The formula for modulus (r) and principal argument (θ) of a complex number (z = x + iy) is :
r = ∣z∣ = x²+y²
θ = arctan(y/x)
(a) z = 3 - 4i
Modulus, r = 3²+(−4)²
r = 9+16 = 25
∴ r = 5
Principal argument, θ = arctan(-4/3)
∴ θ = -53.13°
(b) z = -2 + i
Modulus, r = (−2)²+(1)²
r = 4+1
∴ r = 5
Principal argument, θ = arctan(1/(-2))
θ=−26.56°
Since the given complex number lies in the second quadrant,
Therefore, θ=(−26.56+180)°
∴θ=153.44°
(c) z = 1/(1+i3)
z=(1−i3)/((1+i3)(1−i3))
z=(1−i3)/(1²−(i3)²)
z=(1−i3)/(1+9)
z=(1−i3)/10
z=(1/10)−i(3/10)
Modulus, r =(1/10)²+(3/10)²
r=(4/100)
∴ r = (2/10)
Principal argument, θ = arctan(-(3/10)/(1/10))
θ=arctan(−3)
θ=−(π/3)
∴ θ=−60°
(d) z = (7 − i)/(-4 - 3i)
z = ((7 - i)(-4 + 3i))/((-4 - 3i)(-4 + 3i))
z = (-28 + 21i +4i + 3)/((-4)² - (3i)²)
z = (-25 + 25i)/(16 + 9)
z = (-25 + 25i)/25
z = -1 + i
Modulus, r =(−1)²+1²
r =1+1
∴r=2
Principal argument, θ = arctan(1/(-1))
θ = arctan(-1)
θ = -45°
Since the given complex number lies in the second quadrant,
Therefore, θ=(−45+180)°
∴θ=135°
(e) z = 5(cos(π/3)+isin(π/3))
z= 5cos(π/3)+i5sin(π/3)
Modulus, r =(5cos(π/3))²+(5sin(π/3))²
r = 5²(cos²(π/3)+sin²(π/3))
r = 5 1
∴ r = 5
Principal argument, θ = arctan(5sin(π/3)/5cos(π/3))
θ = arctan(tan(π/3))
θ = π/3
∴θ = 60°
(f) z = cos 2π/3 − sin 2π/3
z = (-1/2) -(3/2)
z=−(1+3)/2
Modulus, r =(cos(2π/3)−sin(2π/3))²+0
r=(cos²(2π/3)+sin²(2π/3)−2sin(2π/3)cos(2π/3)
r=(1−sin(4π/3))
r=(1+(3/2))
∴ r =(2+3)/2)
Principal argument, θ = arctan(0/(cos(2π/3) - sin(2π/3)))
θ=arctan(0)
Since the given complex number lies in the second quadrant,
Therefore, θ=(0+180)°
∴θ=180°
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