The perimeter of a $\Delta ABC$ is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

Quick Solution

Given: Perimeter = 6 × Arithmetic Mean of sin A, sin B, sin C

Using Law of Sines: \( \frac{a}{\sin A} = 2R \), and a = 1 ⇒ \( \sin A = \frac{1}{2R} \)

Assume \( A = 30^\circ \Rightarrow \sin A = \frac{1}{2} \Rightarrow 2R = 2 \)

⇒ b = 2 sin B, c = 2 sin C

Perimeter = \( 1 + b + c = 1 + 2 \sin B + 2 \sin C \)

Mean = \( \frac{\sin A + \sin B + \sin C}{3} \)

Check: \( 1 + 2\sin B + 2\sin C = 6 \cdot \frac{1/2 + \sin B + \sin C}{3} \) ✅

✅ Final Answer: 30°