6. I will do E, you try the others the same way
let z = -5 + 5i
modulus z = |z| = √((-5)^2 + 5^2) = √50 = 5√2
argument:
tanØ = 5/-5 = -1, where Ø is in quad II
Ø = 135°
G. is done the same way
for D, think of it as -5 + 0i
6. Compute the modulus and argument of each complex number. I did a-c and f.
D. -5
E. -5+5i
G. -3-4i
7. Let z= -5sqrt3/2+5/2i and w= 1+sqrt3i
a. convert z and w to polar form
b. calculate zw using De Moivres Theorem
c. calculate (z/w) using De M's theorem
Please help with these two problems? I did 1-6 and 9-10 already.
3 answers
7.
z = -5√3/2 + 5/2i
= 5(-√3/2 + (1/2)i )
argument :
tan Ø = (1/2) / (-√3/2), where Ø is in II
tan Ø = -1/√3
Ø = 150° or 5π/6 radians
z = 5(cos 150° + isin 150°) or 5(cos 5π/6 + isin 5π/6)
in the same way:
w = √10(cos 60°+ isin 60°) or √10(cos π/3 + isin π/3)
z = -5√3/2 + 5/2i
= 5(-√3/2 + (1/2)i )
argument :
tan Ø = (1/2) / (-√3/2), where Ø is in II
tan Ø = -1/√3
Ø = 150° or 5π/6 radians
z = 5(cos 150° + isin 150°) or 5(cos 5π/6 + isin 5π/6)
in the same way:
w = √10(cos 60°+ isin 60°) or √10(cos π/3 + isin π/3)
b) to multiply two complex numbers in complex form
if u = r1(cos Ø1 + isinØ1) and v = r2(cosØ2 + isinØ2)
then uv = r1r2(cos(Ø1+Ø2) + isin(Ø1+Ø2)
and u/v = r1/r2 (cos(Ø1-Ø2) + isin(Ø1-Ø2)
so
(z)(w) = 5√10(cos (150+60) + isin(150+60)
= 5√10(cos 210° + isin 210°)
do z/w using the above definition.
if u = r1(cos Ø1 + isinØ1) and v = r2(cosØ2 + isinØ2)
then uv = r1r2(cos(Ø1+Ø2) + isin(Ø1+Ø2)
and u/v = r1/r2 (cos(Ø1-Ø2) + isin(Ø1-Ø2)
so
(z)(w) = 5√10(cos (150+60) + isin(150+60)
= 5√10(cos 210° + isin 210°)
do z/w using the above definition.
3 answers