How do you multiply # (2+3i)(2-7i) # in trigonometric form?

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Answer 1 · 2018-06-17T13:58:56

#color(blue)((2 + i 3) * (2 - i 7) = 26.25 (0.9524 - i 0.3047)#

#(2 + i 3) * (2 - i7)#

#z_1 * z_2 =( r_1 r_2) (cos (theta_1 + theta_2) + i sin(theta_1 + theta_2))#

#r_1 = sqrt (2^2 + 3^2) = sqrt13#

#r_2 = sqrt(2^2 + 7^2) = sqrt53#

#theta_1 = arctan(3/2) = 56.31^@#

#theta_2 = arctan(-7/2) = -74.05 = 285.95^@, " IV Quadrant"#

#z_1 z_2 = (sqrt13 * sqrt53) (cos(56.31+285.95) + sin(74.05 + 285.95))#

#(2 + i 3) * (2 - i 7) = 26.25 (cos 342.26 + i sin 342.26)#

#(2 + i 3) * (2 - i 7) = 26.25 (0.9524 - i 0.3047)#