The problem shows a right triangle with one angle measuring $60^\circ$, the opposite side labeled as $x$, and the adjacent side (to the angle) given as 37. To solve for $x$, we can use trigonometric ratios. Specifically, the tangent function is appropriate here because we are dealing with the opposite side ($x$) and the adjacent side (37):

$\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{37}$

Now, solve for $x$:

$x = 37 \times \tan(60^\circ)$

We know that $\tan(60^\circ) = \sqrt{3} \approx 1.732$. Substituting this into the equation:

$x = 37 \times 1.732$

Let's compute the value of $x$.The value of $x$ is approximately $64.09$.

Would you like more details on the solution, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the sine and cosine of $60^\circ$ and how do they relate to this problem?
2. How would you solve for $x$ if the adjacent side and hypotenuse were given instead?
3. Can you use the Pythagorean theorem to check the result if the hypotenuse is known?
4. What trigonometric function would you use if the given angle was $30^\circ$ instead?
5. How can you use a calculator to find the tangent of other angles?

Tip: Always ensure your calculator is in the correct mode (degrees or radians) when solving trigonometric problems involving angles!