z = − 7 z=-7 z = − 7
x = R e ( z ) = − 7 x=Re(z)=-7 x = R e ( z ) = − 7
y = I m ( z ) = 0 y=Im(z)=0 y = I m ( z ) = 0
∣ z ∣ = x 2 + y 2 = 7 |z|=\sqrt{x^2+y^2}=7 ∣ z ∣ = x 2 + y 2 = 7
ϕ = a r g ( z ) = π − a r c t a n ( y / ∣ x ∣ ) = π − a r c t a n ( 0 / 7 ) = π − 0 = π \phi =arg(z)=\pi-arctan(y/|x|)=\pi-arctan(0/7)=\pi-0=\pi ϕ = a r g ( z ) = π − a rc t an ( y /∣ x ∣ ) = π − a rc t an ( 0/7 ) = π − 0 = π
z = 7 ( c o s π + i s i n π ) z=7(cos\pi+isin\pi) z = 7 ( cos π + i s inπ )
To find the 6th roots we use this formula
z k = z 6 = ∣ z ∣ 6 ( c o s ϕ + 2 π k 6 + i s i n ϕ + 2 π k 6 ) , k = 0 , 1 , 2 , 3 , 4 , 5 z_k=\sqrt[6]{z}=\sqrt[6]{|z|}(cos {\frac {\phi+2\pi k} 6}+isin{\frac {\phi+2\pi k} 6}), k=0,1,2,3,4,5 z k = 6 z = 6 ∣ z ∣ ( cos 6 ϕ + 2 πk + i s in 6 ϕ + 2 πk ) , k = 0 , 1 , 2 , 3 , 4 , 5
So, let's find all z k z_k z k
z 0 = 7 6 ( c o s π 6 + i s i n π 6 ) z_0=\sqrt[6]{7}(cos {\frac {\pi} 6}+isin{\frac {\pi} 6}) z 0 = 6 7 ( cos 6 π + i s in 6 π )
z 1 = 7 6 ( c o s π 2 + i s i n π 2 ) z_1=\sqrt[6]{7}(cos {\frac {\pi} 2}+isin{\frac {\pi} 2}) z 1 = 6 7 ( cos 2 π + i s in 2 π )
z 2 = 7 6 ( c o s 5 π 6 + i s i n 5 π 6 ) z_2=\sqrt[6]{7}(cos {\frac {5\pi} 6}+isin{\frac {5\pi} 6}) z 2 = 6 7 ( cos 6 5 π + i s in 6 5 π )
z 3 = 7 6 ( c o s 7 π 6 + i s i n 7 π 6 ) z_3=\sqrt[6]{7}(cos {\frac {7\pi} 6}+isin{\frac {7\pi} 6}) z 3 = 6 7 ( cos 6 7 π + i s in 6 7 π )
z 4 = 7 6 ( c o s 3 π 2 + i s i n 3 π 2 ) z_4=\sqrt[6]{7}(cos {\frac {3\pi} 2}+isin{\frac {3\pi} 2}) z 4 = 6 7 ( cos 2 3 π + i s in 2 3 π )
z 5 = 7 6 ( c o s 11 π 6 + i s i n 11 π 6 ) z_5=\sqrt[6]{7}(cos {\frac {11\pi} 6}+isin{\frac {11\pi} 6}) z 5 = 6 7 ( cos 6 11 π + i s in 6 11 π )
Now, let's plot all roots on Argand diagram
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