Why is the argument of a+bi equal to arctan(b/a)?

Think about the point a+bi on the complex plane. Specifically, a is how far along the x (real) axis, and b is how far up the y (imaginary) axis the point is. If you draw a line connecting the origin and the point a+bi then notice that you've constructed a triangle with sides a, b, and sqrt(a^2+b^2). Recall that tan of an angle = opp/adj, applying this to the triangle gives that the angle between the x-axis and the line from the origin is equal to arctan(b/a). This is exactly what the argument of a complex number is, the angle between the x-axis and the line connecting the number and the origin.