$ ext{Evaluate the cosine of an angle using the unit circle in the complex plane} $

$ \text{Given an angle } \theta \text{, we need to find } \cos(\theta) \text{ using the unit circle in the complex plane.} $

$ \text{On the unit circle, the coordinates of a point } P \text{ corresponding to the angle } \theta \text{ are } (\cos(\theta), \sin(\theta)). $

$ \text{Thus, } \cos(\theta) \text{ is simply the x-coordinate.} $

$ \text{For example, if } \theta = \frac{\pi}{3}, \text{ the coordinates on the unit circle are } (\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})). $

$ \cos(\frac{\pi}{3}) = \frac{1}{2}. $

$ ext{Given an angle } heta, ext{ we need to determine } cos( heta) ext{ using the unit circle in the complex plane.} $

$ ext{In the complex plane, the point corresponding to the angle } heta ext{ on the unit circle is } e^{i heta} = cos( heta) + isin( heta). $

$ ext{Therefore, the real part of } e^{i heta} ext{ is } cos( heta). $

$ ext{For instance, if } heta = frac{pi}{4}, ext{ then } e^{ifrac{pi}{4}} = cos(frac{pi}{4}) + isin(frac{pi}{4}). $

$ cos(frac{pi}{4}) = frac{sqrt{2}}{2}. $

$ ext{For an angle } heta ext{, find } cos( heta) ext{ using the unit circle in the complex plane.} $

$ ext{The unit circle in the complex plane is represented by } e^{i heta} = cos( heta) + isin( heta). $

$ cos( heta) ext{ is the real part of } e^{i heta}. $

$ ext{Example: } heta = frac{pi}{6} Rightarrow e^{ifrac{pi}{6}} = cos(frac{pi}{6}) + isin(frac{pi}{6}). $

$ cos(frac{pi}{6}) = frac{sqrt{3}}{2}. $

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