Complex Numbers - Argument of Z

• Feb 8th 2011, 10:24 PM
iFuuZe
Complex Numbers - Argument of Z
Find the argument of z for each of the following in the interval [0,2pi]

i) Z=5-5i
ii) Z=55
iii) Z=-4+8i

i know first off all you plot it on the complex number plane, to find out the quadrant its in. say for i) its quadrant 4. and now im confused what you do next please help.
• Feb 8th 2011, 11:02 PM
Sudharaka
Quote:

Originally Posted by iFuuZe
Find the argument of z for each of the following in the interval [0,2pi]

i) Z=5-5i
ii) Z=55
iii) Z=-4+8i

i know first off all you plot it on the complex number plane, to find out the quadrant its in. say for i) its quadrant 4. and now im confused what you do next please help.

Dear iFuuZe,

Yes, its in the fourth quadrant. The arguement of a complex number is the angle it makes with the real axis when represented in the Argand diagram. Therefore in this case the arguement can be found by $\displaystyle \tan^{-1}\left(\frac{y}{x}\right)=\tan^{-1}\left(\frac{-5}{5}\right)=\tan^{-1}(-1)=-\frac{\pi}{4}$

Hope you understood.
• Feb 9th 2011, 02:37 AM
DrSteve
I think that the easiest way to do this is to plot the complex number in the plane and then form a right triangle by actually drawing the path as you plot the point (for example for 5-5i, you move right 5 then down 5 from the origin - actually draw this path), and then drawing a straight line segment from the origin to the point (for the hypotenuse). Then label all 3 sides (in the previous example, the sides would be 5, 5 and $\displaystyle 5\sqrt{2}$ - no need to worry about negative signs since the picture already shows you what quadrant you're in).

Make sure you know the 30,60,90 and 45,45,90 triangles by heart.

To find a value of arg z, just use the picture and sin, cos or tan to get the reference angle, and then use the quadrant you're in to find a value of arg z.

Note that arg z takes on infinitely many values. If you want the principle argument (which I write Arg z), you may need to add or subtract $\displaystyle 2\pi$ to get a value between $\displaystyle -\pi$ and $\displaystyle \pi$.