Question

question
if \\(cos x = \frac{3}{10}\\), then what is the positive value of \\(cos \frac{1}{2}x\\), in simplest radical form with a rational denominator?

Explanation:

Step1: Recall the half - angle formula for cosine

The half - angle formula for cosine is $\cos\frac{\theta}{2}=\pm\sqrt{\frac{1 + \cos\theta}{2}}$. Since we want the positive value of $\cos\frac{1}{2}x$, we will use the positive square root, so $\cos\frac{x}{2}=\sqrt{\frac{1+\cos x}{2}}$.

Step2: Substitute the given value of $\cos x$

We are given that $\cos x=\frac{3}{10}$. Substitute this value into the formula:
$\cos\frac{x}{2}=\sqrt{\frac{1+\frac{3}{10}}{2}}$

Step3: Simplify the numerator inside the square root

First, simplify $1+\frac{3}{10}$. We know that $1=\frac{10}{10}$, so $1 + \frac{3}{10}=\frac{10 + 3}{10}=\frac{13}{10}$.
Now our formula becomes $\cos\frac{x}{2}=\sqrt{\frac{\frac{13}{10}}{2}}$

Step4: Simplify the fraction inside the square root

Dividing by 2 is the same as multiplying by $\frac{1}{2}$, so $\frac{\frac{13}{10}}{2}=\frac{13}{10}\times\frac{1}{2}=\frac{13}{20}$
So now we have $\cos\frac{x}{2}=\sqrt{\frac{13}{20}}$

Step5: Rationalize the denominator

To rationalize the denominator of $\sqrt{\frac{13}{20}}$, we can rewrite $\frac{13}{20}$ as $\frac{13\times5}{20\times5}=\frac{65}{100}$ (multiplying numerator and denominator by 5 to make the denominator a perfect square).
Then $\sqrt{\frac{65}{100}}=\frac{\sqrt{65}}{\sqrt{100}}=\frac{\sqrt{65}}{10}$

Answer:

$\frac{\sqrt{65}}{10}$