1. **Problem Statement:** Given a right triangle with angle $C = 90^\circ$, angle $B = 60^\circ$, side $AB = 12$, side $AC = b$, and side $BC = a$, find the relationship between the sides using $\cos 60^\circ$. 2. **Recall the cosine definition:** For an angle $\theta$ in a right triangle, $\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$. 3. **Identify sides relative to angle $B=60^\circ$:** - Adjacent side to angle $B$ is $BC = a$. - Hypotenuse is $AB = 12$. 4. **Write the cosine formula for angle $B$:** $$\cos 60^\circ = \frac{a}{12}$$ 5. **Use the known value:** $$\cos 60^\circ = \frac{1}{2}$$ 6. **Set up the equation:** $$\frac{1}{2} = \frac{a}{12}$$ 7. **Solve for $a$: ** Multiply both sides by 12: $$12 \times \frac{1}{2} = 12 \times \frac{a}{12}$$ $$\cancel{12} \times \frac{1}{2} = a \times \cancel{\frac{12}{12}}$$ $$6 = a$$ 8. **Final answer:** $$a = 6$$ This means side $BC$ has length 6 units.