How do you multiply (-5-3i)(3-i) (−5−3i)(3−i) in trigonometric form?

1 Answer
sankarankalyanam
Jun 25, 2018

color(green)((10 - 2 i)* (3 - i) = 18.44 * (-0.9762 - i 0.217)(10−2i)⋅(3−i)=18.44⋅(−0.9762−i0.217)

Explanation:

z_1 * z_2 = (r_1 * r_2) * ((cos theta_1 + theta_2) + i sin (theta_1 + theta_2))z1⋅z2=(r1⋅r2)⋅((cosθ1+θ2)+isin(θ1+θ2))

z_1 = (-5 - 3 i), z_2 = (3 - i)z1=(−5−3i),z2=(3−i)

r_1 = sqrt (-5^2 + -3^2) = sqrt (34)r1=√−52+−32=√34

theta _1 = tan ^-1 (-3/-5) = 210.96^@, " III quadrant"θ1=tan−1(−3−5)=210.96∘, III quadrant

r_2 = sqrt (3^2 + -1^2) = sqrt (10)r2=√32+−12=√10

theta _2 = tan ^-1 (-1/3) = -18.43^@ = 341.57^@, " IV quadrant"θ2=tan−1(−13)=−18.43∘=341.57∘, IV quadrant

z_1 * z_2 = (sqrt34 * sqrt 10) * (cos(210.96 + 341.57) + i (210.96 + 341.57))z1⋅z2=(√34⋅√10)⋅(cos(210.96+341.57)+i(210.96+341.57))

color(green)((10 - 2 i)* (3 - i) = 18.44 * (-0.9762 - i 0.217)(10−2i)⋅(3−i)=18.44⋅(−0.9762−i0.217)

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