How Do You Correctly Convert Complex Numbers to Mod-Arg Form?

[Mentallic]

When given a complex number $z=x+iy$ and transforming this into its mod-arg form giving $rcis\theta$ where $r=\sqrt{(x^2+y^2)}$ and $\theta=arctan(y/x)$, we are assuming that $-\pi/2<\theta<\pi/2$.

What if however a student is asked to convert the complex number -1-i into mod-arg form? If they just start to plug-and-chug they'll quickly end up with the result $\theta=\pi/4$ and all of a sudden they've changed the complex number into it's negative, 1+i.

How does one avoid this dilemma?

 

[Tinyboss]

With the complex log function.

 

[Mentallic]

Thanks

 

[mathman]

When given a complex number $z=x+iy$ and transforming this into its mod-arg form giving $rcis\theta$ where $r=\sqrt{(x^2+y^2)}$ and $\theta=arctan(y/x)$, we are assuming that $-\pi/2<\theta<\pi/2$.

What if however a student is asked to convert the complex number -1-i into mod-arg form? If they just start to plug-and-chug they'll quickly end up with the result $\theta=\pi/4$ and all of a sudden they've changed the complex number into it's negative, 1+i.

How does one avoid this dilemma?

Be careful with the angle domain. You have a semicircle, while it should be a full circle. The signs of x and y have to be taken into account so that you end up in the correct quadrant.