To solve this problem, we can apply the Pythagorean theorem, which states that in a right triangle:
\[ a^2 + b^2 = c^2 \]
where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse.
In this case, we know:
- The length of the leg opposite \(θ\) (let's call it \(a\)) is 12. So, \(a = 12\).
- The hypotenuse \(c\) for a triangle inscribed in the unit circle is 1 (since the hypotenuse equals the radius of the unit circle).
We need to find the length of the other leg (let's call it \(b\)).
Substituting our known values into the Pythagorean theorem:
\[ 12^2 + b^2 = 1^2 \]
Calculating \(12^2\) and \(1^2\):
\[ 144 + b^2 = 1 \]
Now, isolate \(b^2\):
\[ b^2 = 1 - 144 \] \[ b^2 = -143 \]
At this point, we see that there must be a misunderstanding since \(b^2\) cannot be negative in a real triangle formed in a unit circle. Therefore, it is impossible for a leg of length 12 to exist in a right triangle located in a unit circle, whose hypotenuse (the circle's radius) equals 1.
Please double-check the problem's initial conditions or parameters. As presented, they cannot correspond to a valid triangle in a unit circle.