Given that $$ \cos \alpha=\frac{3}{4} $$ and $$ 0

Question

Given that $$ \cos \alpha=\frac{3}{4} $$ 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 ,

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To find $$ \cos \frac{\alpha}{2} $$ given that $$ \cos \alpha = \frac{3}{4} $$, we can use the half-angle formula for cosine:

$$
\cos \frac{\alpha}{2} = \sqrt{\frac{1 + \cos \alpha}{2}}
$$

Since $$ 0 < \alpha < \frac{\pi}{2} $$, we know that $$ \frac{\alpha}{2} $$ will also be in the first quadrant, where cosine is positive. Therefore, we will take the positive square root.

Substituting $$ \cos \alpha = \frac{3}{4} $$ into the formula, we have:

$$
\cos \frac{\alpha}{2} = \sqrt{\frac{1 + \frac{3}{4}}{2}}
$$

First, simplify the expression inside the square root:

$$
1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}
$$

Now substitute this back into the formula:

$$
\cos \frac{\alpha}{2} = \sqrt{\frac{\frac{7}{4}}{2}} = \sqrt{\frac{7}{4} \cdot \frac{1}{2}} = \sqrt{\frac{7}{8}}
$$

Next, simplify $$ \sqrt{\frac{7}{8}} $$:

$$
\sqrt{\frac{7}{8}} = \frac{\sqrt{7}}{\sqrt{8}} = \frac{\sqrt{7}}{2\sqrt{2}} = \frac{\sqrt{7}}{2 \cdot 2^{1/2}} = \frac{\sqrt{7}}{2\sqrt{2}} = \frac{\sqrt{7}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{14}}{4}
$$

Thus, the exact value of $$ \cos \frac{\alpha}{2} $$ is:

$$
\cos \frac{\alpha}{2} = \frac{\sqrt{14}}{4}
$$

Therefore, the final answer is:

$$
\cos \frac{\alpha}{2} = \frac{\sqrt{14}}{4}
$$
$$\cos \frac{\alpha}{2} = \frac{\sqrt{14}}{4}$$

Given that and , determine the exact value of .

(Simplify your answer, including any radicals. Use integers or fractions for any ,

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