z1 = Ö2(cos(-45) + isin(-45)) = Ö2 cis(-45)
by
z2 = 3.6(cos(56.3) + isin(56.3)) = 3.6 cis(56.3)
In all such cases of complex division we require that the z, r in the denominator not be zero.
Thus:
(z1/z2) = (r1 cis(q1)/ r2 cis(q2)) = (r1/ r2) cis (q1 – q2)
Now: (r1/ r2) = (1.414/ 3.6) = 0.39
And we saw previously:
(q1 – q2) = arg(z1) – arg(z2) = (-45) – (56.3) = -101.3
Thus, the basic procedure for division entails dividing the lengths (r’s) and subtracting the angles (q1 – q2).
So:
(z1/ z2) = 0.39 (cos (-101.3) + isin(-101.3))
= 0.39((-0.195) + i(-0.98)) = -0.07 + 0.38i
What about? (1 + i) ¸ Ö3 – i
The first order of business is to get dividend and divisor each into polar form, specifically as a (cis) function:
Then (1 + i) = z1 = x1 + iy1, so arg(z1 ) = arctan (y1/x1)
Further:
arctan (y1/x1) = arctan (1/1) = arctan (1) so q1 = 45 deg
What about r1?
r1= [12 + 12]1/2 = Ö2 = 1.4
so z1 = 1.4 [cos (45) + isin(45)] = 1.4 cis(45)
Now: z2 = Ö3 – i
So arg(z2) = arctan(y2/x2) = arctan(-1/ Ö3) so q2 = (-30 deg)
And for r2: r2 = [(Ö3)2 + (-1)2]1/2 = Ö4 = 2
Then: z2 = 2[cos(-30) +isin(-30)] = 2cis(-30)
We divide: (z1/z2)
Which means dividing the r’s first:
r1/r2 = Ö2/ 2
Then subtract angles: [(q1 – q2) ] = {(45 deg) – (-30 deg)} = 75 degrees
So the end result of the division is:
(z1/z2) = Ö2/ 2 cis(75) = Ö2/ 2 {cos(75) + isin(75)}
= 0.707{cos(75) + isin(75)}
Since cos(75) = 0.258 and sin(75) =0.966, we have:
(z1/z2) = 0.707[(0.258) + i(0.966)] = 0.183 + 0.683i
Another very convenient way to express complex numbers is in the exponential form.
Thus, we can write: cos(q) + isin(q) = r exp (iq )
Thus, the previous numbers we divided (z1 and z2) may be expressed:
z1 = Ö2 [cos (45) + isin(45)] = Ö2 exp (i p/4)
z2 = 2[cos(-30) +isin(-30)] = 2 exp(i (-p/6))
Problems for the Math Maven:
1) Express each of the following end results in the form: r exp(iq)::
a) (2 + 3i)(1 – 2i)
b) (1 + i) (1- i)
c) (1 + Ö-3)2
2) Plot the results of (b) and (c) on the same Argand diagram and obtain the resultant. Check algebraically!
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