# How do you multiply e^(( pi )/ 4 i) * e^( pi/2 i )  in trigonometric form?

Nov 4, 2016

${e}^{\frac{\pi}{4} i} \cdot {e}^{\frac{\pi}{2} i} = - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i$

#### Explanation:

As ${e}^{\frac{\pi}{4} i} = \cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)$

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

${e}^{\frac{\pi}{4} i} \cdot {e}^{\frac{\pi}{2} i}$

= $\left(\cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right) \cdot \left(\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right)$

= $\cos \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right) + i \cos \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right) + {i}^{2} \sin \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)$

= $\cos \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right) + i \left\{\cos \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)\right\} + \sin \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right) - \sin \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)$

= $\left\{\cos \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right) - \sin \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)\right\} + i \left\{\cos \left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)\right\} + \sin \left(\frac{\pi}{2}\right) \cos \left(\frac{\pi}{4}\right)$

= $\cos \left(\frac{\pi}{2} + \frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{2} + \frac{\pi}{4}\right)$

= $\cos \left(\frac{3 \pi}{4}\right) + i \sin \left(\frac{3 \pi}{4}\right)$

= $- \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i$