In a right triangle, the two non-right angles $A$ and $C$ are complementary, meaning that $A + C = 90^\circ$. This fact gives us the relationship:

$\sin(A) = \cos(C) \quad \text{and} \quad \cos(A) = \sin(C)$

Now, let's solve the two parts of the problem:

1. If $\sin(A) = \frac{24}{25}$, what is $\cos(C)$?

Since $\sin(A) = \cos(C)$ (because $A + C = 90^\circ$): $\cos(C) = \sin(A) = \frac{24}{25}$

2. If $\cos(C) = \frac{20}{29}$, what is $\sin(A)$?

Again, since $\sin(A) = \cos(C)$: $\sin(A) = \cos(C) = \frac{20}{29}$

Thus, the two values are:

1. $\cos(C) = \frac{24}{25}$
2. $\sin(A) = \frac{20}{29}$

Would you like a more detailed explanation of complementary angles and their trigonometric identities? Here are five related questions you can explore:

1. What is the relationship between sine and cosine for complementary angles in any right triangle?
2. How can you find the hypotenuse of a right triangle if you are given one side and one angle?
3. How do you calculate the tangent of an angle if you know its sine or cosine value?
4. If $\cos(A) = \frac{7}{25}$, how can you find $\sin(A)$?
5. How do you solve for angle $A$ given $\sin(A) = \frac{24}{25}$?

Tip: In right triangles, the Pythagorean identity $\sin^2(A) + \cos^2(A) = 1$ is a helpful tool for calculating missing trigonometric values.