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

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Answer 1 · 2018-06-17T13:58:56

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

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

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

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

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

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

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

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

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

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

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