Inverse Euler Formulas

• The inverse Euler formulas allow us to write the cosine and sine function in terms of complex exponentials:
  $$\cos\theta = \frac{e^{j\theta} + e^{-j\theta}}{2},$$
  and
  $$\sin\theta = \frac{e^{j\theta} - e^{-j\theta}}{2j}.$$

• This can be shown by adding and subtracting two complex exponentials with the same frequency but opposite in sign,
  

  $$e^{j\theta}+e^{-j\theta} = \cos\theta+j\sin\theta+\cos\theta - j\sin\theta$$

  $$= 2\cos\theta,$$

  
  and
  

  $$e^{j\theta}-e^{-j\theta} = \cos\theta+j\sin\theta-\cos\theta+j\sin\theta$$

  $$= 2j\sin\theta.$$

  
  

• A real cosine signal is actually composed of two complex exponential signals:
  1. one with a positive frequency
  2. one with a negative frequency