If we have tan x = √3/3, I would expect ...

Well, for starters, √3 > 1, and -1 ≤ sin(x) ≤ 1. You're also forgetting some of the rules of trig; tangent, sine, and cosine are all bound by some relationship.

If it helps, draw the right triangle. If the tan(x) is 1/√3 (I simplified it). Then the opposite side must be 1, and the adjacent side must be √3. That makes the hypotenuse:

h² = 1² + (√3)²
h² = 4
h = 2

So opposite is 1, adjacent is √3, and hypotenuse is 2. Then...

sin(x) = opposite/hypotenuse = 1/2
cos(x) = adjacent/hypotenuse = √(3)/2

Tan(x) is equal to sin(x)/cos(x).

And it is true that tan(x) is equal to √3/3

But you can't assume that sin(x) is √3 and cos(x) is 3


Sin(x)/cos(x) results in √3/3, but the individual values of each are not equal to those.


Like 2/4=1/2

But is 2=1?

No

tangent is a RATIO of sine to cosine.

For example, suppose a/b = 1/2. You don't know the actual values of a and b; just that their ratio is 1 to 2.
a could be 1, in which case b = 2.
a could be 2, in which case b = 4
etc.