given that $\\cos\\alpha = \\frac{1}{5}$ and $0 < \\alpha < \\frac{\\pi}{2}$, determine the exact value of…

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{1}{5}). Substitute (\cos\alpha) into the formula: [ \begin{align*} \cos\frac{\alpha}{2}&=\sqrt{\frac{1+\frac{1}{5}}{2}}\ &=\sqrt{\frac{\frac{5 + 1}{5}}{2}}\ &=\sqrt{\frac{\frac{6}{5}}{2}}\ &=\sqrt{\frac{6}{10}}\ &=\sqrt{\frac{3}{5}}\ &=\frac{\sqrt{3}}{\sqrt{5}}\ &=\frac{\sqrt{3}\times\sqrt{5}}{\sqrt{5}\times\sqrt{5}}\ &=\frac{\sqrt{15}}{5} \end{align*} ]

Answer:

(\frac{\sqrt{15}}{5})