In , let the sides opposite be . Given find angle . A. B. C. D.

Triangle Solving Solution – Problem 22: find angle

In $\triangle ABC$ , let the sides opposite $A,B,C$ be $a,b,c$ . Given $\sin(B-C)+\sin A=\frac{3}{2},\qquad b=\sqrt{3}\,c,$ find angle $C$ .

A. $\frac{\pi}{6}$ B. $\frac{\pi}{3}$ C. $\frac{\pi}{4}$ D. $\frac{\pi}{2}$

Step 1. Since $\sin A=\sin(B+C)$ , we have $\sin(B-C)+\sin A=\sin(B-C)+\sin(B+C)=2\sin B\cos C=\frac{3}{2}.$

Step 2. From $b=\sqrt{3}\,c$ and the Law of Sines, $\frac{b}{c}=\frac{\sin B}{\sin C}$ , so $\sin B=\sqrt{3}\sin C$ .

Step 3. Substituting into Step 1 gives $2\sqrt{3}\sin C\cos C=\frac{3}{2}$ , i.e. $\sin 2C=\frac{\sqrt{3}}{2}$ .

Step 4. Thus $2C=\frac{\pi}{3}$ or $\frac{2\pi}{3}$ , so $C=\frac{\pi}{6}$ or $\frac{\pi}{3}$ .

Step 5. If $C=\frac{\pi}{3}$ , then $\sin B=\sqrt{3}\sin C=\frac{3}{2}>1$ , impossible. Hence $C=\frac{\pi}{6}$ , so the correct choice is A.

Chain A: Law of Sines approach

Set up side-angle relations [2 pts]: States and correctly advances the key derivation steps
Substitute and simplify [2 pts]: Substitutes correctly and simplifies accurately
Handle multiple cases / admissibility [1 pt]: Considers branches and rejects invalid cases
Conclusion and verification [1 pt]: States the conclusion and checks against constraints
Final answer [1 pt]: Gives the correct final result (for multiple-choice, include the option letter)

Copies formulas without concrete substitution or derivation
Guesses the answer / provides only a conclusion with no reasoning
Uses an approach incompatible with the problem conditions, leading to an invalid conclusion

Computation error [-1]: Incorrect algebraic/trigonometric manipulation
Logical gap [-1]: Missing a key equivalence step or a necessary condition check
Nonstandard final statement [-1]: Missing units/range/option letter or wrong answer format