Triangle Solving Solution – Problem 25: Find the sine of the smallest angle

Question

In a right triangle, the cosines of its three interior angles form an arithmetic progression. Find the sine of the smallest angle.

A. $\frac{3}{5}$ B. $\frac{4}{5}$ C. $\frac{\sqrt{5}}{5}$ D. $\frac{2\sqrt{5}}{5}$

Step-by-step solution

Step 1. Let $A<B<C=\frac{\pi}{2}$ . Then $\cos C=0$ , and the arithmetic-progression condition gives $\cos C+\cos A=2\cos B$ , i.e. $\cos A=2\cos B$ .

Step 2. Since $A+B=\frac{\pi}{2}$ , we have $\cos B=\sin A$ . Thus $\cos A=2\sin A$ .

Step 3. Hence $\tan A=\frac{1}{2}$ . With $A\in\left(0,\frac{\pi}{2}\right)$ , we obtain $\sin A=\frac{\sqrt{5}}{5}$ .

Step 4. Therefore the correct choice is C.

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 [2 pts]: Considers branches and rejects invalid cases
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 25: Find the sine of the smallest angle