Trigonometric ratio of an angle in a right triangle.

1. Identify the problem type: This is a trigonometry problem involving a right-angled triangle.
2. Recall the definition of tangent: In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.
3. Identify the angle of interest: The problem asks for $\tan(67^\circ)$.
4. Identify the sides of the triangle relative to the angle $67^\circ$:
- The side opposite the $67^\circ$ angle is the side with length 12.
- The side adjacent to the $67^\circ$ angle (and not the hypotenuse) is the side with length 5.
- The hypotenuse is the side with length 13.
5. Apply the tangent definition: Using the definition of tangent, we have:
$$\tan(67^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}$$