Since the terminal side of angle \\alpha\ passes through the point \P\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right)\,
we have \x= \frac{1}{2}\ and \y= \frac{\sqrt{3}}{2}\,
thus \r=1\,
therefore \\cos \alpha= \frac{1}{2}\.
Hence, the answer is \$\boxed{\frac{1}{2}}$.
From the given information that the terminal side of angle \\alpha\ passes through the point \P\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right)\, we can easily calculate the value of \OP=r\, and then, based on the definition of trigonometric functions for any angle, we obtain the answer.
This question examines the definition of trigonometric functions for any angle, where calculating the value of \OP=r\ based on the coordinates of point \P\ is key to solving the problem.