To solve for $x$ using trigonometric ratios, we'll use the given right triangle and the provided angle $37^\circ$. The right triangle has a hypotenuse of $13$, an adjacent side $x$ to the $37^\circ$ angle, and an opposite side of $12$.

We'll use the cosine trigonometric ratio, which relates the adjacent side and the hypotenuse:

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Here, $\theta = 37^\circ$, the adjacent side is $x$, and the hypotenuse is $13$:

$\cos(37^\circ) = \frac{x}{13}$

To solve for $x$:

$x = 13 \cdot \cos(37^\circ)$

We need the value of $\cos(37^\circ)$. Using a calculator, we find:

$\cos(37^\circ) \approx 0.7986$

Substitute this value back into the equation:

$x = 13 \cdot 0.7986$ $x \approx 10.382$

Therefore, $x \approx 10.38$.

Would you like more details or have any questions?

Here are some related questions:

1. How can we verify the solution for $x$ using the sine or tangent ratios?
2. What is the sine ratio and how can it be used to solve for the opposite side?
3. How do trigonometric ratios relate to the unit circle?
4. Can you explain why the cosine ratio involves the adjacent side and hypotenuse?
5. What are some practical applications of trigonometric ratios in real life?
6. How would you solve for the hypotenuse if the adjacent side and the angle were given?
7. How can you use the tangent ratio to find unknown sides in a right triangle?
8. What is the significance of angle of elevation and depression in trigonometry?

Tip: When using trigonometric ratios, ensure your calculator is set to the correct mode (degrees or radians) based on the angle measurement you are using.