Answer :
To solve for the values of [tex]\(\cos(C)\)[/tex] and [tex]\(\sin(A)\)[/tex] in a right triangle [tex]\(ABC\)[/tex] given the complementary angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex]:
### Step-by-Step Solution:
1. Identify the Relationship Between Angles:
In any right triangle, the two acute angles are complementary, meaning [tex]\(\angle A + \angle C = 90^\circ\)[/tex].
2. Use the Complementary Angle Property:
For two complementary angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] in a right triangle, the sine of one angle is equal to the cosine of the other:
[tex]\[ \sin(A) = \cos(90^\circ - A) = \cos(C) \][/tex]
3. Given Value for [tex]\(\sin(A)\)[/tex]:
It is given that:
[tex]\[ \sin(A) = \frac{24}{25} \][/tex]
4. Determine [tex]\(\cos(C)\)[/tex]:
Using the complementary angle property mentioned above:
[tex]\[ \cos(C) = \sin(A) = \frac{24}{25} \][/tex]
Therefore, [tex]\(\cos(C) = 0.96\)[/tex].
5. Given Value for [tex]\(\cos(C)\)[/tex]:
Although there seems to be a typo in the question stating [tex]\(\cos(C) = \frac{20}{20}\)[/tex] (which simplifies to 1, but it should probably reference the correct complementary relationship):
We previously established that:
[tex]\[ \cos(C) = \frac{24}{25} \][/tex]
6. Reaffirm [tex]\(\sin(A)\)[/tex]:
Therefore, once again we see:
[tex]\[ \sin(A) = \frac{24}{25} \][/tex]
Which reaffirms that [tex]\(\sin(A) = 0.96\)[/tex].
### Final Answer:
If [tex]\(\sin(A) = \frac{24}{25}\)[/tex], then the value of [tex]\(\cos(C) = 0.96\)[/tex].
Additionally, confirming [tex]\(\cos(C)\)[/tex] correctly as [tex]\(\frac{24}{25}\)[/tex] (or 0.96), reaffirms that [tex]\(\sin(A)= 0.96\)[/tex].
So:
- If [tex]\(\sin(A) = \frac{24}{25}\)[/tex], the value of [tex]\(\cos(C)\)[/tex] is [tex]\(\boxed{0.96}\)[/tex].
- Given that [tex]\(\cos(C) = 0.96\)[/tex], the value of [tex]\(\sin(A)\)[/tex] is [tex]\(\boxed{0.96}\)[/tex].
### Step-by-Step Solution:
1. Identify the Relationship Between Angles:
In any right triangle, the two acute angles are complementary, meaning [tex]\(\angle A + \angle C = 90^\circ\)[/tex].
2. Use the Complementary Angle Property:
For two complementary angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] in a right triangle, the sine of one angle is equal to the cosine of the other:
[tex]\[ \sin(A) = \cos(90^\circ - A) = \cos(C) \][/tex]
3. Given Value for [tex]\(\sin(A)\)[/tex]:
It is given that:
[tex]\[ \sin(A) = \frac{24}{25} \][/tex]
4. Determine [tex]\(\cos(C)\)[/tex]:
Using the complementary angle property mentioned above:
[tex]\[ \cos(C) = \sin(A) = \frac{24}{25} \][/tex]
Therefore, [tex]\(\cos(C) = 0.96\)[/tex].
5. Given Value for [tex]\(\cos(C)\)[/tex]:
Although there seems to be a typo in the question stating [tex]\(\cos(C) = \frac{20}{20}\)[/tex] (which simplifies to 1, but it should probably reference the correct complementary relationship):
We previously established that:
[tex]\[ \cos(C) = \frac{24}{25} \][/tex]
6. Reaffirm [tex]\(\sin(A)\)[/tex]:
Therefore, once again we see:
[tex]\[ \sin(A) = \frac{24}{25} \][/tex]
Which reaffirms that [tex]\(\sin(A) = 0.96\)[/tex].
### Final Answer:
If [tex]\(\sin(A) = \frac{24}{25}\)[/tex], then the value of [tex]\(\cos(C) = 0.96\)[/tex].
Additionally, confirming [tex]\(\cos(C)\)[/tex] correctly as [tex]\(\frac{24}{25}\)[/tex] (or 0.96), reaffirms that [tex]\(\sin(A)= 0.96\)[/tex].
So:
- If [tex]\(\sin(A) = \frac{24}{25}\)[/tex], the value of [tex]\(\cos(C)\)[/tex] is [tex]\(\boxed{0.96}\)[/tex].
- Given that [tex]\(\cos(C) = 0.96\)[/tex], the value of [tex]\(\sin(A)\)[/tex] is [tex]\(\boxed{0.96}\)[/tex].
