Why in case of a complex number z=x+iy the principle argument is different from the general argument?
I got to know that the principle argument of the complex no.is from -pi to +pi
But the general argument is from 0 to 2pi. Why??
Is this true for any complex no.or only for a particular type of complex numbers??
When we talk about complex number, then the range of the principal argument of $\theta$ in: $$z=r(\cos(\theta)+i\sin(\theta))$$ is in $[-\pi,\pi]$.
When the complex number is in the first or the second quadrant, then we have: $$\theta\in[0,\pi]$$ When the complex number belongs to the thrid and fouth quadrant, we consider $\theta$ to be negative, or: $$\theta\in[-\pi,0]$$ In fact this doesn't change the value of sine and cosine function. In particular: $$\cos(-\phi)=\cos(\phi)$$ and: $$\sin(-\phi)=-\sin(\phi)$$