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.