To determine the length of the other leg of the triangle given that the leg opposite to angle \( \theta \) has a length of 12, we can use the Pythagorean Theorem.
The Pythagorean Theorem states:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the legs of the triangle. In this case, we have:
- Leg opposite \( \theta \) (let's call it \( a \)): 12
- The other leg (let's call it \( b \)): unknown
- The hypotenuse (let's call it \( c \)): since the right triangle is inscribed in the unit circle, \( c = 1 \) (the length of the radius of the unit circle).
Plugging in the known values, we get:
\[ 12^2 + b^2 = 1^2 \]
Calculating \( 12^2 \):
\[ 144 + b^2 = 1 \]
Now, rearranging to solve for \( b^2 \):
\[ b^2 = 1 - 144 \] \[ b^2 = -143 \]
Since \( b^2 \) cannot be negative in the context of real numbers, it indicates that a leg of length 12 cannot exist in a right triangle inscribed in the unit circle (where the hypotenuse must be 1). Therefore, the given length of the opposite leg (12) exceeds the hypotenuse (1) in a valid right triangle on the unit circle.
Thus, under these constraints, no valid length for the other leg exists.
In conclusion, the provided information leads to a contradiction, and the other leg cannot be computed as specified.