AS/A2 Pure 4 Pure 5

P6 Topic 1: Complex numbers
Straight line loci 1: arg z = q
The locus of z when arg z = q, where q is a fixed known angle. If z=x+iy is represented on an Argand diagram by the vector going from the origin to the point P with coordinates (x,y), then the equation arg z = q requires the vector z to make an angle of q with the positive x axis (the real axis). So the locus is a line segment from the origin at an angle q to the x-axis. In the diagram below you can vary the value of q. Clearly, P must lie on the half-line shown.

Summary Cabri Geometre interactive geometry.