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# How do you evaluate  e^( ( pi)/4 i) - e^( ( pi)/3 i) using trigonometric functions?

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#### Explanation

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#### Explanation:

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Oct 17, 2017

$\frac{\sqrt{2} - 1}{2} - i \cdot \frac{\sqrt{3} - \sqrt{2}}{2}$

#### Explanation:

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

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

=$\frac{\sqrt{2}}{2} + i \cdot \frac{\sqrt{2}}{2}$

${e}^{\frac{\pi}{3} \cdot i}$

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

=$\frac{1}{2} + i \cdot \frac{\sqrt{3}}{2}$

Hence,

${e}^{\frac{\pi}{4} \cdot i} - {e}^{\frac{\pi}{3} \cdot i} = \frac{\sqrt{2} - 1}{2} + i \cdot \frac{\sqrt{2} - \sqrt{3}}{2}$

=$\frac{\sqrt{2} - 1}{2} - i \cdot \frac{\sqrt{3} - \sqrt{2}}{2}$

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