To find \(\sin X\) and \(\tan Y\) for the right triangle \(XYZ\), we use the following definitions from trigonometry:

- \(\sin X\) is the ratio of the side opposite to angle \(X\) (which is \(YZ\)) to the hypotenuse \(XY\).
- \(\tan Y\) is the ratio of the side opposite to angle \(Y\) (which is \(XZ\)) to the side adjacent to angle \(Y\) (which is \(YZ\)).

### 1. Calculating \(\sin X\):

\[
\sin X = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{YZ}{XY} = \frac{3}{\sqrt{34}}
\]

To rationalize the denominator:

\[
\sin X = \frac{3}{\sqrt{34}} \times \frac{\sqrt{34}}{\sqrt{34}} = \frac{3\sqrt{34}}{34}
\]

So, \(\sin X = \frac{3\sqrt{34}}{34}\).

### 2. Calculating \(\tan Y\):

\[
\tan Y = \frac{\text{opposite}}{\text{adjacent}} = \frac{XZ}{YZ} = \frac{5}{3}
\]

So, \(\tan Y = \frac{5}{3}\).

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How would you find the value of \(\cos X\) and \(\sin Y\)?
2. What is the value of \(\cos Y\) and \(\tan X\)?
3. Can you find the angle measures for \(X\) and \(Y\) using \(\sin X\) and \(\tan Y\)?
4. How would \(\sin X\) and \(\tan Y\) change if the sides of the triangle were scaled by a factor of 2?
5. Can you verify \(\sin^2 X + \cos^2 X = 1\) using the values from this triangle?

**Tip:** Always simplify trigonometric ratios as much as possible, and consider rationalizing the denominator if needed.

Learn how to find sin X and tan Y in a right triangle XYZ using trigonometric ratios. Step-by-step solution with detailed calculations and rationalization of denominators. Suitable for high school students studying trigonometry.

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