trig
posted by trig .
ok I'm trying to solve for an angle knowing adjacent and opposite of a triangle
so I do tangent
by definition
tan theta = a^1 o
how do I algebraically solve for theta

Do you mean adjacent SIDE? and opposite ANGLE?
If so, use the law of sines to get the angle B that is opposite the adjacent side (b).
Let A be the angle for which you know the adjacent side length, a. Then
a/sin A = b/sin B
solve for sin B and angle B
Then use C = 180  A  B degrees
There will be two possible values of B if obtuse angles are allowed. If so, you have been given the ambiguous sidesideangle (SSA) set of known quantities.