Question

given that \\(\cos \alpha = \frac{1}{5}\\) and \\(0 < \alpha < \frac{\pi}{2}\\), determine the exact value of \\(\cos \frac{\alpha}{2}\\).\\(\cos \frac{\alpha}{2} = \square\\)(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Recall Half - Angle Formula

The half - angle formula for cosine is \(\cos\frac{\theta}{2}=\pm\sqrt{\frac{1 + \cos\theta}{2}}\). We need to determine the sign of \(\cos\frac{\alpha}{2}\). Given that \(0\lt\alpha\lt\frac{\pi}{2}\), then \(0\lt\frac{\alpha}{2}\lt\frac{\pi}{4}\). Since the cosine function is positive in the first quadrant, \(\cos\frac{\alpha}{2}\gt0\), so we use the positive square root: \(\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}\).

Step2: Substitute \(\cos\alpha\) into the Formula

We know that \(\cos\alpha = \frac{1}{5}\). Substitute this value into the formula \(\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}\):

$$
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(To rationalize the denominator, we multiply the numerator and denominator inside the square root by 5: \(\sqrt{\frac{3}{5}}=\frac{\sqrt{3\times5}}{5}=\frac{\sqrt{15}}{5}\))

Answer:

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