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Multiply the two complex numbers (9-4i)*(2-i)(9−4i)⋅(2−i) using FOILFOIL Method.
(9-4i)*(2-i)(9−4i)⋅(2−i)
rArr18-9i-8i+4i^2⇒18−9i−8i+4i2
rArr 18-17i+4*(-1)⇒18−17i+4⋅(−1)
Note: i^2=(-1)i2=(−1)
rArr 18-17i-4⇒18−17i−4
rArr14-17i⇒14−17i
Hence, color(red)((9-4i)*(2-i)=14-17i(9−4i)⋅(2−i)=14−17i Intermediate Result 1
Convert this intermediate result to Trigonometric Form.
For a complex number in standard form: color(blue)(z=a+biz=a+bi,
r=sqrt(a^2+b^2r=√a2+b2
theta =tan^(-1)(b/a)θ=tan−1(ba)
cos (theta) = a/rcos(θ)=ar
rArr a=r*cos(theta)⇒a=r⋅cos(θ)
sin(theta)=b/rsin(θ)=br
rArr b=r*sin(theta)⇒b=r⋅sin(θ)
:. z=r*[cos(theta)+i*sin(theta)]
Using Intermediate Result 1,
r=sqrt(a^2+b^2
rArr sqrt(14^2+(-17)^2
rArr sqrt(196+289
rArr sqrt(485
:. r ~~ 22.02272
theta = tan^(-1)(b/a)
rArr tan^(-1)((-17)/14)
theta=tan^(-1)(-1.21429)
theta ~~-50.52763939^@
theta ~~ -50.5^@
Hence,
color(red)(z=22.0*[cos(-50.5^@)+i*sin(-50.5^@)]
Hope it helps.
