given that \\( \\cos \\alpha = \\frac { 3 } { 5 } \\) and \\( 0 < \\alpha < \\frac { \\pi } { 2 } \\)…

Explanation:

Step1: Determine the quadrant of (\frac{\alpha}{2})

Since (0 < \alpha<\frac{\pi}{2}), then (0 < \frac{\alpha}{2}<\frac{\pi}{4}). So (\frac{\alpha}{2}) is in the first - quadrant, and (\cos\frac{\alpha}{2}>0).

Step2: Use the half - angle formula for cosine

The half - angle formula for cosine is (\cos\frac{\theta}{2}=\sqrt{\frac{1 + \cos\theta}{2}}). Here (\theta=\alpha) and (\cos\alpha=\frac{3}{5}). Substitute (\cos\alpha=\frac{3}{5}) into the formula: [ \begin{align*} \cos\frac{\alpha}{2}&=\sqrt{\frac{1+\frac{3}{5}}{2}}\ &=\sqrt{\frac{\frac{5 + 3}{5}}{2}}\ &=\sqrt{\frac{\frac{8}{5}}{2}}\ &=\sqrt{\frac{8}{5\times2}}\ &=\sqrt{\frac{4}{5}}\ &=\frac{2}{\sqrt{5}}\ &=\frac{2\sqrt{5}}{5} \end{align*} ]

Answer:

(\frac{2\sqrt{5}}{5})