Answer to Question #117368 in Algebra for jay

Question #117368
Find the modulus and the principal argument of each of the given complex numbers.
(a) 3 − 4i,

(b) −2 + i,

(c) 1/1 + i √3

(d) (7 − i)/(-4 - 3i)

(e) 5(cos π/3 + isin π/3)

(f) cos 2π/3 − sin 2π/3.
1
Expert's answer
2020-06-03T19:01:23-0400

The formula for modulus (r) and principal argument (θ) of a complex number (z = x + iy) is :


r = ∣z∣ = x²+y²\sqrt{\smash[b]{x² + y²}}

θ = arctan(y/x)


(a) z = 3 - 4i

Modulus, r = 3²+(4)²\sqrt{\smash[b]{3² + (-4)²}}

​ r = 9+16\sqrt{\smash[b]{9 + 16}} = 25\sqrt{\smash[b]{25}}

∴ r = 5


Principal argument, θ = arctan(-4/3)

θ = -53.13°

(b) z = -2 + i

Modulus, r = (2)²+(1)²\sqrt{\smash[b]{(-2)² + (1)²}}

r = 4+1\sqrt{\smash[b]{4 + 1}}

∴ r = 5\sqrt{\smash[b]{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)1/(1 + i\sqrt{\smash[b]{3}})


z=(1i3)/((1+i3)(1i3))z = (1 - i\sqrt{\smash[b]{3}})/((1 + i\sqrt{\smash[b]{3}})(1 - i\sqrt{\smash[b]{3}}))


z=(1i3)/(1²(i3)²)z = (1 - i\sqrt{\smash[b]{3}})/(1² - (i\sqrt{\smash[b]{3}})²)


z=(1i3)/(1+9)z = (1 - i\sqrt{\smash[b]{3}})/(1 + 9)


z=(1i3)/10z = (1 - i\sqrt{\smash[b]{3}})/10


z=(1/10)i(3/10)z = (1/10) - i(\sqrt{\smash[b]{3}}/10)



Modulus, r =(1/10)²+(3/10)²\sqrt{\smash[b]{(1/10)² + (\sqrt{\smash[b]{3}}/10)²}}

r=(4/100)r = \sqrt{\smash[b]{(4/100)}}


∴ r = (2/10)


Principal argument, θ = arctan(-(3/10)/(1/10)(\sqrt{\smash[b]{3}}/10)/(1/10))


θ=arctan(3)(- \sqrt{\smash[b]{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\sqrt{\smash[b]{(-1)² + 1²}}

r =1+1\sqrt{\smash[b]{1 + 1}}


r=2\sqrt{\smash[b]{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\sqrt{\smash[b]{(5cos(π/3))² + (5sin(π/3))²}}

r = 5²(cos²(π/3)+sin²(π/3))\sqrt{\smash[b]{5²(cos²(π/3)+sin²(π/3))}}

r = 5 1\sqrt{\smash[b]{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)(\sqrt{\smash[b]{3}}/2)


z=(1+3)/2z = - (1 + \sqrt{\smash[b]{3}})/2


Modulus, r =(cos(2π/3)sin(2π/3))²+0\sqrt{\smash[b]{(cos(2π/3) - sin(2π/3))² + 0}}


r=(cos²(2π/3)+sin²(2π/3)2sin(2π/3)cos(2π/3)r = \sqrt{\smash[b]{(cos²(2π/3) + sin²(2π/3) - 2sin(2π/3)cos(2π/3)}}


r=(1sin(4π/3))r = \sqrt{\smash[b]{(1 - sin(4π/3))}}


r=(1+(3/2))r = \sqrt{\smash[b]{(1 + (\sqrt{\smash[b]{3}}/2)) }}


∴ r =(2+3)/2)\sqrt{\smash[b]{(2 + \sqrt{\smash[b]{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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