Let's break down the problem step-by-step.
1. Given: The sine of angle A is expressed as:
[tex]\[
\sin(A) = \frac{24}{2 \pi}
\][/tex]
Since angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary in a right triangle ([tex]\(A + C = 90^\circ\)[/tex]), we have the relationship:
[tex]\[
\sin(A) = \cos(C)
\][/tex]
2. Calculating [tex]\(\cos(C)\)[/tex] using the given [tex]\(\sin(A)\)[/tex]:
Substitute the given value for [tex]\(\sin(A)\)[/tex]:
[tex]\[
\cos(C) = \sin(A) = \frac{24}{2 \pi}
\][/tex]
To simplify:
[tex]\[
\cos(C) = \frac{24}{2 \pi} = \frac{24}{6.283185307} \approx 3.819718634205488
\][/tex]
Thus, the value of [tex]\(\cos(C)\)[/tex] is:
[tex]\[
\cos(C) \approx 3.819718634205488
\][/tex]
3. Given: The cosine of angle [tex]\(C\)[/tex] is expressed as:
[tex]\[
\cos(C) = \frac{20}{28}
\][/tex]
We need to find [tex]\(\sin(A)\)[/tex]. Again, using the relationship between complementary angles in a right triangle:
[tex]\[
\sin(A) = \cos(C)
\][/tex]
Substitute the given value for [tex]\(\cos(C)\)[/tex], and simplify:
[tex]\[
\sin(A) = \frac{20}{28} = \frac{20}{28} \approx 0.7142857142857143
\][/tex]
Thus, the value of [tex]\(\sin(A)\)[/tex] is:
[tex]\[
\sin(A) \approx 0.7142857142857143
\][/tex]
Final Answers:
[tex]\[
\cos(C) \approx 3.819718634205488
\][/tex]
[tex]\[
\sin(A) \approx 0.7142857142857143
\][/tex]
