z 1 = 1 + i 3 z_1=1+i\sqrt{3} z 1 = 1 + i 3
∣ z 1 ∣ = 1 + 3 = 2 |z_1|=\sqrt{1+3}=2 ∣ z 1 ∣ = 1 + 3 = 2
c o s θ 1 = 1 / 2 , s i n θ 1 = 3 / 2 cos\theta_1=1/2, sin\theta_1=\sqrt{3}/2 cos θ 1 = 1/2 , s in θ 1 = 3 /2
θ 1 = π / 3 \theta_1=\pi/3 θ 1 = π /3
z 2 = 2 i z_2=2i z 2 = 2 i
∣ z 2 ∣ = 2 |z_2|=2 ∣ z 2 ∣ = 2
c o s θ 2 = 0 , s i n θ 2 = 1 cos\theta_2=0, sin\theta_2=1 cos θ 2 = 0 , s in θ 2 = 1
θ 2 = π / 2 \theta_2=\pi/2 θ 2 = π /2
From Argand diagram:
a r g ( z 1 + z 2 ) = θ 1 + θ 2 − θ 1 2 = π 3 + π / 2 − π / 3 2 = arg(z_1+z_2)=\theta_1+\frac {\theta_2-\theta_1}{2}=\frac {\pi}{3}+\frac {\pi/2-\pi/3}{2}= a r g ( z 1 + z 2 ) = θ 1 + 2 θ 2 − θ 1 = 3 π + 2 π /2 − π /3 =
= π 3 + π 12 = 5 π 12 =\frac {\pi}{3}+\frac {\pi}{12}=\frac {5\pi}{12} = 3 π + 12 π = 12 5 π
t a n ( 5 π / 12 ) = t a n ( a r g ( z 1 + z 2 ) ) = 3 + 2 1 = 3 + 2 tan(5\pi/12)=tan(arg(z_1+z_2))=\frac {\sqrt{3}+2}{1}=\sqrt{3}+2 t an ( 5 π /12 ) = t an ( a r g ( z 1 + z 2 )) = 1 3 + 2 = 3 + 2
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