The image depicts a right triangle $XYZ$, where angle $Z$ is $90^\circ$, angle $X$ is $60^\circ$, and angle $Y$ is $30^\circ$. Side $XY$ is labeled as 4 units.

We are asked to find the value of $\tan(Y)$, which is $\tan(30^\circ)$.

Step-by-step Solution:

For a right triangle: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ For $\theta = 30^\circ$, the side opposite to $Y$ is $XZ$, and the side adjacent to $Y$ is $YZ$.

Using trigonometric ratios for a 30-60-90 triangle:

• The side opposite $30^\circ$ (side $XZ$) is half of the hypotenuse $XY$, so: $XZ = \frac{XY}{2} = \frac{4}{2} = 2$
• The side opposite $60^\circ$ (side $YZ$) is $\frac{\sqrt{3}}{2}$ times the hypotenuse, so: $YZ = \frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3}$

Now, we can compute $\tan(30^\circ)$: $\tan(30^\circ) = \frac{XZ}{YZ} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}}$ Rationalizing the denominator: $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Thus, the value of $\tan(Y)$ is $\frac{\sqrt{3}}{3}$.

The correct answer is: $\frac{\sqrt{3}}{3}$ (second option).

Would you like more details on this, or have any questions? Here are five related questions you might find helpful:

1. How do you calculate sine and cosine for a 30-degree angle?
2. What are the properties of a 30-60-90 triangle?
3. How do you rationalize denominators in trigonometry?
4. What is the difference between sine, cosine, and tangent functions?
5. How can you use trigonometry to find missing side lengths in a right triangle?

Tip: Memorizing the side ratios of a 30-60-90 triangle can simplify solving many trigonometric problems!