Question

The perimeter of a triangle \( \triangle ABC \) is 6 times the arithmetic mean of the sines of its angles. If the side \( a \) is 1, then the angle \( A \) is:

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

Hint:

Use the Law of Sines and the given conditions to express the perimeter and the angles of the triangle in terms of one another.

Answer 1

The Correct Option is A

Step 1: Understand the given condition.
We are given that the perimeter of the triangle is 6 times the arithmetic mean of the sines of its angles. Let the angles of the triangle be \( A \), \( B \), and \( C \). The perimeter \( P \) of the triangle is the sum of the sides, which is: \[ P = a + b + c. \] The arithmetic mean of the sines of the angles is: \[ \frac{\sin A + \sin B + \sin C}{3}. \]

Step 2: Use the relationship between perimeter and the sines of the angles.
According to the given condition, we have: \[ P = 6 \times \frac{\sin A + \sin B + \sin C}{3}. \] Since we are given that the side \( a = 1 \), we need to find the angle \( A \).

Step 3: Use the Law of Sines to express the sides.
The Law of Sines states that: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. \] Since \( a = 1 \), we have: \[ \frac{1}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. \]

Step 4: Simplify the equation.
We can express the perimeter \( P \) in terms of \( \sin A \), \( \sin B \), and \( \sin C \), using the fact that \( a = 1 \). After simplifying the equation, we get the result that the angle \( A \) must be \( \frac{\pi}{6} \).

Step 5: Conclusion.
Thus, the angle \( A \) is \( \frac{\pi}{6} \), and the correct answer is (a).