Answers:
16. sin(A) = a/12
17. tan(A) = a/9
18. sin(B) = 3/4 (reduced from 9/12)
19. cos(B) = a/12
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Step-by-step explanation:
The sine of an angle is equal to the ratio of the opposite side over the hypotenuse. For reference angle A, the opposite side is BC = a. The hypotenuse is always the longest side. The hypotenuse is AB = 12.
sin(angle) = opposite/hypotenuse
sin(A) = BC/AB
sin(A) = a/12
Similarly,
sin(B) = opposite/hypotenuse = AC/AB = 9/12 = 3/4.
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Tangent is the ratio of opposite over adjacent
tan(angle) = opposite/adjacent
tan(A) = BC/AC
tan(A) = a/9
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Cosine is the ratio of adjacent over hypotenuse
cos(angle) = adjacent/hypotenuse
cos(B) = BC/AB
cos(B) = a/12
we see that this is the same result as sin(A). It turns out that sin(A) = cos(B) if and only if A+B = 90. In other words, the sin(A) = cos(B) when A and B are the acute angles of a right triangle.
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Side note: you could use the Pythagorean theorem to solve for the value of 'a'. All of this assumes that triangle ACB is a right triangle.