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The image you provided shows a multiple-choice question asking for the correct expression for \(\sin \theta\). Let's analyze the options:

- **The sine function** is defined as:
  \[
  \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
  \]
  In a right triangle, this means that \(\sin \theta\) is the ratio of the side opposite to angle \(\theta\) to the hypotenuse of the triangle.

In the coordinate plane (polar coordinates), the hypotenuse \(r\) is the distance from the origin to the point \((x, y)\), and the opposite side is the \(y\)-coordinate of the point.

So, \(\sin \theta = \frac{y}{r}\).

### Now, let's analyze the choices:
1. **Option 1**: \( \frac{y}{x} \) — incorrect.
2. **Option 2**: \( \frac{x}{y} \) — incorrect.
3. **Option 3**: \( \frac{x}{r} \) — incorrect.
4. **Option 4**: \( \frac{y}{r} \) — correct.

### The correct answer is the last option: \( \frac{y}{r} \).

Would you like any additional explanation or details on this? Here are some related questions you might find helpful:

1. How is the sine function used in trigonometric identities?
2. What is the difference between sine in a right triangle and in the unit circle?
3. How do you find the value of \(r\) in a coordinate plane?
4. How are the sine, cosine, and tangent functions related?
5. What are the applications of the sine function in physics?

**Tip**: Always visualize trigonometric functions with triangles or coordinate planes to better understand their relationships!

sin θ = ______. Identify the correct expression for sin θ from the options provided.

Learn how to find the sine of an angle using the correct expression from a right triangle and in a coordinate plane. This trigonometry problem includes core concepts like the sine function formula and the Pythagorean theorem. Ideal for students in Grades 9-11.

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