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The problem provided asks to find the value of the expression \( (\sin 2x) \cdot \tan \frac{x}{2} \), given that \( \sin x = \frac{24}{26} \) and \( x \) is an acute angle.

### Step 1: Simplify \( \sin x \)
We know that:
\[
\sin x = \frac{24}{26} = \frac{12}{13}
\]
Since \( x \) is acute, we can find \( \cos x \) using the identity:
\[
\sin^2 x + \cos^2 x = 1
\]
Substituting \( \sin x = \frac{12}{13} \):
\[
\left(\frac{12}{13}\right)^2 + \cos^2 x = 1
\]
\[
\frac{144}{169} + \cos^2 x = 1
\]
\[
\cos^2 x = 1 - \frac{144}{169} = \frac{25}{169}
\]
\[
\cos x = \frac{5}{13}
\]

### Step 2: Calculate \( \sin 2x \)
We use the double angle identity for sine:
\[
\sin 2x = 2 \sin x \cos x
\]
Substitute \( \sin x = \frac{12}{13} \) and \( \cos x = \frac{5}{13} \):
\[
\sin 2x = 2 \cdot \frac{12}{13} \cdot \frac{5}{13} = \frac{120}{169}
\]

### Step 3: Find \( \tan \frac{x}{2} \)
We use the half-angle identity for tangent:
\[
\tan \frac{x}{2} = \frac{1 - \cos x}{\sin x}
\]
Substituting \( \cos x = \frac{5}{13} \) and \( \sin x = \frac{12}{13} \):
\[
\tan \frac{x}{2} = \frac{1 - \frac{5}{13}}{\frac{12}{13}} = \frac{\frac{8}{13}}{\frac{12}{13}} = \frac{8}{12} = \frac{2}{3}
\]

### Step 4: Calculate the final expression
Now, we calculate \( (\sin 2x) \cdot \tan \frac{x}{2} \):
\[
(\sin 2x) \cdot \tan \frac{x}{2} = \frac{120}{169} \cdot \frac{2}{3} = \frac{240}{507} = \frac{80}{169}
\]

Thus, the value of the expression is:
\[
\boxed{\frac{80}{169}}
\]

Would you like any further details on this solution? 

Here are some additional related questions:
1. How do you find the cosine of an angle given the sine?
2. What are the double angle identities in trigonometry?
3. How do you derive the half-angle identity for tangent?
4. What is the relationship between sine, cosine, and tangent?
5. Can you calculate the exact value of trigonometric expressions without a calculator?

**Tip:** When working with trigonometric identities, it's essential to remember the Pythagorean identities to switch between sine and cosine easily!

It is known that x is an acute angle and satisfies sin x = 24/26. Find the value of (sin 2x) * tan (x/2).

Learn how to solve the trigonometric problem where sin x = 24/26, and you're asked to find the value of (sin 2x) * tan (x/2). This solution uses double angle and half-angle identities along with the Pythagorean theorem. Suitable for students in Grades 10-12.

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