find tangent of angle in right triangle

1. Identify the definition of tangent: In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

1. Locate the angle $62^{\circ}$: Looking at the provided triangle, the angle $62^{\circ}$ is at the top vertex.

3. Identify the opposite and adjacent sides relative to $62^{\circ}$:
- The side opposite to the $62^{\circ}$ angle is the horizontal leg, which has a length of $15$.
- The side adjacent to the $62^{\circ}$ angle is the vertical leg, which has a length of $8$.
- The side with length $17$ is the hypotenuse.

4. Calculate the tangent: Using the definition from step 1, we substitute the values identified in step 3:
$$\tan(62^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{15}{8}$$

1. Compare with the given options: The calculated value $\frac{15}{8}$ matches option C.