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

Triangle Solving Solution – Problem 24: find

Question

In $\triangle ABC$ , let the sides opposite $A,B,C$ be $a,b,c$ . If $(a^{2}+c^{2}-b^{2})\tan B=\sqrt{3}ac,$ find $\cos 5B$ .

A. $\frac{1}{2}$ B. $\pm\frac{\sqrt{3}}{2}$ C. $\frac{\sqrt{3}}{2}$ D. $\pm\frac{1}{2}$

Step-by-step solution

Step 1. By the Law of Cosines, $a^{2}+c^{2}-b^{2}=2ac\cos B$ . Substituting into the condition gives $2ac\cos B\tan B=\sqrt{3}ac$ .

Step 2. Cancelling $ac>0$ , we get $2\sin B=\sqrt{3}$ , so $\sin B=\frac{\sqrt{3}}{2}$ .

Step 3. Since $0<B<\pi$ , $B=\frac{\pi}{3}$ or $B=\frac{2\pi}{3}$ .

Step 4. Then $\cos 5B=\cos\frac{5\pi}{3}=\frac12$ or $\cos\frac{10\pi}{3}=\cos\frac{4\pi}{3}=-\frac12$ . Hence $\cos 5B=\pm\frac12$ , so the correct choice is D.

Final answer

Marking scheme

1. Checkpoints (max 7 pts total)

Chain A: Combined Law of Sines and Cosines 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)

2. Zero-credit items

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

3. Deductions

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

Triangle Solving – Problem 24: find