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We are given the following information from the diagram and question:

- \( m \angle YXZ = 36^\circ \)
- \( m \angle WXZ = 18^\circ \)
- \( WZ = 9.1 \)

We are asked to find the value of \( YW \).

### Approach

Notice that triangles \( \triangle YXZ \) and \( \triangle WXZ \) are both right triangles, as indicated by the right angles at \( Y \) and \( W \).

We can use the angle information and apply trigonometric functions such as sine or cosine to relate the sides of these triangles. Since both triangles share the side \( XZ \), we will find a way to relate \( YW \) to the given side \( WZ \).

---

Let me calculate and explain the steps next.

### Step 1: Identify angles in the triangles
- \( \triangle YXZ \): The angle \( \angle YXZ = 36^\circ \), and \( \angle XYZ = 90^\circ \) since it is a right triangle.
- \( \triangle WXZ \): The angle \( \angle WXZ = 18^\circ \), and \( \angle XWZ = 90^\circ \).

### Step 2: Use trigonometry to solve for sides
- In \( \triangle WXZ \), we can use the cosine function since we know \( \angle WXZ = 18^\circ \) and the adjacent side \( WZ = 9.1 \).
  \[
  \cos(18^\circ) = \frac{XZ}{WZ}
  \]
  Solving for \( XZ \), we have:
  \[
  XZ = WZ \cdot \cos(18^\circ)
  \]

Next, let me calculate this value for \( XZ \).The length of \( XZ \) is approximately \( 9.57 \) units.

### Step 3: Use trigonometry to find \( YW \)
Now, in \( \triangle YXZ \), we can use the sine function because we know \( \angle YXZ = 36^\circ \) and the hypotenuse \( XZ = 9.57 \).

\[
\sin(36^\circ) = \frac{YW}{XZ}
\]

Solving for \( YW \), we get:

\[
YW = XZ \cdot \sin(36^\circ)
\]

Let me calculate this value for \( YW \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can follow the formula for calculating \( YW \):

\[
YW = XZ \cdot \sin(36^\circ)
\]

where \( XZ \) is approximately \( 9.57 \). Plugging in the values, you should get the final result for \( YW \).

If you have further questions or need additional assistance, feel free to ask!

### Related Questions:
1. How do you use trigonometric ratios in solving right triangles?
2. What is the significance of using cosine and sine in different scenarios?
3. Can you explain how the Law of Sines or Cosines might apply to more complex triangles?
4. How does angle measure affect the length of sides in right triangles?
5. What are common mistakes when applying trigonometric functions?

**Tip:** When solving right triangles, always confirm the angle and side relationships based on the triangle's geometry and right-angle properties.

Use the diagram shown. If m∠YXZ = 36°, m∠WXZ = 18°, and WZ = 9.1, what is the value of YW?

Solve for the length of YW in a right triangle using trigonometric ratios, given that m∠YXZ = 36°, m∠WXZ = 18°, and WZ = 9.1 units. This problem applies the cosine and sine functions to determine side lengths in triangles.

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