!
2 " 3i( ) "2 + i( )
c)
!
1+ 2i
3" 4i+2 " i
5i
d)
!
i "1( )4
Find the value of the magnitude (ρ) and the argument (θ) for expressions f)-g) e) 1+i
f)
!
"2
1+ i 3
g)
!
3 " i( )6
Use the polar form (ρeiθ) to prove the equalities h) - j):
h)
!
5i
2 + i=1+ 2i
i)
!
"1+ i( )7
= "8 1+ i( ) j)
!
cos" + isin"( )n
= cos n"( ) + isin n"( ) {this equality is known as DeMoivre’s theorem} k) Without using a calculator, fill in the two smallest, positive values of the angles
(express as multiples of π) corresponding to the values of tan(θ) in the following table:
tan(θ) Θ (two values, expressed as multiples of π)
0
!
3 /3 1
!
3
!
±" -
!
3 -1
-
!
3 /3