Let's solve the problem step-by-step.
1. We know that in a right triangle, the sum of the two acute angles is [tex]\(90^\circ\)[/tex], making them complementary angles. Angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary.
2. In a right triangle, the sine of one of the acute angles is equal to the cosine of the other acute angle, and vice versa. This is because:
[tex]\[
\sin(A) = \cos(C) \quad \text{and} \quad \cos(A) = \sin(C)
\][/tex]
3. Given [tex]\(\sin(A) = \frac{24}{25}\)[/tex], since angle A and angle C are complementary, we can infer that:
[tex]\[
\cos(C) = \sin(A)
\][/tex]
Therefore,
[tex]\[
\cos(C) = \frac{24}{25}
\][/tex]
4. Similarly, given [tex]\(\cos(C) = \frac{20}{29}\)[/tex], we use the complementary property:
[tex]\[
\sin(A) = \cos(C)
\][/tex]
Thus,
[tex]\[
\sin(A) = \frac{20}{29}
\][/tex]
So the detailed solutions are:
1. If [tex]\(\sin(A) = \frac{24}{25}\)[/tex], the value of [tex]\(\cos(C)\)[/tex] is [tex]\(\frac{24}{25}\)[/tex].
2. If [tex]\(\cos(C) = \frac{20}{29}\)[/tex], the value of [tex]\(\sin(A)\)[/tex] is [tex]\(\frac{20}{29}\)[/tex].
Therefore, filling in the boxes with the correct values, we get:
If [tex]\(\sin (A) = \frac{24}{25}\)[/tex], the value of [tex]\(\cos (C) = \boxed{\frac{24}{25}}\)[/tex]
If [tex]\(\cos (C) = \frac{20}{29}\)[/tex], the value of [tex]\(\sin (A) = \boxed{\frac{20}{29}}\)[/tex]
