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To find the indicated side \(a\) of the right triangle given the side \(12\) and an angle of \(30^\circ\), we can use trigonometric ratios. Specifically, we will use the sine function, as it relates the opposite side of an angle in a right triangle to the hypotenuse.

Given:
- The side adjacent to the \(30^\circ\) angle (let's call this side \(c\)) is \(12\).
- We need to find the opposite side \(a\) to the \(30^\circ\) angle.

For a right triangle with an angle of \(30^\circ\):

\[
\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

We know that:

\[
\sin(30^\circ) = \frac{1}{2}
\]

So, 

\[
\frac{a}{12} = \frac{1}{2}
\]

Solving for \(a\):

\[
a = 12 \cdot \frac{1}{2} = 6
\]

Therefore, the indicated side \(a\) is \(6\).

Now let's verify it using the given expression:

\[
a = \frac{12}{\sqrt{?}}
\]

Since we found \(a = 6\), we can substitute it into the equation to find the value under the square root:

\[
6 = \frac{12}{\sqrt{?}}
\]

Solving for the unknown under the square root:

\[
6 \sqrt{?} = 12
\]

\[
\sqrt{?} = 2
\]

\[
? = 4
\]

Thus, the value under the square root should be 4.

Would you like more details or have any other questions? Here are some related questions for further practice:

1. What is the length of the hypotenuse in this triangle?
2. How do you find the length of side \(b\) using trigonometric ratios?
3. What is the relationship between the sides of a 30°-60°-90° triangle?
4. How can you use the Pythagorean theorem to verify the lengths of the sides?
5. What is the cosine of the 30° angle in this triangle?
6. How would you calculate the area of this triangle?
7. If the angle were 45° instead of 30°, how would the side lengths change?
8. What other methods can be used to find the side lengths in a right triangle?

**Tip:** Always double-check your calculations and reasoning when working with trigonometric functions, especially in applied problems.

Learn how to find the opposite side \( a \) of a right triangle given the adjacent side 12 and angle 30°. This problem uses trigonometric ratios and the sine function to determine the side length. Suitable for high school students studying trigonometry.

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