# How do you multiply e^(( 11 pi )/ 8 i) * e^( pi/2 i )  in trigonometric form?

Jan 3, 2016

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

#### Explanation:

${e}^{\frac{11 \pi}{8} i} \cdot {e}^{\left(\frac{\pi}{2}\right) i}$

${e}^{\frac{11 \pi}{8} i} = \cos \left(\frac{11 \pi}{8}\right) + i \sin \left(\frac{11 \pi}{8}\right)$

${e}^{\left(\frac{\pi}{2}\right) i} = \cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)$

${e}^{\frac{11 \pi}{8} i} \cdot {e}^{\left(\frac{\pi}{2}\right) i}$

$= \left(\cos \left(\frac{11 \pi}{8}\right) + i \sin \left(\frac{11 \pi}{8}\right)\right) \cdot \left(\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right)$

$= \cos \left(\frac{11 \pi}{8} + \frac{\pi}{2}\right) + i \sin \left(\frac{11 \pi}{8} + \frac{\pi}{2}\right)$

$= \cos \left(\frac{11 \pi}{8} + \frac{4 \pi}{8}\right) + i \sin \left(\frac{11 \pi}{8} + \frac{4 \pi}{4}\right)$

$= \cos \left(\left(11 + 4\right) \frac{\pi}{8}\right) + i \sin \left(\left(11 + 4\right) \frac{\pi}{8}\right)$

$= \cos \left(\frac{15 \pi}{8}\right) + i \sin \left(\frac{15 \pi}{8}\right)$

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:
${e}^{\frac{11 \pi}{8} i} \cdot {e}^{\left(\frac{\pi}{2}\right) i}$
$= {e}^{\left(\frac{11 \pi}{8} + \frac{\pi}{2}\right) i}$ using exponent rule ${a}^{m} \cdot {a}^{n} = {a}^{m + n}$
$= {e}^{\left(\frac{15 \pi}{8}\right) i}$

Use the Euler's formula

$= \cos \left(\frac{15 \pi}{8}\right) + i \sin \left(\frac{15 \pi}{8}\right)$ Answer.