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

### Step 1: Find \(\alpha\)
In a right triangle, the sum of the angles is \(90^\circ\) (for the right angle at \(B\)) plus the two acute angles \(\alpha\) and \(\theta\). Therefore:
\[
\alpha = 90^\circ - \theta
\]

From the previous calculation, we found \(\tan(\theta) = \frac{4}{3}\), so:
\[
\theta = \arctan\left(\frac{4}{3}\right)
\]

Thus:
\[
\alpha = 90^\circ - \arctan\left(\frac{4}{3}\right)
\]

### Step 2: Use the half-angle identity
The half-angle identity for sine is:
\[
\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \cos(\alpha)}{2}}
\]

First, we need to find \(\cos(\alpha)\). Using the identity \(\cos(\alpha) = \sin(\theta)\), and knowing:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{10} = 0.8
\]

Thus:
\[
\cos(\alpha) = 0.8
\]

Now, apply the half-angle identity:
\[
\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - 0.8}{2}} = \sqrt{\frac{0.2}{2}} = \sqrt{0.1} = \frac{\sqrt{10}}{10}
\]

So, \(\sin\left(\frac{\alpha}{2}\right) = \frac{\sqrt{10}}{10}\).

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 \(\cos(\alpha)\) using the sides of the triangle?
2. What is the exact value of \(\tan\left(\frac{\alpha}{2}\right)\)?
3. How can you calculate \(\cos\left(\frac{\alpha}{2}\right)\)?
4. What is the sine of angle \(\theta\) in this triangle?
5. How would you calculate the length of the altitude from \(A\) to side \(BC\)?
6. What are the trigonometric identities for double angles \(\alpha\) and \(\theta\)?
7. How do you find the circumradius of triangle \(ABC\)?
8. What is the relationship between \(\sin(\alpha)\) and \(\sin(\theta)\) in this triangle?

**Tip:** When working with trigonometric identities, always double-check the values you obtain for angles to ensure that you are using the correct identity for the problem at hand.

Learn how to calculate sin (α/2) in a right triangle using the half-angle identity for sine. Step-by-step solution includes finding the angle α, applying trigonometric identities, and using the right triangle properties. Suitable for advanced high school students and those preparing for trigonometry exams.

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