Hi Jonathan. That question has no answer. Look at a sine wave: all the values of f(x) = sin(x) vary between -1 and 1, because that's the range of the sine function. Your question says sin(theta) = -3, so that's not possible.

But, if you meant to say sin(theta) = -0.3, this is how you would solve it.

The sine is the ratio of the opposite side to the hypotenuse. Consider a "unit circle" (that's a circle with radius 1). The radius is the hypotenuse, so in this case, the opposite side is -0.3. That's the "y" value on a normal graph, which means either quadrant III or quadrant IV. There are in fact two angles (two values of theta) in which sin(theta) = -0.3.

Now if the hypotenuse of a right triangle is 1 and the opposite side is 0.3 (we can ignore the negative sign for the moment), then Pythagoras can tell us the length of the adjacent (x) side. 0.3² + x² = 1². When I use my calculator to solve this, I get x = ± 0.95.

Now we know:
y = opposite = -0.3 and
x = adjacent = +/- 0.95 and
r = hypotenuse = 1

We can calculate any of the trigonometric ratios, just be considering their definitions:
tan = opposite / adjacent.
cos = adjacent / hypotenuse.
cot = adjacent / opposite.
csc = hypotenuse / opposite.
sec = hypotenuse / adjacent.

So you can find all the values. Just remember that some of these will have two answers (because of the +/- on the adjacent length) and some will only have one value (those that don't use the adjacent side).

Hope this helps,
Stephen La Rocque.