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The question asks to find the length of side \( AC \) in a right triangle \( \triangle ABC \), where:

- \( \angle B = 90^\circ \),
- \( \angle A = 60^\circ \),
- \( BC = 6 \, \text{cm} \).

### Solution:

Since \( \triangle ABC \) is a right triangle and \( \angle A = 60^\circ \), we can use trigonometric ratios to solve for \( AC \).

1. In a right triangle, \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle A = 60^\circ \), the opposite side is \( BC \) and the hypotenuse is \( AC \).

\[
\sin(60^\circ) = \frac{BC}{AC}
\]
\[
\sin(60^\circ) = \frac{6}{AC}
\]
Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), we can plug this in:

\[
\frac{\sqrt{3}}{2} = \frac{6}{AC}
\]

2. Solve for \( AC \):

\[
AC = \frac{6 \times 2}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3} \, \text{cm}
\]

Thus, the correct answer is \( \boxed{4\sqrt{3} \, \text{cm}} \), which corresponds to option \( d \).

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Would you like more details on how the trigonometric functions were applied? Here are five related questions to consider:

1. What is the Pythagorean theorem, and can it be applied here?
2. How do you calculate the length of the third side of a triangle using trigonometry?
3. What are the properties of a 30-60-90 triangle?
4. How can you use cosine instead of sine to solve the same problem?
5. What is the general formula for solving for sides in right triangles with any angle?

**Tip:** Always check the given angle before selecting a trigonometric ratio to ensure you're using the correct opposite, adjacent, or hypotenuse.

In ΔABC, ∠B = 90°, ∠A = 60°, BC = 6 cm, find AC.

Learn how to calculate the length of side AC in a right triangle where ∠A = 60°, ∠B = 90°, and side BC = 6 cm using trigonometric ratios. Suitable for high school students.

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