To calculate $\cos\left(\frac{\theta}{2}\right)$ for the given right triangle, we'll first need to find the value of $\theta$, the angle at $C$. Then, we'll use the half-angle identity to determine $\cos\left(\frac{\theta}{2}\right)$.

Step 1: Find $\theta$

Using trigonometry in the right triangle $ABC$, we can calculate $\theta$ using the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{6} = \frac{4}{3}$

To find $\theta$, take the arctan (inverse tangent): $\theta = \arctan\left(\frac{4}{3}\right)$

Step 2: Use the half-angle identity

The half-angle identity for cosine is: $\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}$

First, we need to find $\cos(\theta)$ using the sides of the triangle: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{10} = 0.6$

Now, apply the half-angle identity: $\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + 0.6}{2}} = \sqrt{\frac{1.6}{2}} = \sqrt{0.8} = \sqrt{\frac{4}{5}} = \frac{2\sqrt{5}}{5}$

Thus, $\cos\left(\frac{\theta}{2}\right) = \frac{2\sqrt{5}}{5}$.

Would you like more details on any of these steps or do you have further questions?

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Here are 8 related questions for further exploration:

1. How do you calculate $\sin(\theta)$ using the sides of a triangle?
2. What is the relationship between the angles $\alpha$ and $\theta$ in this triangle?
3. How can you use the Pythagorean theorem to verify the hypotenuse in this triangle?
4. What is the exact value of $\tan\left(\frac{\theta}{2}\right)$?
5. How do you calculate the area of triangle $ABC$?
6. What is the length of the median from point $A$ to side $BC$?
7. How would you calculate $\cos(\alpha)$ given the sides of the triangle?
8. What are the sine and cosine of $\frac{\alpha}{2}$?

Tip: When dealing with right triangles, always start by identifying the sides relative to the angles you need to calculate (opposite, adjacent, and hypotenuse). This helps in selecting the correct trigonometric ratio.