To find [tex]\(\sin A + \sin B\)[/tex] for the complementary angles [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] in a right triangle [tex]\(ABC\)[/tex] with [tex]\(\cos A = 0.83\)[/tex] and [tex]\(\cos B = 0.55\)[/tex]:
1. Calculating [tex]\(\sin A\)[/tex]:
- We use the Pythagorean identity: [tex]\(\sin^2(A) + \cos^2(A) = 1\)[/tex].
- Given [tex]\(\cos A = 0.83\)[/tex]:
[tex]\[
\sin^2(A) = 1 - \cos^2(A)
\][/tex]
[tex]\[
\sin^2(A) = 1 - (0.83)^2
\][/tex]
[tex]\[
\sin^2(A) = 1 - 0.6889
\][/tex]
[tex]\[
\sin^2(A) = 0.3111
\][/tex]
[tex]\[
\sin A = \sqrt{0.3111} \approx 0.558
\][/tex]
2. Calculating [tex]\(\sin B\)[/tex]:
- Similarly, using [tex]\(\cos B = 0.55\)[/tex]:
[tex]\[
\sin^2(B) = 1 - \cos^2(B)
\][/tex]
[tex]\[
\sin^2(B) = 1 - (0.55)^2
\][/tex]
[tex]\[
\sin^2(B) = 1 - 0.3025
\][/tex]
[tex]\[
\sin^2(B) = 0.6975
\][/tex]
[tex]\[
\sin B = \sqrt{0.6975} \approx 0.835
\][/tex]
3. Adding [tex]\(\sin A\)[/tex] and [tex]\(\sin B\)[/tex]:
[tex]\[
\sin A + \sin B \approx 0.558 + 0.835 = 1.393
\][/tex]
Thus, [tex]\(\sin A + \sin B \approx 1.393\)[/tex].
By comparing this result with the provided answer choices, the closest and appropriate option is:
[tex]\[
\boxed{1.38}
\][/tex]
