How do you multiply (−1−7i)(−3−4i) in trigonometric form?

1 Answer
Shiva Prakash M V
Feb 23, 2018

Thus,
(−1−7i)(−3−4i)=10√2cis7π4

Explanation:

Let,z1=−1−7i,

Re(z1)=−1, Im(z1)=−7

r1=√(−1)2+(−7)2=√50=5√2

θ1=tan−1(−7−1)=π+tan−1(7)

Let,z2=−3−4i

Re(z2)=−3, Im(z2)=−4

r2=√(−3)2+(−4)2=√25=5

θ2=tan−1(−4−3)=π+tan−1(43)

z1=r1cisθ1
z2=r2cisθ2

z1z2=(r1cisθ1)(r2cisθ2)

z1z2=r1r2cisθ1cisθ2
By De-Moivre's theorem

cisθ1cisθ2=cis(θ1+θ2)

Thus,

z1z2=r1r2cis(θ1+θ2)

Substituting,

r1r2=5√2×5=10√2
θ1+θ2=π+tan−1(7)+π+tan−1(43)
=2π+tan−1(7)+tan−1(43)

tan−1(7)+tan−1(43)=tan−1(7+431−7×43)

7+431−7×43=7×3+43−7×4=21+43−28=25−25=1−1
tan−1(7)+tan−1(43)=tan−1(1−1)=2π−tan−1(1)

tan−1(1)=π4

2π−tan−1(1)=2π−π4=7π4

r1cisθ1r2cisθ2=10√2cis7π4

Thus,
(−1−7i)(−3−4i)=10√2cis7π4

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