To find the sum of [tex]\(\sin A\)[/tex] and [tex]\(\sin B\)[/tex] where [tex]\(\cos A = 0.83\)[/tex] and [tex]\(\cos B = 0.55\)[/tex], we can use the Pythagorean identity for trigonometric functions in a right triangle. Let's go through the steps:
1. Recall the Pythagorean identity:
[tex]\[
\cos^2 \theta + \sin^2 \theta = 1
\][/tex]
2. Using this identity, solve for [tex]\(\sin A\)[/tex]:
[tex]\[
\cos^2 A + \sin^2 A = 1
\][/tex]
[tex]\[
(0.83)^2 + \sin^2 A = 1
\][/tex]
[tex]\[
0.6889 + \sin^2 A = 1
\][/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]
3. Similarly, solve for [tex]\(\sin B\)[/tex]:
[tex]\[
\cos^2 B + \sin^2 B = 1
\][/tex]
[tex]\[
(0.55)^2 + \sin^2 B = 1
\][/tex]
[tex]\[
0.3025 + \sin^2 B = 1
\][/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]
4. Sum up [tex]\(\sin A\)[/tex] and [tex]\(\sin B\)[/tex]:
[tex]\[
\sin A + \sin B \approx 0.5578 + 0.8352 = 1.393
\][/tex]
Therefore, the correct answer is:
D. 1.38
