Question

If A, B, C are angles of a triangle such that $\cot \frac{A}{2} = 3 \tan \frac{C}{2}$ then $\sin A, \sin B, \sin C$ are in}

• A. Arithmetic Progression
• B. Geometric Progression
• C. Harmonic Progression
• D. Arithmetic Geometric Progression

Hint:

If sides $a, b, c$ are in A.P., then $\sin A, \sin B, \sin C$ are always in A.P. due to the Sine Rule.

Answer 1

The Correct Option is A

Step 1: Concept
In a triangle, $A+B+C = 180^{\circ}$. Use the half-angle cotangent identity $\cot \frac{A}{2} = \sqrt{\frac{s(s-a)}{(s-b)(s-c)}}$.

Step 2: Meaning
Given $\cot \frac{A}{2} = 3 \tan \frac{C}{2}$, we can write $\cot \frac{A}{2} \cot \frac{C}{2} = 3$.

Step 3: Analysis
Using half-angle formulas: $\frac{s(s-a)}{(s-b)(s-c)} \cdot \frac{s(s-c)}{(s-a)(s-b)} = 3^{2}$ is not efficient. Instead, use $\frac{s}{s-b} = 3 \implies s = 3s - 3b \implies 2s = 3b \implies a+b+c = 3b \implies a+c = 2b$.

Step 4: Conclusion
Since $a+c = 2b$, the sides $a, b, c$ are in A.P. By the Sine Rule, $\sin A, \sin B, \sin C$ are also in A.P. Final Answer: (A)