https://file.aiquickdraw.com/mathbot/file/1726771152737int7zydw.jpg

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!

Given right triangle XYZ with angle Z = 90°, angle X = 60°, angle Y = 30°, and side XY = 4 units, what is the value of tan(Y)?

Learn how to find the tangent of a 30° angle in a right triangle XYZ, with angle Z = 90°, angle X = 60°, and side XY = 4 units. This solution demonstrates key trigonometric concepts and the use of 30-60-90 triangle properties. Suitable for students in Grades 9-12.

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