How do you multiply $e^{\frac{π}{2} i} \cdot e^{3 \frac{π}{2} i}$ $e^(( pi )/ 2 i) * e^( 3 pi/2 i )$ in trigonometric form?

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How do you multiply $e^{\frac{π}{2} i} \cdot e^{3 \frac{π}{2} i}$ $e^(( pi )/ 2 i) * e^( 3 pi/2 i )$ in trigonometric form?

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Answer 1 · 2017-09-25T14:18:47

The answer is $= 1$ $=1$

Explanation:

We apply Euler's identity

$e^{i θ} = cos θ + i sin θ$ $e^(itheta)=costheta +i sin theta$

$i^{2} = - 1$ $i^2=-1$

$e^{a} \cdot e^{b} = e^{a + b}$ $e^a *e^b=e^(a+b)$

So,

$e^{i \frac{π}{2}} = cos (\frac{π}{2}) + i sin (\frac{π}{2}) = 0 + 1 \cdot i = i$ $e^(ipi/2)=cos(pi/2)+isin(pi/2)=0+1*i=i$

$e^{i 3 \frac{π}{2}} = cos (\frac{3}{2} π) + i sin (\frac{3}{2} π) = 0 - 1 \cdot i = - i$ $e^(i3pi/2)=cos(3/2pi)+isin(3/2pi)=0-1*i=-i$

Therefore,

$e^{i \frac{π}{2}} \cdot e^{i 3 \frac{π}{2}} = i \cdot - i = - i^{2} = 1$ $e^ (ipi/2)* e^(i3pi/2)=i*-i=-i^2=1$

We can also perform

$e^{i \frac{π}{2}} \cdot e^{i 3 \frac{π}{2}} = e^{(i \frac{π}{2} + i 3 \frac{π}{2})} = e^{i 4 \frac{π}{2}} = e^{i 2 π}$ $e^(ipi/2)*e^(i3pi/2)=e^((ipi/2+i3pi/2))=e^(i4pi/2)=e^(i2pi)$

$= cos (2 π) + i sin (2 π)$ $=cos(2pi)+isin(2pi)$

$= 1 + i \cdot 0 = 1$ $=1+i*0=1$

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How do you multiply $e^{\frac{π}{2} i} \cdot e^{3 \frac{π}{2} i}$ $e^(( pi )/ 2 i) * e^( 3 pi/2 i )$ in trigonometric form?

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Explanation:

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