How do you multiply $(5 + 3 i) (3 + 2 i)$ $(5+3i)(3+2i)$ in trigonometric form?

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How do you multiply $(5 + 3 i) (3 + 2 i)$ $(5+3i)(3+2i)$ in trigonometric form?

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Answer 1 · 2017-11-13T08:53:47

$(5 + 3 i) (3 + 2 i) \approx 21.02 (cos (64.65) + i sin (64.65))$ $(5+3i)(3+2i)~~21.02(cos(64.65)+isin(64.65))$

Explanation:

$a + b i$ $a+bi$ in trig form is $r (cos θ + i sin θ$ $r(cos\theta+isin\theta$ ), where:

$r = \sqrt{a^{2} + b^{2}}$ $r=sqrt(a^2+b^2)$
$θ = ∣ ∣ ∣ arctan (\frac{b}{a}) ∣ ∣ ∣$ $\theta=abs(arctan(b/a))$

$(5 + 3 i) (3 + 2 i) ≅ (r_{1} ({cos θ}_{1} + i {sin θ}_{1})) (r_{2} ({cos θ}_{2} + i {sin θ}_{2}))$ $(5+3i)(3+2i)~=(r_1(cos\theta_1+isin\theta_1))(r_2(cos\theta_2+isin\theta_2))$
$= r_{1} r_{2} (cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2}))$ $=r_1r_2(cos(\theta_1+\theta_2)+isin(\theta_1+\theta_2))$
$= \sqrt{5^{2} + 3^{2}} \sqrt{3^{2} + 2^{2}} (cos (∣ ∣ ∣ arctan (\frac{3}{5}) ∣ ∣ ∣ + ∣ ∣ ∣ arctan (\frac{2}{3}) ∣ ∣ ∣) + i sin (∣ ∣ ∣ arctan (\frac{3}{5}) ∣ ∣ ∣ + ∣ ∣ ∣ arctan (\frac{2}{3}) ∣ ∣ ∣))$ $=sqrt(5^2+3^2)sqrt(3^2+2^2)(cos(abs(arctan(3/5))+abs(arctan(2/3)))+isin(abs(arctan(3/5))+abs(arctan(2/3))))$
$\approx \sqrt{34} \sqrt{13} (cos (64.65) + i sin (64.65))$ $~~sqrt(34)sqrt(13)(cos(64.65)+isin(64.65))$
$= \sqrt{442} (cos (64.65) + i sin (64.65))$ $=sqrt(442)(cos(64.65)+isin(64.65))$
$\approx 21.02 (cos (64.65) + i sin (64.65))$ $~~21.02(cos(64.65)+isin(64.65))$

Given ${tan θ}_{1} = \frac{3}{5}$ $tantheta_1=3/5$ and ${tan θ}_{1} = \frac{2}{3}$ $tantheta_1=2/3$

$tan θ = \frac{\frac{3}{5} + \frac{2}{3}}{1 - \frac{3}{5} \times \frac{2}{3}} = \frac{\frac{19}{15}}{\frac{3}{5}} = \frac{19}{9}$ $tantheta=(3/5+2/3)/(1-3/5xx2/3)=(19/15)/(3/5)=19/9$

and in exact form we can write product as

$\sqrt{442} (cos (arctan (\frac{19}{9})) + i sin ((arctan (\frac{19}{9})))$ $sqrt442(cos(arctan(19/9))+isin((arctan(19/9)))$

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How do you multiply $(5 + 3 i) (3 + 2 i)$ $(5+3i)(3+2i)$ in trigonometric form?

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How do you multiply (5+3i)(3+2i) (5+3i)(3+2i) in trigonometric form?

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How do you multiply $(5 + 3 i) (3 + 2 i)$ $(5+3i)(3+2i)$ in trigonometric form?