To solve the problem, we need to use some properties of right triangles and trigonometric identities involving complementary angles.
1. Understanding Complementary Angles:
- Angles \(A\) and \(C\) in the right triangle \(ABC\) are complementary. This means:
[tex]\[
A + C = 90^\circ
\][/tex]
- Therefore, \(\sin(A)\) and \(\cos(C)\) are related:
[tex]\[
\sin(A) = \cos(90^\circ - A) = \cos(C)
\][/tex]
2. Given Information:
- We are given:
[tex]\[
\sin(A) = \frac{24}{25}
\][/tex]
3. Find \(\cos(C)\):
- Since \(\sin(A) = \cos(C)\) (as \(A\) and \(C\) are complementary angles), we substitute:
[tex]\[
\cos(C) = \sin(A) = \frac{24}{25}
\][/tex]
So, the value of \(\cos(C)\) is \(\frac{24}{25}\).
4. Given Incorrect Statement Analysis:
- We need to understand the point of confusion:
[tex]\[
\cos(C) = \frac{20}{20} = 1
\][/tex]
- This statement is incorrect because \(\cos\) of an angle in a right triangle cannot be greater than 1, and it does not match the given \(\sin(A)\) value.
5. Confirming Values:
The values provided are:
[tex]\[
\cos(C) = 0.96 \quad \text{and} \quad \sin(A) = 0.96
\][/tex]
These values align since:
[tex]\[
\frac{24}{25} \approx 0.96
\][/tex]
Hence, we can now populate the required boxes based on the details:
- The value of \(\cos(C) = \frac{24}{25}\)
- The value of \(\sin(A) = 0.96\)
So the answers in numerical form are:
[tex]\[
\boxed{\frac{24}{25}}
\][/tex]
[tex]\[
\boxed{0.96}
\][/tex]
