How do you multiply $e^{\frac{11 π}{8} i} \cdot e^{\frac{π}{2} i}$ $e^(( 11 pi )/ 8 i) * e^( pi/2 i )$ in trigonometric form?

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How do you multiply $e^{\frac{11 π}{8} i} \cdot e^{\frac{π}{2} i}$ $e^(( 11 pi )/ 8 i) * e^( pi/2 i )$ in trigonometric form?

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Answer 1 · 2016-01-03T12:49:35

Euler formula $e^{i θ} = cos (θ) + i sin (θ)$ $e^(itheta) = cos(theta)+isin(theta)$ that would convert to trigonometric form. When we multiply trigonometric form we add the angles and multiply the modulus.

Explanation:

$e^{\frac{11 π}{8} i} \cdot e^{(\frac{π}{2}) i}$ $e^((11pi)/8i)*e^((pi/2)i)$

$e^{\frac{11 π}{8} i} = cos (\frac{11 π}{8}) + i sin (\frac{11 π}{8})$ $e^((11pi)/8i) = cos((11pi)/8) + isin((11pi)/8)$

$e^{(\frac{π}{2}) i} = cos (\frac{π}{2}) + i sin (\frac{π}{2})$ $e^((pi/2)i) = cos(pi/2) + isin(pi/2)$

$e^{\frac{11 π}{8} i} \cdot e^{(\frac{π}{2}) i}$ $e^((11pi)/8i)*e^((pi/2)i)$

$= (cos (\frac{11 π}{8}) + i sin (\frac{11 π}{8})) \cdot (cos (\frac{π}{2}) + i sin (\frac{π}{2}))$ $= (cos((11pi)/8) + isin((11pi)/8))*(cos(pi/2) + isin(pi/2))$

$= cos (\frac{11 π}{8} + \frac{π}{2}) + i sin (\frac{11 π}{8} + \frac{π}{2})$ $=cos((11pi)/8+pi/2)+isin((11pi)/8+pi/2)$

$= cos (\frac{11 π}{8} + \frac{4 π}{8}) + i sin (\frac{11 π}{8} + \frac{4 π}{4})$ $=cos((11pi)/8+(4pi)/8)+isin((11pi)/8+(4pi)/4)$

$= cos ((11 + 4) \frac{π}{8}) + i sin ((11 + 4) \frac{π}{8})$ $=cos((11+4)pi/8) + isin((11+4)pi/8)$

$= cos (\frac{15 π}{8}) + i sin (\frac{15 π}{8})$ $=cos((15pi)/8)+isin((15pi)/8)$

Note: The question said in trigonometric form so converted to trigonometric form and multiplied. Using Euler's form would be easy multiply and then convert, the choice is yours.

Alternate method:
$e^{\frac{11 π}{8} i} \cdot e^{(\frac{π}{2}) i}$ $e^((11pi)/8i)*e^((pi/2)i)$
$= e^{(\frac{11 π}{8} + \frac{π}{2}) i}$ $=e^(((11pi)/8+pi/2)i)$ using exponent rule $a^{m} \cdot a^{n} = a^{m + n}$ $a^m*a^n=a^(m+n)$
$= e^{(\frac{15 π}{8}) i}$ $=e^(((15pi)/8)i)$

Use the Euler's formula

$= cos (\frac{15 π}{8}) + i sin (\frac{15 π}{8})$ $=cos((15pi)/8) + isin((15pi)/8)$ Answer.

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How do you multiply $e^{\frac{11 π}{8} i} \cdot e^{\frac{π}{2} i}$ $e^(( 11 pi )/ 8 i) * e^( pi/2 i )$ in trigonometric form?

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