How do you divide ( i-2) / (i -6 )i−2i−6 in trigonometric form?

1 Answer
sankarankalyanam
Jun 25, 2018

color(maroon)((-2 + i) / (-6 + i) = 0.3676 ( 0.9957 - i 0.2942)−2+i−6+i=0.3676(0.9957−i0.2942)

Explanation:

z_1 / z_2 = (|r_1| / |r_2|) (cos (theta_1 - theta_2) + i sin (theta_1 - theta_2))z1z2=(|r1||r2|)(cos(θ1−θ2)+isin(θ1−θ2))

z_1 = -2 + i , z_2 = -6 + i z1=−2+i,z2=−6+i

|r_1| = sqrt(-2^2 + 1^2) = sqrt 5|r1|=√−22+12=√5

theta_1 = tan ^ (-1) (1/-2) = 153.43 ^@ " II Quadrant"θ1=tan−1(1−2)=153.43∘ II Quadrant

|r_2| = sqrt(-6^2 + (1)^2) = sqrt 37|r2|=√−62+(1)2=√37

theta_2 = tan ^-1 (1/ -6) = 170.54^@ , " II Quadrant"θ2=tan−1(1−6)=170.54∘, II Quadrant

z_1 / z_2 = |sqrt(5/37)| * (cos (153.43- 170.54) + i sin (143.43 - 170.54))z1z2=∣∣∣√537∣∣∣⋅(cos(153.43−170.54)+isin(143.43−170.54))

color(maroon)((-2 + i) / (-6 + i) = 0.3676 ( 0.9957 - i 0.2942)−2+i−6+i=0.3676(0.9957−i0.2942)

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