(a) 3 − 4i
z = 3 − 4 i z=3-4i z = 3 − 4 i ∣ z ∣ = ( 3 ) 2 + ( − 4 ) 2 = 5 |z|=\sqrt{(3)^2+(-4)^2}=5 ∣ z ∣ = ( 3 ) 2 + ( − 4 ) 2 = 5 A r g ( z ) = arctan ( − 4 3 ) = − arctan ( 4 3 ) Arg(z)=\arctan(-{4\over 3})=-\arctan({4\over 3}) A r g ( z ) = arctan ( − 3 4 ) = − arctan ( 3 4 )
(b) −2 + i
z = − 2 + i z=-2+i z = − 2 + i ∣ z ∣ = ( − 2 ) 2 + ( 1 ) 2 = 5 |z|=\sqrt{(-2)^2+(1)^2}=\sqrt{5} ∣ z ∣ = ( − 2 ) 2 + ( 1 ) 2 = 5 A r g ( z ) = π − arctan ( 1 2 ) Arg(z)=\pi-\arctan({1\over 2}) A r g ( z ) = π − arctan ( 2 1 )
(c) 1/(1 + i√3)
z = 1 1 + i 3 = 1 − i 3 4 z={1\over 1+i\sqrt{3}}={1-i\sqrt{3}\over 4} z = 1 + i 3 1 = 4 1 − i 3 ∣ z ∣ = ( 1 4 ) 2 + ( − 3 4 ) 2 = 1 2 |z|=\sqrt{({1\over 4})^2+(-{\sqrt{3}\over 4})^2}={1\over 2} ∣ z ∣ = ( 4 1 ) 2 + ( − 4 3 ) 2 = 2 1 A r g ( z ) = − π 3 Arg(z)=-{\pi\over 3} A r g ( z ) = − 3 π
(d) (7 − i)/(−4 − 3i)
z = 7 − i − 4 − 3 i = 1 25 ( 7 − i ) ( − 4 + 3 i ) = − 1 + i z={7-i\over -4-3i}={1\over 25}(7-i)(-4+3i)=-1+i z = − 4 − 3 i 7 − i = 25 1 ( 7 − i ) ( − 4 + 3 i ) = − 1 + i ∣ z ∣ = ( − 1 ) 2 + ( 1 ) 2 = 2 |z|=\sqrt{(-1)^2+(1)^2}=\sqrt{2} ∣ z ∣ = ( − 1 ) 2 + ( 1 ) 2 = 2 A r g ( z ) = π − π 4 = 3 π 4 Arg(z)=\pi-{\pi\over 4}={3\pi\over 4} A r g ( z ) = π − 4 π = 4 3 π
(e) 5(cos π/3 + isin π/3)
z = 5 ( cos ( π 3 ) + i sin ( π 3 ) ) z=5(\cos({\pi\over 3})+i\sin({\pi\over 3})) z = 5 ( cos ( 3 π ) + i sin ( 3 π )) ∣ z ∣ = 5 |z|=5 ∣ z ∣ = 5 A r g ( z ) = π 3 Arg(z)={\pi\over 3} A r g ( z ) = 3 π
(f) cos 2π/3 −sin 2π/3.
z = cos ( 2 π 3 ) − sin ( 2 π 3 ) = − 1 2 − 3 2 z=\cos({2\pi\over 3})-\sin({2\pi\over 3})=-{1\over 2}-{\sqrt{3}\over 2} z = cos ( 3 2 π ) − sin ( 3 2 π ) = − 2 1 − 2 3 ∣ z ∣ = 1 + 3 2 |z|={1+\sqrt{3}\over 2} ∣ z ∣ = 2 1 + 3 A r g ( z ) = π Arg(z)=\pi A r g ( z ) = π
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