Find the argument of (i) `z=-8i`        (ii) `z= 1/2 - isqrt(3)/2`

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tiburtius eNotes educator| Certified Educator

For complex number `z=x+iy` we calculate argument `arg(z)=varphi`  by using the following formula

`tan varphi=y/x`

(i) In this case we have `"Re"(z)=0` so tangent is undefined but we know that can only happen for `arg(z)=pi/2` or `arg(z)=(3pi)/2` and since `"Im"(z)<0` our argument is `arg(z)=(3pi)/2`


`tan varphi=(-sqrt3/2)/(1/2)=-sqrt3`

Hence we have `varphi=(2pi)/3` or `varphi=(5pi)/3` . Since our number `z` is in 4th quadrant argument is `arg(z)=(5pi)/3` ` `