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To solve for \(x\) in the given right triangle, we can use the tangent function, which relates the opposite side to the adjacent side for a given angle in a right triangle.

Given:
- The angle \(37^\circ\)
- The side adjacent to the angle (base) is \(8\) cm
- The opposite side to the angle (height) is \(x\)

We use the tangent function:

\[
\tan(37^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{8}
\]

We know that:

\[
\tan(37^\circ) \approx 0.75355
\]

So,

\[
0.75355 = \frac{x}{8}
\]

Solving for \(x\):

\[
x = 8 \cdot 0.75355
\]

\[
x \approx 6.0284
\]

Rounding to the nearest hundredth:

\[
x \approx 6.03 \text{ cm}
\]

Thus, \(x\) is approximately \(6.03\) cm.

Would you like more details or have any other questions? Here are some related questions for further practice:

1. How would you solve for the hypotenuse in this triangle?
2. What is the sine of the \(37^\circ\) angle in this triangle?
3. How can you use the Pythagorean theorem to verify the lengths of the sides?
4. What is the cosine of the \(37^\circ\) angle in this triangle?
5. How do you find the area of this triangle?
6. If the angle were \(45^\circ\) instead of \(37^\circ\), how would the side lengths change?
7. What other trigonometric functions can be used to find \(x\)?
8. How would you find the length of the other side of the triangle?

**Tip:** Always ensure your calculator is in the correct mode (degrees or radians) based on the angle measurement given in the problem.

Learn how to solve for x in a right triangle with a 37-degree angle using the tangent function. This problem involves trigonometric concepts and calculations. Suitable for high school students.

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