Hi Jocelyn,

By your notation and the fact that you are to use the Pythagorean Theorem only I think you have a right triangle with angles A, B and C where C is the right angle and sides of length a, b and c as in the diagram below.

If this is correct the the Pythagorean Theorem tell us that

a² + b² = c²

and hence if you know two of a, b and c you can find the third. For example if a = 9 and c = 15 then

9² + b² = 15² or 81 + b² = 225
Hence b² = 225 - 81 = 144 and thus b = 12.

Similarly if you know the definitions of the trig functions you can solve for the trig functions in your examples. For example if sinA = 12/15 then a = 12 and c = 15. From the calculation above using Pythagoras Theorem you know that b = 12 and thus

secB = c/a = 15/9 = 5/3.

I hope this helps,
Penny