How do you multiply  (4+5i)(-3+7i)  in trigonometric form?

1 Answer
Feb 28, 2018

$\left(4 + 5 i\right) \left(- 3 + 7 i\right) = 48.765 c i s 2.317$
$\left(4 + 5 i\right) \left(- 3 + 7 i\right) = - 33.111 + 35.842 i$

Explanation:

Let
${z}_{1} = 4 + 5 i$
${z}_{2} = - 3 + 7 i$
We need to find

$z = {z}_{1} {z}_{2}$
${r}_{1} = \sqrt{{4}^{2} + {5}^{2}} = \sqrt{16 + 25} = \sqrt{41}$
$r 1 = \sqrt{41}$
${\theta}_{1} = {\tan}^{-} 1 \left(\frac{5}{4}\right)$
${z}_{1} = {r}_{1} c i s {\theta}_{1}$
${z}_{1} = \sqrt{41} c i s \left({\tan}^{-} 1 \left(\frac{5}{4}\right)\right)$

${r}_{2} = \sqrt{{\left(- 3\right)}^{2} + {7}^{2}} = \sqrt{9 + 49} = \sqrt{58}$
${r}_{2} = \sqrt{58}$
${\theta}_{2} = {\tan}^{-} 1 \left(\frac{7}{-} 3\right)$
${z}_{2} = {r}_{2} c i s {\theta}_{2}$
${z}_{2} = \sqrt{58} c i s \left({\tan}^{-} 1 \left(\frac{7}{-} 3\right)\right)$

$z = {z}_{1} {z}_{2}$
$z = \left(\sqrt{41} c i s \left({\tan}^{-} 1 \left(\frac{5}{4}\right)\right)\right) \left(\sqrt{58} c i s \left({\tan}^{-} 1 \left(\frac{7}{-} 3\right)\right)\right)$

$z = \sqrt{41} \sqrt{58} c i s \left({\tan}^{-} 1 \left(\frac{5}{4}\right)\right) c i s \left({\tan}^{-} 1 \left(\frac{7}{-} 3\right)\right)$
$\sqrt{41} \sqrt{58} = \sqrt{41 \times 58} = \sqrt{2378}$
By De-Moivre's Theorem

$c i s \left({\tan}^{-} 1 \left(\frac{5}{4}\right)\right) c i s \left({\tan}^{-} 1 \left(\frac{7}{-} 3\right)\right) = c i s \left({\tan}^{-} 1 \left(\frac{5}{4}\right) + {\tan}^{-} 1 \left(\frac{7}{-} 3\right)\right)$
${\tan}^{-} 1 \left(\frac{5}{4}\right) + {\tan}^{-} 1 \left(\frac{7}{-} 3\right) = {\tan}^{-} 1 \left(\frac{\frac{5}{4} + \frac{7}{-} 3}{1 - \frac{5}{4} \times \frac{7}{-} 3}\right)$

$\frac{5}{4} + \frac{7}{-} 3 = \frac{5 \times - 3 + 7 \times 4}{4 \times - 3} = \frac{- 15 + 28}{-} 12 = \frac{13}{-} 12$
$z = r c i s \theta$

$r = \sqrt{2378}$
$\theta = {\tan}^{-} 1 \left(\frac{13}{-} 12\right)$
$z = \sqrt{2378} c i s \left({\tan}^{-} 1 \left(\frac{13}{-} 12\right)\right)$
$r = 48.765$
${\tan}^{-} 1 \left(\frac{13}{-} 12\right)$
adjacent side is negative,
the angle lies in the second quadrant
${\tan}^{-} 1 \left(\frac{13}{-} 12\right) = \pi - {\tan}^{-} 1 \left(\frac{13}{12}\right)$
${\tan}^{-} 1 \left(\frac{13}{12}\right) = 0.825$
expressed in radians
${\tan}^{-} 1 \left(\frac{13}{12}\right) = \pi - 0.825$
$\theta = 2.317$

$\cos 0.825 = - 0.679$
since the complex number is in 2nd quadrant
$\sin 0.825 = 0.735$

$r = 48.765$
$x = r \cos \theta = 48.765 \times \left(- 0.679\right)$
$x = - 33.111$

$y = r \sin \theta = 48.765 \times 0.735$
$y = 35.842$
$z = x + i y$

$z = \left(- 33.111\right) + i \left(35.842\right)$

Thus,
$\left(4 + 5 i\right) \left(- 3 + 7 i\right) = 48.765 c i s 2.317$
$\left(4 + 5 i\right) \left(- 3 + 7 i\right) = - 33.111 + 35.842 i$