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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

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.

Tangent is the ratio of opposite over adjacent

tan(angle) = opposite/adjacent

tan(A) = BC/AC

tan(A) = a/9

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.

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.

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