Question

With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If \( a = 1 \), then \( \angle A = \)

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{6} \)

Hint:

In problems involving the perimeter and angles of a triangle, using trigonometric identities and symmetry can simplify calculations significantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We use the Sine Rule: \( \frac{\text{a}}{\sin A} = \frac{\text{b}}{\sin B} = \frac{\text{c}}{\sin C} = 2\text{R} \). Perimeter is \( \text{a} + \text{b} + \text{c} = 2\text{R}(\sin A + \sin B + \sin C) \).
Step 2: Detailed Explanation:
Arithmetic mean of sines = \( \frac{\sin A + \sin B + \sin C}{3} \).
Given: \( \text{a} + \text{b} + \text{c} = 6 \left( \frac{\sin A + \sin B + \sin C}{3} \right) \).
\( 2\text{R}(\sin A + \sin B + \sin C) = 2(\sin A + \sin B + \sin C) \).
This implies \( 2\text{R} = 2 \Rightarrow \text{R} = 1 \).
From Sine Rule: \( \text{a} = 2\text{R}\sin A \).
Substituting \( \text{a} = 1 \) and \( 2\text{R} = 2 \):
\( 1 = 2 \sin A \Rightarrow \sin A = 1/2 \).
\( A = \pi/6 \).
Step 3: Final Answer:
The angle \( A \) is \( \frac{\pi}{6} \).