A tree is 8 meters tall, and its shadow is 10 meters long. The distance from the top of the tree to the tip of the shadow (hypotenuse) is 12.81 meters. What is the sine of the angle between the tree and the ground?

To find the sine of the angle (\(\theta\)) between the tree and the ground, we can use the definition of sine in a right triangle:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case:
- The **opposite side** is the height of the tree = 8 meters.
- The **hypotenuse** is the distance from the top of the tree to the tip of the shadow = 12.81 meters.

Now, we calculate \(\sin(\theta)\):

\[
\sin(\theta) = \frac{8}{12.81} \approx 0.6245
\]

Thus, the sine of the angle is approximately **0.6245**.

Would you like more details or have any other questions? Here are five related questions to explore further:

1. How can you find the angle itself using the sine function?
2. What is the cosine of the same angle using the information provided?
3. Can you calculate the tangent of the angle?
4. How would the values change if the tree were taller?
5. What is the significance of using trigonometry in real-world problems like this?

**Tip:** When solving right triangles, identifying the opposite, adjacent, and hypotenuse sides for an angle is key to choosing the correct trigonometric ratio (sine, cosine, or tangent).

Learn how to find the sine of the angle between a tree and the ground, given that the tree is 8 meters tall, its shadow is 10 meters long, and the hypotenuse is 12.81 meters. This problem explores trigonometric ratios and the Pythagorean Theorem. Suitable for Grades 9-10.

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