



 
Hi Austin, To express 1 + i in the form r e^{i } = r (cos() + i sin()) I think of the geometry. In the complex plane plot the point 1 + i. The modulus r of p = i + i is the distance from O to P. Since PQO is a right triangle Pythagoras theorem tells you that r = √2. The argument of P is the angle, measured counterclockwise from the positive real axis to the line segment OP. Here = 3/4 . Cos( 3/4 ) = 1/√2 and sin(3/4 ) = 1/√2. Thus You don't need to use the diagram. The modulus of x + iy is √(x^{2} + y^{2}) which for i + i is
Also x = r cos () and y = r sin() so 1 = √2 cos () and 1 = √2 sin (). This gives = 3/4 . Penny  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 