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This problem involves a right triangle, and we need to find the side labeled \( x \). From the given information:

- The hypotenuse is \( 37 \) units.
- One of the angles is \( 30^\circ \).
- \( x \) is the side adjacent to the \( 30^\circ \) angle.

We can use the cosine trigonometric function to solve for \( x \) because the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse:

\[
\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{37}
\]

Now, solve for \( x \):

\[
x = 37 \cdot \cos(30^\circ)
\]

The value of \( \cos(30^\circ) \) is \( \frac{\sqrt{3}}{2} \). Substituting this into the equation:

\[
x = 37 \cdot \frac{\sqrt{3}}{2} \approx 37 \cdot 0.866
\]

Let me calculate the value of \( x \).The value of the side labeled \( x \) is approximately \( 32.04 \) units.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How would you find the opposite side of the triangle using trigonometry?
2. What is the sine of a \( 30^\circ \) angle, and how can it help in solving for other sides?
3. Can you apply the Pythagorean theorem to verify the solution?
4. How does the angle size affect the ratio of the sides in a right triangle?
5. How would the solution change if the angle were \( 45^\circ \) instead of \( 30^\circ \)?

**Tip:** When solving right triangles, always decide which trigonometric function to use based on the given angle and the sides you know.

Find the side labeled x in the right triangle where the hypotenuse is 37, the adjacent side is x, and the angle is 30 degrees.

Learn how to solve for the side labeled x in a right triangle with a hypotenuse of 37 and a 30° angle. This problem covers key trigonometry concepts and demonstrates how to use the cosine function to find unknown sides. Suitable for high school students.

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