Question

watch the video and then solve the problem given below.
click here to watch the video.
given that \\( \cos \alpha = - \frac { 1 } { 3 } \\) 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: 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

The half - angle formula for cosine is \(\cos\frac{\theta}{2}=\sqrt{\frac{1 + \cos\theta}{2}}\). Here \(\theta=\alpha\) and \(\cos\alpha=\frac{1}{3}\).
Substitute \(\cos\alpha=\frac{1}{3}\) into the formula:

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Answer:

\(\frac{\sqrt{6}}{3}\)