need help in complex numbers argument Any help appreciated please,
use an argand diagram to find, in the form a+bi, the complex numbers which satisfy the following pairs of equations.
arg(z+2)=1/2π, argz=2/3π
Thanks

$z= -2 +2\sqrt{3} i$ -- to see why,  first note that $arg(z+2)=\frac{1}{2}\pi$ means $z+2$ is on the positive imaginary axis, so $z$ is (in rectangular terms) 2 units left of that, so $a=-2$ (where $z=a+bi$).  Now thinking of $z$ in polar form, the given condition $arg(z)=\frac{2}{3}\pi$ means $z=r e^{\frac{2}{3}\pi i}$ for whatever value of $r$ puts $z$ 2 units left of the vertical axis -- but noting the 30-60-90 triangles associated with $z$ (e.g. drop a vertical from $z$ down to the negative real axis), it must be the case that $r=4$.  Converting to rectangular gives the answer above, or equivalently (again noting 30-60-90 triangle stuff), the vertical leg of the triangle is $\sqrt{3}$ times the length of the horizontal leg.
