To solve the problem, note that in a right triangle, the sum of the angles is [tex]\(90^\circ\)[/tex]. Therefore, angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary, which means:
[tex]\[A + C = 90^\circ\][/tex]
In trigonometry, we know that for complementary angles ([tex]\(\theta\)[/tex] and [tex]\(90^\circ - \theta\)[/tex]):
[tex]\[\sin(A) = \cos(90^\circ - A)\][/tex]
[tex]\[\cos(C) = \cos(90^\circ - C)\][/tex]
### Step-by-Step Solution
1. Calculate [tex]\(\cos(C)\)[/tex] when [tex]\(\sin(A) = \frac{24}{25}\)[/tex]:
Since [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary:
[tex]\[\sin(A) = \cos(C)\][/tex]
Given that:
[tex]\[\sin(A) = \frac{24}{25}\][/tex]
Therefore:
[tex]\[\cos(C) = \frac{24}{25}\][/tex]
So, the value of [tex]\(\cos(C)\)[/tex] is:
[tex]\[
\cos(C) = \frac{24}{25} = 0.96
\][/tex]
2. Calculate [tex]\(\sin(A)\)[/tex] when [tex]\(\cos(C) = \frac{20}{29}\)[/tex]:
Again, since [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary:
[tex]\[\cos(C) = \sin(90^\circ - C)\][/tex]
Given that:
[tex]\[\cos(C) = \frac{20}{29}\][/tex]
This implies:
[tex]\[\sin(A) = \frac{20}{29}\][/tex]
So, the value of [tex]\(\sin(A)\)[/tex] is:
[tex]\[
\sin(A) = \frac{20}{29} \approx 0.6896551724137931
\][/tex]
Therefore, the solutions are:
1. [tex]\(\cos(C) = 0.96\)[/tex] when [tex]\(\sin(A) = \frac{24}{25}\)[/tex]
2. [tex]\(\sin(A) = 0.6896551724137931\)[/tex] when [tex]\(\cos(C) = \frac{20}{29}\)[/tex]
