EE102 Systems and Signals (Discussion)


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Euler's Equation and Applications

Euler's equation states: \fbox{$ e^{i\theta} = \text{cos}(\theta) + i.\text{sin}(\theta) $}

Some simple special cases would be:

$\displaystyle \theta = 0$ :    $\displaystyle e^{i0} = 1$  
$\displaystyle \theta = \frac{\pi}{2}$ :    $\displaystyle e^{i\frac{\pi}{2}} = i$  
:    $\displaystyle e^{i\pi} = -1$  
$\displaystyle \theta = -\frac{\pi}{2}$ :    $\displaystyle e^{-i\frac{\pi}{2}} = -i$  

Euler's equation is very useful in deriving many trigonometric identities. We notice, that

$\displaystyle (e^{i\theta})^* =$   cos$\displaystyle (\theta) - i$sin$\displaystyle (\theta) = e^{-i\theta}$

where $ ^*$ denotes complex conjugation.

Hence we get the following formulas:

cos$\displaystyle (\theta)$ $\displaystyle =$ Re$\displaystyle [e^{i\theta}] = \frac{e^{i\theta} + e^{-i\theta}}{2}$  
sin$\displaystyle (\theta)$ $\displaystyle =$ Im$\displaystyle [e^{i\theta}] = \frac{e^{i\theta} - e^{-i\theta}}{2i}$  

Euler's equation is very useful in deriving many trigonometric identities. First let's derive the formulas for the cosine and sine of a sum of two angles. We have

$\displaystyle e^{i(\theta_1+\theta_2)} = e^{i\theta_1}\cdot e^{i\theta_2}$

Hence,
cos$\displaystyle (\theta_1+\theta_2) + i$sin$\displaystyle (\theta_1+\theta_2)$ $\displaystyle =$ $\displaystyle [$cos$\displaystyle (\theta_1)+i$sin$\displaystyle (\theta_1)].[$cos$\displaystyle (\theta_2)+i$sin$\displaystyle (\theta_2)]$  
  $\displaystyle =$ $\displaystyle [$cos$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)-$sin$\displaystyle (\theta_1)$sin$\displaystyle (\theta_2)] + i [$sin$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)+$cos$\displaystyle (\theta_1)$sin$\displaystyle (\theta_2)]$  

Equating the real and imaginary parts on both sides, we get the identities
cos$\displaystyle (\theta_1+\theta_2)$ $\displaystyle =$ cos$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)-$sin$\displaystyle (\theta_1)$sin$\displaystyle (\theta_2)$  
sin$\displaystyle (\theta_1+\theta_2)$ $\displaystyle =$ sin$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)+$cos$\displaystyle (\theta_1)$sin$\displaystyle (\theta_2)$  

Replacing $ \theta_2$ by $ -\theta_2$ in the above identities, we get

cos$\displaystyle (\theta_1-\theta_2)$ $\displaystyle =$ cos$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)+$sin$\displaystyle (\theta_1)$sin$\displaystyle (\theta_2)$  
sin$\displaystyle (\theta_1-\theta_2)$ $\displaystyle =$ sin$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)-$cos$\displaystyle (\theta_1)$sin$\displaystyle (\theta_2)$  

Using the above two sets of identities, we get

cos$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)$ $\displaystyle =$ $\displaystyle \frac{\text{cos}(\theta_1+\theta_2)+\text{cos}(\theta_1-\theta_2)}{2}$  
sin$\displaystyle (\theta_1)$sin$\displaystyle (\theta_2)$ $\displaystyle =$ $\displaystyle \frac{\text{cos}(\theta_1-\theta_2)-\text{cos}(\theta_1+\theta_2)}{2}$  
sin$\displaystyle (\theta_1)$cos$\displaystyle (\theta_2)$ $\displaystyle =$ $\displaystyle \frac{\text{sin}(\theta_1+\theta_2)+\text{sin}(\theta_1-\theta_2)}{2}$  

next up previous contents
Next: First Order Linear Differential Up: Complex Numbers Previous: Complex Numbers

visitors since January 7., 2002. -

Sankaran Panchapagesan
2002-06-05