To solve for [tex]\(\sin A + \sin B\)[/tex] given that [tex]\(\cos A = 0.83\)[/tex] and [tex]\(\cos B = 0.55\)[/tex], we can use the trigonometric identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
for each angle [tex]\(\theta\)[/tex].
First, we find [tex]\(\sin A\)[/tex]. Given [tex]\(\cos A = 0.83\)[/tex]:
[tex]\[
\sin^2(A) + \cos^2(A) = 1
\][/tex]
[tex]\[
\sin^2(A) + 0.83^2 = 1
\][/tex]
[tex]\[
\sin^2(A) + 0.6889 = 1
\][/tex]
[tex]\[
\sin^2(A) = 1 - 0.6889
\][/tex]
[tex]\[
\sin^2(A) = 0.3111
\][/tex]
[tex]\[
\sin(A) = \sqrt{0.3111} \approx 0.5578
\][/tex]
Next, we find [tex]\(\sin B\)[/tex]. Given [tex]\(\cos B = 0.55\)[/tex]:
[tex]\[
\sin^2(B) + \cos^2(B) = 1
\][/tex]
[tex]\[
\sin^2(B) + 0.55^2 = 1
\][/tex]
[tex]\[
\sin^2(B) + 0.3025 = 1
\][/tex]
[tex]\[
\sin^2(B) = 1 - 0.3025
\][/tex]
[tex]\[
\sin^2(B) = 0.6975
\][/tex]
[tex]\[
\sin(B) = \sqrt{0.6975} \approx 0.8352
\][/tex]
Finally, we add [tex]\(\sin A\)[/tex] and [tex]\(\sin B\)[/tex]:
[tex]\[
\sin A + \sin B \approx 0.5578 + 0.8352 = 1.392928045119877
\][/tex]
By observing the options given:
A. [tex]\(0.28\)[/tex]
B. [tex]\(1\)[/tex]
C. [tex]\(0.38\)[/tex]
D. [tex]\(1.38\)[/tex]
The closest option to our calculated sum of [tex]\(\sin A + \sin B\)[/tex] is:
[tex]\[
\boxed{1.38}
\][/tex]
