Find the value of $cos( heta)$ using the unit circle in the complex plane when $ heta = frac{pi}{3}$.

First, understand that on the unit circle, a point corresponding to an angle $\theta$ can be represented as $e^{i\theta} = \cos(\theta) + i\sin(\theta)$.

For $\theta = \frac{\pi}{3}$,

$e^{i\frac{\pi}{3}} = \cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\pi}{3}\right)$.

We know that $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ and $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$.

Hence, $e^{i\frac{\pi}{3}} = \frac{1}{2} + i\frac{\sqrt{3}}{2}$.

So, $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.

On the unit circle in the complex plane, the coordinates of a point corresponding to an angle $ heta$ are $(cos( heta), sin( heta))$.

For $ heta = frac{pi}{3}$,

We need $cosleft(frac{pi}{3}
ight)$.

Using the unit circle, the $x$-coordinate at $ heta = frac{pi}{3}$ is $cosleft(frac{pi}{3}
ight)$.

We know from trigonometric values that $cosleft(frac{pi}{3}
ight) = frac{1}{2}$.

So, $cosleft(frac{pi}{3}
ight) = frac{1}{2}$.

The value of $cosleft(frac{pi}{3}
ight)$ on the unit circle:

$cosleft(frac{pi}{3}
ight) = frac{1}{2}$.

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