How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{4 π}{3} i}$ $e^( ( pi)/4 i) - e^( ( 4 pi)/3 i)$ using trigonometric functions?

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How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{4 π}{3} i}$ $e^( ( pi)/4 i) - e^( ( 4 pi)/3 i)$ using trigonometric functions?

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Answer 1 · 2016-11-14T08:38:17

$\frac{sin (π)}{4} + i \frac{cos (π)}{4}$ $sin ( pi)/4 + i cos ( pi)/4$
- $\frac{sin (4 π)}{3} + i \frac{cos (4 π)}{3}$ $sin ( 4 pi)/3 + i cos ( 4 pi)/3$

Explanation:

In complex terms, $R (sin θ + i cos θ)$ $R ( sintheta + i costheta )$ = $R e^{i θ}$ $Re ^ (i theta)$
where $R$ $R$ is the modulus and $θ$ $theta$ is the argument.

So
$e^{\frac{π}{4} i} - e^{\frac{4 π}{3} i}$ $e^( ( pi)/4 i) - e^( ( 4 pi)/3 i)$ = $1 (\frac{sin (π)}{4} + i \frac{cos (π)}{4})$ $1 ( sin ( pi)/4 + i cos ( pi)/4 )$
- $1 (\frac{sin (4 π)}{3} + i \frac{cos (4 π)}{3})$ $1 ( sin ( 4 pi)/3 + i cos ( 4 pi)/3 )$

= $\frac{sin (π)}{4} + i \frac{cos (π)}{4}$ $sin ( pi)/4 + i cos ( pi)/4$
- $\frac{sin (4 π)}{3} + i \frac{cos (4 π)}{3}$ $sin ( 4 pi)/3 + i cos ( 4 pi)/3$

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How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{4 π}{3} i}$ $e^( ( pi)/4 i) - e^( ( 4 pi)/3 i)$ using trigonometric functions?

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How do you evaluate $e^{\frac{π}{4} i} - e^{\frac{4 π}{3} i}$ $e^( ( pi)/4 i) - e^( ( 4 pi)/3 i)$ using trigonometric functions?