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Given the point \((-20, 48)\) on the terminal side of the angle \(\theta\), let's find \(\sin \theta\).

### Step-by-step Solution:

1. **Identify the coordinates:** The coordinates of the point are \(x = -20\) and \(y = 48\).

2. **Find the hypotenuse:** The hypotenuse \(r\) of the right triangle formed can be found using the Pythagorean theorem:
   \[
   r = \sqrt{x^2 + y^2} = \sqrt{(-20)^2 + 48^2} = \sqrt{400 + 2304} = \sqrt{2704}.
   \]
   Simplifying this, we get:
   \[
   r = 52.
   \]

3. **Calculate \(\sin \theta\):** By definition, \(\sin \theta = \frac{y}{r}\):
   \[
   \sin \theta = \frac{48}{52}.
   \]
   Simplify this fraction:
   \[
   \sin \theta = \frac{12}{13}.
   \]

So, the exact value of \(\sin \theta\) is \(\frac{12}{13}\).

Would you like more details or have any questions?

### Additional Related Questions

1. What is the cosine of the angle \(\theta\) given the same point?
2. How would you find \(\tan \theta\) for this point?
3. If the point were in a different quadrant, how would the sign of \(\sin \theta\) change?
4. What is the relationship between the coordinates of a point and the trigonometric functions?
5. How would you find \(\sin \theta\) if only the angle measure was given?

### Tip
When working with trigonometric functions, always remember to consider the quadrant in which the point lies to determine the correct signs of the functions.

Suppose that θ is in standard position and the given point is on the terminal side of θ. Give the exact value of the indicated trig function for θ (-20, 48); Find sin θ.

Learn how to calculate sin(θ) for the point (-20, 48) in standard position on the terminal side of θ using trigonometry and the Pythagorean theorem. Suitable for high school students in Grades 10-12.

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