https://file.aiquickdraw.com/mathbot/file/17260197180560quzakdo.jpg

The image shows a right triangle with the following dimensions:

- The side opposite the angle \( \theta \) (which we'll refer to as the opposite side) is 32 meters.
- The adjacent side (next to \( \theta \)) is 60 meters.
- The hypotenuse (the side opposite the right angle) is 68 meters.

### To find the angle \( \theta \):
We can use the trigonometric ratios. Since we know the lengths of the opposite side and the adjacent side, we can use the tangent function:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{32}{60}
\]

Now, calculate the angle \( \theta \):

\[
\theta = \arctan\left(\frac{32}{60}\right)
\]

Let's calculate this:

\[
\theta \approx \arctan(0.5333) \approx 28.07^\circ
\]

Thus, the angle \( \theta \) is approximately \( 28.07^\circ \).

### Would you like more details on this solution or have any other questions?

### Related Questions:
1. How would you find \( \theta \) using the sine or cosine functions instead?
2. What would the value of \( \theta \) be if the opposite side was 40 meters instead of 32 meters?
3. How can you use the Pythagorean theorem to verify the lengths given in the triangle?
4. What are the other two trigonometric ratios (sine and cosine) for the angle \( \theta \)?
5. How would the triangle change if \( \theta \) was a different angle?

### Tip:
Always double-check your trigonometric ratio calculations by considering alternative methods or checking with known values.

Find the angle \( \theta \) in a right triangle with side lengths of 32 meters (opposite), 60 meters (adjacent), and 68 meters (hypotenuse).

Learn how to find the angle \( \theta \) in a right triangle with side lengths of 32m, 60m, and 68m using the tangent function. Detailed step-by-step solution for high school students.

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