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

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

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Answer 1 · 2018-06-17T13:58:56

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

Explanation:

$(2 + i 3) \cdot (2 - i 7)$ $(2 + i 3) * (2 - i7)$

$z_{1} \cdot z_{2} = (r_{1} r_{2}) (cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2}))$ $z_1 * z_2 =( r_1 r_2) (cos (theta_1 + theta_2) + i sin(theta_1 + theta_2))$

$r_{1} = \sqrt{2^{2} + 3^{2}} = \sqrt{13}$ $r_1 = sqrt (2^2 + 3^2) = sqrt13$

$r_{2} = \sqrt{2^{2} + 7^{2}} = \sqrt{53}$ $r_2 = sqrt(2^2 + 7^2) = sqrt53$

$θ_{1} = arctan (\frac{3}{2}) = {56.31}^{\circ}$ $theta_1 = arctan(3/2) = 56.31^@$

$θ_{2} = arctan (- \frac{7}{2}) = - 74.05 = {285.95}^{\circ}, IV Quadrant$ $theta_2 = arctan(-7/2) = -74.05 = 285.95^@, " IV Quadrant"$

$z_{1} z_{2} = (\sqrt{13} \cdot \sqrt{53}) (cos (56.31 + 285.95) + sin (74.05 + 285.95))$ $z_1 z_2 = (sqrt13 * sqrt53) (cos(56.31+285.95) + sin(74.05 + 285.95))$

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

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

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

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