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.