James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A60
appendix h Complex Numbers
The polar form of complex numbers gives insight into multiplication and division. Let
z 1 − r 1 scos 1 1 i sin 1 d
z 2 − r 2 scos 2 1 i sin 2 d
z
¨¡+¨
Im
¨
¨¡
z¡
Re
be two complex numbers written in polar form. Then
z 1 z 2 − r 1 r 2 scos 1 1 i sin 1 dscos 2 1 i sin 2 d
− r 1 r 2 fscos 1 cos 2 2 sin 1 sin 2 d 1 issin 1 cos 2 1 cos 1 sin 2 dg
Therefore, using the addition formulas for cosine and sine, we have
1 z 1 z 2 − r 1 r 2 fcoss 1 1 2 d 1 i sins 1 1 2 dg
z¡z
FIGURE 6
This formula says that to multiply two complex numbers we multiply the moduli and add
the arguments. (See Figure 6.)
A similar argument using the subtraction formulas for sine and cosine shows that to
divide two complex numbers we divide the moduli and subtract the arguments.
Im
0
¨
_¨
1
r
FIGURE 7
r
1
z
z
Re
z 1
z 2
− r 1
r 2
fcoss 1 2 2 d 1 i sins 1 2 2 dg z 2 ± 0
In particular, taking z 1 − 1 and z 2 − z (and therefore 1 − 0 and 2 − ), we have the
following, which is illustrated in Figure 7.
If z − rscos 1 i sin d, then
1
z − 1 r
scos 2 i sin d.
EXAMPLE 5 Find the product of the complex numbers 1 1 i and s3 2 i in polar
form.
SOLUTIOn From Example 4 we have
1 1 i − s2 Scos 4 4D 1 i sin
and
s3 2 i − 2FcosS2 6D1i sinS2 6DG
Im
0
œ„2
2
z=1+i
2œ„2
π
12
zw
Re
w=œ„3-i
So, by Equation 1,
s1 1 idss3 2 id − 2s2 FcosS 4 2 6D 1 i sinS 4 2 6DG
− 2s2 Scos 12 1 i sin 12D
FIGURE 8
This is illustrated in Figure 8.
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