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To solve the problem, we are asked to evaluate \( g\left(\frac{\theta}{2}\right) \), given \( g(x) = \cos(x) \), and to use the figure (a circle) to help solve the expression. Let's follow these steps based on the half-angle formulas:

### Step-by-Step Solution

1. **Identify the Half-Angle Formula:**
   The cosine half-angle formula is given by:
   \[
   \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
   \]
   The sign \( \pm \) depends on the quadrant of the angle \( \frac{\theta}{2} \).

2. **Extract the Information from the Figure:**
   We are given a unit circle with a radius \( r = \sqrt{7} \) (as deduced from the equation \( x^2 + y^2 = 7 \)). From the point on the circle \( (a, \sqrt{6}) \), we know that \( y = \sqrt{6} \), and we need to find the \( x \)-coordinate using the circle's equation.

3. **Find the \( x \)-coordinate:**
   Using the equation of the circle:
   \[
   x^2 + y^2 = 7
   \]
   Substitute \( y = \sqrt{6} \):
   \[
   x^2 + (\sqrt{6})^2 = 7 \implies x^2 + 6 = 7 \implies x^2 = 1
   \]
   Therefore, \( x = \pm 1 \). Given the point is in the second quadrant, \( x = -1 \).

4. **Calculate \( \cos(\theta) \):**
   From the definition of cosine in a unit circle, \( \cos(\theta) = \frac{x}{r} \). Since \( x = -1 \) and \( r = \sqrt{7} \):
   \[
   \cos(\theta) = \frac{-1}{\sqrt{7}}
   \]

5. **Apply the Half-Angle Formula:**
   Now, substitute \( \cos(\theta) = \frac{-1}{\sqrt{7}} \) into the half-angle formula:
   \[
   \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \frac{-1}{\sqrt{7}}}{2}}
   \]
   Simplifying inside the square root:
   \[
   \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{\frac{\sqrt{7} - 1}{\sqrt{7}}}{2}} = \pm \sqrt{\frac{\sqrt{7} - 1}{2\sqrt{7}}}
   \]
   
6. **Determine the Sign:**
   Since \( \frac{\theta}{2} \) is in the first quadrant (all angles from 0 to 90° are in Quadrant I), cosine is positive. Therefore, we take the positive value:
   \[
   \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{\sqrt{7} - 1}{2\sqrt{7}}}
   \]

Thus, the answer is \( \sqrt{\frac{\sqrt{7} - 1}{2\sqrt{7}}} \).

### Would you like additional details, or do you have any questions on this process? Here are some follow-up questions you might find helpful:
1. How do we apply the cosine half-angle formula in different quadrants?
2. What happens if the point was in a different quadrant?
3. Could we calculate sine instead using a similar method?
4. How does the half-angle formula simplify for specific angles like \( \theta = 180^\circ \)?
5. What is the geometric meaning of the half-angle formula?

**Tip:** Always check the quadrant of the angle when using half-angle formulas, as it determines the sign!

Use the figure to evaluate the following function, given that g(x) = cos(x), and use a Half-angle formula to solve for g(θ/2).

This problem involves using the half-angle formula for cosine to evaluate g(θ/2) given g(x) = cos(x) and a point on the unit circle. The solution includes finding the x-coordinate from the equation of the circle and applying the cosine half-angle formula.

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