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.
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
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)
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.
To compute the modulus and argument of complex numbers, we can use the following formulas:
1. Modulus: The modulus (or absolute value) of a complex number z = a + bi is given by:
|z| = sqrt(a^2 + b^2)
2. Argument: The argument (or angle) of a complex number z = a + bi is given by:
arg(z) = tan^(-1)(b/a)
Now let's solve the given problems step by step:
6. Compute the modulus and argument of each complex number:
D. -5:
- Modulus: |z| = sqrt((-5)^2 + 0^2) = sqrt(25) = 5
- Argument: arg(z) = tan^(-1)(0/(-5)) = tan^(-1)(0) = 0
E. -5 + 5i:
- Modulus: |z| = sqrt((-5)^2 + 5^2) = sqrt(50) = 5√2
- Argument: arg(z) = tan^(-1)(5/(-5)) = tan^(-1)(-1) = -π/4 (in the fourth quadrant)
G. -3 - 4i:
- Modulus: |z| = sqrt((-3)^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5
- Argument: arg(z) = tan^(-1)(-4/(-3)) = tan^(-1)(4/3) ≈ 53.13°
7. Let z = -5√3/2 + 5/2i and w = 1 + √3i:
a. Convert z and w to polar form:
To convert a complex number from rectangular form (a + bi) to polar form (r*e^(iθ)), we use the following formulas:
- Modulus: r = |z| = sqrt(a^2 + b^2)
- Argument: θ = arg(z) = tan^(-1)(b/a)
z:
- Modulus: r = |z| = sqrt((-5√3/2)^2 + (5/2)^2) = sqrt(75/4) = 5√3/2
- Argument: θ = arg(z) = tan^(-1)((5/2) / (-5√3/2)) = tan^(-1)(-1/√3) = -π/6
w:
- Modulus: r = |w| = sqrt(1^2 + (√3)^2) = sqrt(4) = 2
- Argument: θ = arg(w) = tan^(-1)((√3) / 1) = tan^(-1)(√3) ≈ π/3
b. Calculate zw using De Moivre's Theorem:
De Moivre's Theorem states that for any complex number z in polar form (r*e^(iθ)) and any positive integer n, we have:
(z)^n = r^n * e^(i*nθ)
Using this theorem:
zw = (5√3/2 * 2) * e^(i*(-π/6 + π/3)) = 5√3 * e^(i*π/6) * e^(i*π/3) = 5√3 * e^(i*π/2) = 5√3i
c. Calculate (z/w) using De Moivre's Theorem:
(z/w) = (5√3/2 * 1/2) * e^(i*(-π/6 - π/3)) = 5√3/4 * e^(i*(-π/6 - π/3)) = 5√3/4 * e^(i*(-π/2)) = -5√3i/4
I hope this helps! Let me know if you have any further questions.