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To find the length of \( AC \) in the given right triangle \( \triangle ABC \), we will use trigonometry. The angle \( \angle BAC \) is given as \(20^\circ\), \( AB = 2 \), and \( \angle ABC \) is \(90^\circ\).

We need to find \( AC \), which is the hypotenuse of the triangle.

Using the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

Here, \(\theta = 20^\circ\), the opposite side is \( AB = 2 \), and the hypotenuse is \( AC \).

So, we can write:
\[ \sin(20^\circ) = \frac{2}{AC} \]

Rearranging for \( AC \):
\[ AC = \frac{2}{\sin(20^\circ)} \]

Let's calculate this using \(\sin(20^\circ) \approx 0.3420\):
\[ AC = \frac{2}{0.3420} \approx 5.85 \]

Thus, \( AC \approx 5.85 \) (rounded to the nearest hundredth).

**Answer: \( AC = 5.85 \)**

Would you like any further details or have any questions?

Here are some related questions you might find interesting:
1. What is the length of \( BC \) using the given information?
2. How do you find \(\cos(20^\circ)\) for the given triangle?
3. How do you use the tangent function to find \( BC \)?
4. What is the length of the altitude from point \( B \) to hypotenuse \( AC \)?
5. How can you verify the calculated length of \( AC \) using the Pythagorean theorem?
6. How do you find the length of \( AC \) using the cosine function?
7. What are the other trigonometric ratios (sec, csc, cot) for \( 20^\circ \)?
8. How do you find the area of \( \triangle ABC \)?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) based on the angle measure given in trigonometric calculations.

Learn how to calculate the length of AC in right triangle ABC using trigonometry. Detailed step-by-step solution using the sine function and trigonometric ratios. Suitable for high school students.

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