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In right triangle [tex]\(ABC\)[/tex], [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are complementary angles, and [tex]\(\sin A = \frac{8}{9}\)[/tex]. What is [tex]\(\cos B\)[/tex]?
A. [tex]\(\frac{8 \sqrt{17}}{17}\)[/tex] B. [tex]\(\frac{8}{9}\)[/tex] C. [tex]\(\frac{\sqrt{17}}{9}\)[/tex] D. [tex]\(\frac{\sqrt{17}}{8}\)[/tex]
To determine [tex]\( \cos B \)[/tex] in a right triangle [tex]\( ABC \)[/tex], where [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are complementary angles (meaning their sum is [tex]\(90^\circ\)[/tex]), and given that [tex]\( \sin A = \frac{8}{9} \)[/tex], follow these steps:
1. Understand Complementary Angles: Since [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are complementary, [tex]\(\angle A + \angle B = 90^\circ\)[/tex]. This relationship implies that [tex]\(\cos B = \sin A\)[/tex]. This is a crucial property in trigonometry for complementary angles that states: in a right triangle, the sine of one angle is equal to the cosine of the other angle.
2. Given Information: We know from the problem that [tex]\( \sin A = \frac{8}{9} \)[/tex]. Using the complementary angle property: [tex]\[
\cos B = \sin A
\][/tex]
3. Substitute the Known Value: Replace [tex]\(\sin A\)[/tex] with the given value: [tex]\[
\cos B = \frac{8}{9}
\][/tex]
Therefore, by substituting the given values and using the properties of complementary angles in a right triangle, we conclude that: [tex]\[
\cos B = \frac{8}{9}
\][/tex]
So, the correct answer is: [tex]\[
\boxed{\frac{8}{9}}
\][/tex]
Hence, the correct option is: [tex]\[
\boxed{B}
\][/tex]
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