How do you simplify $5 (cos π + i sin π) \cdot 2 (cos (\frac{3 π}{4}) + i sin (\frac{3 π}{4}))$ $5(cospi+isinpi)*2(cos((3pi)/4)+isin((3pi)/4))$ and express the result in rectangular form?

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How do you simplify $5 (cos π + i sin π) \cdot 2 (cos (\frac{3 π}{4}) + i sin (\frac{3 π}{4}))$ $5(cospi+isinpi)*2(cos((3pi)/4)+isin((3pi)/4))$ and express the result in rectangular form?

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Answer 1 · 2017-05-17T11:30:01

$5 \sqrt{2} (1 - i) .$ $5sqrt2(1-i).$

Explanation:

${5 (cos π + i sin π)} {2 (cos (3 \frac{π}{4}) + i sin (3 \frac{π}{4}))}$ ${5(cospi+isinpi)}{2(cos(3pi/4)+isin(3pi/4))}$

$= {5 (- 1 + 0 i)} {2 (- cos (\frac{π}{4}) + i sin (\frac{π}{4}))}$ $={5(-1+0i)}{2(-cos(pi/4)+isin(pi/4))}$

$= - 5 {2 (- \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \cdot i)}$ $=-5{2(-1/sqrt2+1/sqrt2*i)}$

$= 5 \sqrt{2} (1 - i),$ $=5sqrt2(1-i),$

Otherwise, knowing that,

$r (cos α + i sin α) \cdot R (cos β + i sin β) = (r R) (cos (α + β) + i sin (α + β)),$ $r(cosalpha+isinalpha)*R(cosbeta+isinbeta)=(rR)(cos(alpha+beta)+isin(alpha+beta)),$

we have, the Reqd. Value

$= (5 \times 2) {cos (π + 3 \frac{π}{4}) + i sin (π + 3 \frac{π}{4})}$ $=(5xx2){cos(pi+3pi/4)+isin(pi+3pi/4)}$

$= 10 {cos (2 π - \frac{π}{4}) + i sin (2 π - \frac{π}{4})}$ $=10{cos(2pi-pi/4)+isin(2pi-pi/4)}$

$= 10 {\frac{cos π}{4} - i \frac{sin π}{4}}$ $=10{cospi/4-isinpi/4}$

$= 10 (\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}})$ $=10(1/sqrt2-i/sqrt2)$

$= 5 \sqrt{2} (1 - i),$ $=5sqrt2(1-i),$ as derived earlier!

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How do you simplify $5 (cos π + i sin π) \cdot 2 (cos (\frac{3 π}{4}) + i sin (\frac{3 π}{4}))$ $5(cospi+isinpi)*2(cos((3pi)/4)+isin((3pi)/4))$ and express the result in rectangular form?

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Explanation:

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How do you simplify $5 (cos π + i sin π) \cdot 2 (cos (\frac{3 π}{4}) + i sin (\frac{3 π}{4}))$ $5(cospi+isinpi)*2(cos((3pi)/4)+isin((3pi)/4))$ and express the result in rectangular form?