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To solve the problem shown in the image, we need to express the trigonometric functions \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) in terms of the sides of the right triangle.

In the image, the right triangle is labeled as follows:
- \(x\) is the adjacent side to the angle \(\theta\),
- \(y\) is the opposite side to the angle \(\theta\),
- \(z\) is the hypotenuse.

Using the definitions of sine, cosine, and tangent for a right triangle, we have:

1. **Sine (\(\sin \theta\))**:
   \[
   \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{z}
   \]

2. **Cosine (\(\cos \theta\))**:
   \[
   \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{z}
   \]

3. **Tangent (\(\tan \theta\))**:
   \[
   \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}
   \]

Thus, the exact expressions are:

- \(\sin \theta = \frac{y}{z}\)
- \(\cos \theta = \frac{x}{z}\)
- \(\tan \theta = \frac{y}{x}\)

Would you like further clarification on these concepts or how to apply them? 

Here are 5 related questions to expand your understanding:
1. How do you find the hypotenuse if you only know the opposite and adjacent sides?
2. What is the relationship between the Pythagorean theorem and trigonometric functions?
3. How would these trigonometric ratios change if the triangle were not a right triangle?
4. How can you find the angle \(\theta\) if you know the values of sine, cosine, or tangent?
5. What are the reciprocal trigonometric functions for sine, cosine, and tangent?

**Tip**: Always check that the sum of squares of the opposite and adjacent sides equals the square of the hypotenuse, according to the Pythagorean theorem.

Find the exact expressions for sin θ, cos θ, and tan θ for the given right triangle with sides labeled x, y, and z.

Learn how to find the exact values for sin θ, cos θ, and tan θ for a right triangle. This problem involves understanding the trigonometric ratios and applying them to a triangle where the sides are labeled x, y, and z. Perfect for high school students learning trigonometry.

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