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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]$\cos (C)=\frac{20}{29}$[/tex], the value of [tex]$\sin (A)=$[/tex] [tex]$\square$[/tex]



Answer :

GinnyAnswer
Given that \( A \) and \( C \) are complementary angles in a right triangle, we know that \( A + C = 90^\circ \). This implies that \( \sin(A) = \cos(C) \) and \( \cos(A) = \sin(C) \).

1. If \(\sin (A) = \frac{24}{25}\):

Since \( A \) and \( C \) are complementary angles, \(\cos (C)\) is equal to \(\sin (A)\).
Therefore, \(\cos (C) = \frac{24}{25}\).

So, the value of \(\cos (C)\) is \( \boxed{\frac{24}{25}} \).

2. If \(\cos (C) = \frac{20}{29}\):

Again, because \( A \) and \( C \) are complementary angles, \(\sin (A)\) is equal to \(\cos (C)\).
So, \(\sin (A) = \frac{20}{29}\).

Thus, the value of [tex]\(\sin (A)\)[/tex] is [tex]\( \boxed{\frac{20}{29}} \)[/tex].

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