Let's go through the steps to solve the problem:
### Part 1: Finding [tex]\(\cos(C)\)[/tex] given [tex]\(\sin(A)=\frac{24}{25}\)[/tex]
Given: [tex]\(\sin(A) = \frac{24}{25}\)[/tex]
In a right triangle, the sine of angle [tex]\(A\)[/tex] can be described as the ratio of the length of the opposite side to the hypotenuse. This means:
- Opposite side (to angle [tex]\(A\)[/tex]) = 24
- Hypotenuse = 25
To find [tex]\(\cos(C)\)[/tex], we first need to determine the adjacent side. Using the Pythagorean theorem:
[tex]\[ (\text{Adj})^2 + (\text{Opposite})^2 = (\text{Hypotenuse})^2 \][/tex]
[tex]\[ (\text{Adj})^2 + 24^2 = 25^2 \][/tex]
[tex]\[ (\text{Adj})^2 + 576 = 625 \][/tex]
[tex]\[ (\text{Adj})^2 = 625 - 576 \][/tex]
[tex]\[ (\text{Adj})^2 = 49 \][/tex]
[tex]\[ \text{Adj} = \sqrt{49} = 7 \][/tex]
So, the adjacent side (to angle [tex]\(A\)[/tex]) is 7.
Since angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary in a right triangle, [tex]\(\sin(A) = \cos(C)\)[/tex].
Therefore, [tex]\(\cos(C) = \frac{\text{Adj}}{\text{Hypotenuse}} = \frac{7}{25} = 0.28\)[/tex]
[tex]\(\cos(C) = \boxed{0.28}\)[/tex]
### Part 2: Finding [tex]\(\sin(A)\)[/tex] given [tex]\(\cos(C)=\frac{20}{29}\)[/tex]
Given: [tex]\(\cos(C) = \frac{20}{29}\)[/tex]
In a right triangle, the cosine of angle [tex]\(C\)[/tex] can be described as the ratio of the length of the adjacent side to the hypotenuse. This means:
- Adjacent side (to angle [tex]\(C\)[/tex]) = 20
- Hypotenuse = 29
To find [tex]\(\sin(A)\)[/tex], we first need to determine the opposite side. Using the Pythagorean theorem:
[tex]\[ (\text{Opposite})^2 + (\text{Adj})^2 = (\text{Hypotenuse})^2 \][/tex]
[tex]\[ (\text{Opposite})^2 + 20^2 = 29^2 \][/tex]
[tex]\[ (\text{Opposite})^2 + 400 = 841 \][/tex]
[tex]\[ (\text{Opposite})^2 = 841 - 400 \][/tex]
[tex]\[ (\text{Opposite})^2 = 441 \][/tex]
[tex]\[ \text{Opposite} = \sqrt{441} = 21 \][/tex]
So, the opposite side (to angle [tex]\(C\)[/tex]) is 21.
Since angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary in a right triangle, [tex]\(\cos(C) = \sin(A)\)[/tex].
Therefore, [tex]\(\sin(A) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{21}{29} \approx 0.7241379310344828\)[/tex]
[tex]\(\sin(A) = \boxed{0.7241379310344828}\)[/tex]
So, we have:
- [tex]\(\cos(C) = 0.28\)[/tex]
- [tex]\(\sin(A) = 0.7241379310344828\)[/tex]
