To solve this problem, we analyze the relationship between the given functions $f(x) = \cos x$ and $g(x) = \sin x$.

Step-by-Step Solution:

1. Relationship Between $\cos x$ and $\sin x$:

   • Recall the trigonometric identity: $\sin x = \cos \left(x - \frac{\pi}{2}\right).$
   • This shows that $g(x) = \sin x$ can be viewed as $f(x) = \cos x$ shifted horizontally to the right by $\frac{\pi}{2}$.

2. Type of Transformation:

   • Horizontal shifts are horizontal translations in the graph of $f(x)$.
   • Therefore, the graph of $g(x) = \sin x$ is a horizontal translation of the graph of $f(x) = \cos x$.

Correct Answer:

(A) The graph of $g$ is a horizontal translation of the graph of $f$.

Would you like further clarification or a detailed graph of the functions?

Here are 5 related questions for further exploration:

1. How do you derive transformations of trigonometric functions mathematically?
2. What are the effects of vertical and horizontal translations on the graph of a function?
3. How does the identity $\sin x = \cos(x - \frac{\pi}{2})$ generalize to other trigonometric transformations?
4. What happens to the graph of $f(x) = \cos x$ under a vertical translation or dilation?
5. How can you identify transformations using phase shifts in trigonometric equations?

Tip: Always remember that horizontal translations of trigonometric functions correspond to phase shifts in their equations.