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Let's solve the problem by applying trigonometry to the right triangle.

The given triangle has:
- \(\angle W = 57^\circ\),
- \(WY = 65\),
- Hypotenuse \(w = 8\),
- Unknown sides \(x\), \(y\), and \(z\).

Here’s how we can find the lengths of the missing sides:

### Step 1: Use Trigonometric Ratios
For right triangles, we use the following trigonometric functions:
- **Sine**: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
- **Cosine**: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
- **Tangent**: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

#### To find \(y\) (the adjacent side to \(57^\circ\)):
We will use the cosine function:
\[
\cos(57^\circ) = \frac{y}{65}
\]
\[
y = 65 \cdot \cos(57^\circ)
\]

#### To find \(x\) (the opposite side to \(57^\circ\)):
We will use the sine function:
\[
\sin(57^\circ) = \frac{x}{65}
\]
\[
x = 65 \cdot \sin(57^\circ)
\]

#### To find \(z\) (the remaining side):
The Pythagorean theorem can be applied:
\[
z = \sqrt{y^2 + x^2}
\]

I'll now calculate these values.Here are the calculated values:

- \(y = 35.4\) (adjacent side)
- \(x = 54.5\) (opposite side)
- \(z = 65.0\) (hypotenuse)

These values represent the missing sides of the triangle based on the given conditions.

Would you like any further clarification or a detailed explanation?

Here are 5 related questions:
1. How would the values change if the angle \(W\) were larger?
2. How can the Pythagorean theorem be applied to verify the values of \(x\) and \(y\)?
3. What happens to the trigonometric ratios if the angle is \(90^\circ\)?
4. Can you explain why \(\sin(57^\circ)\) and \(\cos(57^\circ)\) are sufficient to solve the problem?
5. How would you solve this if the hypotenuse were unknown?

**Tip:** Always ensure your calculator is set to degrees when working with trigonometric functions involving angles in degrees.

Hallar las longitudes de W, X, Y, y Z que se muestran en la siguiente figura si WY = 65. Redondear las respuestas a la décima más cercana.

Learn how to find the lengths of sides in a right triangle using trigonometric ratios such as sine and cosine, with the help of the Pythagorean theorem. This problem involves finding the sides W, X, Y, and Z when WY = 65 and angle W = 57°. Suitable for students in Grades 9-12.

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