Complex numbers, arguments for -i, -1 + sqrt3i

[veritas]

My questions is this:

Determine the argument for

a) -i
b)-1 + V3i (V = square root)

The answer in my textbook for question a) is 3pi/2, but i don't understand how they got to this answer. Please show me the steps in detail, and if it's the same steps for question b), no need to explain it, thank you.

 

[arthur ohlsten]

I am not sure if this is what you need

sketch a coordinate system with the x axis as reals, and the y axis as imaginaries, i
a)
then -i is a point on the "y" axis at -1
and the angle ,measured from the +x axis is 270 degrees or 3pi/2
then the complex number is cos@+i sin@

b)
on the above axis mark the point -1 on the x axis. Sketch a vertical line to the height sqrt3 , and make a mark.
we have a right triangle adjacent side 1 opposite side sqrt3.
sketch the hypoteneuse of the above right triangle and mark it 2
then the angle from the x axis is 180-60 or 120 degrees
120 degrees =2/3 pi

or the complex number is cos 2pi/3 +i sin 2pi/3

Arthur

 

[veritas]

then the angle from the x axis is 180-60 or 120 degrees

Thank you Arthur, your answer helped a lot. And to find the degrees you did sqrt3/1 and then tan -1 of sqrt3 to find the 60 degrees, and since it's on the second quadrant you did 180 - 60, right?

 

[arthur ohlsten]

Yes with explanation
the triangle is in the second quadrant because adjacent side - and opposite side +i

the hypoteneuse squared is then [sqrt 3]^2+1^2
hypoteneuse =+/- 2 , but hypoteneuse is always positive

ratio of adjacent side to hypoteneuse is 1/2, and I know sin of 30 degrees = 1/2
then "top" angle is 30 degrees and adjacent angle =60 degrees.

then angle from plus x axis is 180-60=120 degrees

not as elegant as your approach, but it gets me there
YOU ARE CORRECT
Arthur