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To solve this problem, we need to find \( \sin \left( \frac{\theta}{2} \right) \) based on the given right triangle \( \triangle ABC \), where:

- \( AB = 6 \)
- \( BC = 8 \)
- \( AC \) is the hypotenuse.

### Step 1: Find \( \theta \) using trigonometry
Since \( \triangle ABC \) is a right triangle with \( \angle B = 90^\circ \), we can find \( \theta \) by using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} = \frac{6}{8} = \frac{3}{4}.
\]
So, \( \theta = \tan^{-1}\left(\frac{3}{4}\right) \).

### Step 2: Find the hypotenuse \( AC \)
Using the Pythagorean theorem:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.
\]

### Step 3: Find \( \sin(\theta) \)
Now, we calculate \( \sin(\theta) \) using:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{6}{10} = \frac{3}{5}.
\]

### Step 4: Use the half-angle identity for \( \sin \left( \frac{\theta}{2} \right) \)
The half-angle formula for sine is:
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}.
\]
We need to find \( \cos(\theta) \) to proceed.

### Step 5: Find \( \cos(\theta) \)
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{8}{10} = \frac{4}{5}.
\]

### Step 6: Substitute into the half-angle formula
Now, substitute \( \cos(\theta) = \frac{4}{5} \) into the half-angle formula:
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \frac{4}{5}}{2}} = \sqrt{\frac{\frac{1}{5}}{2}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}}.
\]

### Answer
The correct answer is:
**a. \( \frac{1}{\sqrt{10}} \)**

Would you like further clarification on any step?

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#### Related Questions
1. How would you find \( \cos(\frac{\theta}{2}) \) for this triangle?
2. What is the exact value of \( \theta \) in degrees?
3. How would you find \( \tan(\frac{\theta}{2}) \) using a half-angle formula?
4. How does the Pythagorean identity apply to verify \( \sin(\theta) \) and \( \cos(\theta) \)?
5. Can we use any other trigonometric identity to find \( \sin(\frac{\theta}{2}) \)?

#### Tip:
For half-angle identities, always remember to determine the correct sign based on the angle’s quadrant, even though in this case, \( \theta \) lies in the first quadrant.

For the above diagram, find sin(θ/2), if 0° < θ < 360°.

Learn how to calculate sin(θ/2) for a right triangle using trigonometric identities and half-angle formulas. This problem involves calculating tan(θ), determining hypotenuse length with the Pythagorean theorem, and applying the half-angle identity. Suitable for Grades 10-12 students.

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