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Saturday, 14 May 2011 00:00

Sum of Squares Using Euler's Formula

By  Gideon
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Euler's formula makes life much easier when it comes to deriving trigonometric identities.

 

Note, first of all, that \[ |e^{i\alpha}| = 1, \]gives us:

\[ \left( \text{Re} (e^{i\alpha}) + i\text{Im}(e^{i\alpha}) \right) \left( \text{Re}(e^{i\alpha}) – i\text{Im}(e^{i\alpha}) \right) \]

\[ = \left( \cos{\alpha} + i\sin{\alpha} \right) \left( \cos{\alpha} - i\sin{\alpha} \right) \]

\[ = \cos^{2}{\alpha} + \sin^{2}{\alpha} = 1. \]

Read 2846 times Last modified on Tuesday, 01 April 2014 19:12
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