To find the least positive value of $x$ for which

$\cos\left(\frac{x}{2} + \frac{\pi}{3}\right) = \frac{1}{\sqrt{2}}$

we'll proceed with the following steps.

Step 1: Set up the Equation

We know that:

$\cos\left(\frac{x}{2} + \frac{\pi}{3}\right) = \frac{1}{\sqrt{2}}$

The angle for which $\cos(\theta) = \frac{1}{\sqrt{2}}$ is $\theta = \frac{\pi}{4}$ and $\theta = \frac{7\pi}{4}$ (or in general, $\theta = \frac{\pi}{4} + 2n\pi$ or $\theta = \frac{7\pi}{4} + 2n\pi$ for integer $n$).

Step 2: Set Up Possible Equations for $\frac{x}{2} + \frac{\pi}{3}$

Using the values of $\theta$ that satisfy the equation, we have:

1. $\frac{x}{2} + \frac{\pi}{3} = \frac{\pi}{4} + 2n\pi$
2. $\frac{x}{2} + \frac{\pi}{3} = \frac{7\pi}{4} + 2n\pi$

where $n$ is an integer.

Step 3: Solve for $x$

Let's solve each case for $x$ to find the least positive solution.

Case 1: $\frac{x}{2} + \frac{\pi}{3} = \frac{\pi}{4}$

1. Subtract $\frac{\pi}{3}$ from both sides: $\frac{x}{2} = \frac{\pi}{4} - \frac{\pi}{3}$

2. Find a common denominator and simplify: $\frac{\pi}{4} - \frac{\pi}{3} = \frac{3\pi - 4\pi}{12} = -\frac{\pi}{12}$

3. Multiply by 2 to solve for $x$: $x = -\frac{\pi}{6}$

Since we are looking for the least positive $x$, this solution is not applicable here.

Case 2: $\frac{x}{2} + \frac{\pi}{3} = \frac{7\pi}{4}$

1. Subtract $\frac{\pi}{3}$ from both sides: $\frac{x}{2} = \frac{7\pi}{4} - \frac{\pi}{3}$

2. Simplify by finding a common denominator: $\frac{7\pi}{4} - \frac{\pi}{3} = \frac{21\pi - 4\pi}{12} = \frac{17\pi}{12}$

3. Multiply by 2 to solve for $x$: $x = \frac{17\pi}{6}$

Conclusion

The least positive value of $x$ that satisfies the equation is:

$x = \frac{17\pi}{6}$

Would you like a more detailed explanation of any step, or have any other questions?

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Here are some related questions to expand on this topic:

1. What other trigonometric functions have periodic solutions, and how do we find them?
2. How do we solve equations involving both sine and cosine functions?
3. How can we determine all possible solutions in a given interval?
4. What are some practical applications of trigonometric equations in physics or engineering?
5. How can we use inverse trigonometric functions to solve similar equations?

Tip: For trigonometric equations, using known values and symmetries of the trigonometric functions often helps simplify and solve complex expressions.