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Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

A right triangle [tex]$ABC$[/tex] has complementary angles [tex]$A$[/tex] and [tex]$C$[/tex].

If [tex]\sin (A) = \frac{24}{25}[/tex], the value of [tex]\cos (C) =[/tex] [tex]$\square$[/tex]

If [tex]\cos (C) = \frac{20}{25}[/tex], the value of [tex]\sin (A) =[/tex] [tex]$\square$[/tex]



Answer :

GinnyAnswer
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]

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