1. In a triangle ABC, if \(\sin A + \sin B + \sin C = \frac{3\sqrt{3}}{2}\), then the triangle can be equilateral.
2. In a triangle ABC, if \(\cos A + \cos B + \cos C = \frac{3}{2}\), then the triangle can be equilateral.
Which of the statements given above is/are correct?
Analysing Statement 1: Sine Sum Condition
Statement 1 says: In a triangle ABC, if \(\sin A + \sin B + \sin C = \frac{3\sqrt{3}}{2}\), then the triangle can be equilateral.
We know that for the angles \(A, B, C\) of any triangle, the sum of their sines has a maximum value.
- The identity for the sum of sines in a triangle is: \(\sin A + \sin B + \sin C = 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}\).
- It is a known result that the maximum value of \(4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}\) for the angles of a triangle is \(\frac{3\sqrt{3}}{2}\).
- This maximum value is achieved only when the triangle is equilateral, meaning \(A = B = C = \frac{\pi}{3}\) (or 60 degrees).
- Let's verify this for an equilateral triangle: \(\sin 60^{\circ} + \sin 60^{\circ} + \sin 60^{\circ} = \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}\).
- Therefore, if \(\sin A + \sin B + \sin C\) equals its maximum value \(\frac{3\sqrt{3}}{2}\), the triangle must be equilateral. Statement 1 is correct.
Analysing Statement 2: Cosine Sum Condition
Statement 2 says: In a triangle ABC, if \(\cos A + \cos B + \cos C = \frac{3}{2}\), then the triangle can be equilateral.
Similar to the sine sum, the cosine sum of the angles of a triangle also has a known range and condition for its maximum.
- For the angles \(A, B, C\) of a triangle, we have the identity: \(\cos A + \cos B + \cos C = 1 + 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}\).
- It's a known result that the sum \(\cos A + \cos B + \cos C\) is always less than or equal to \(\frac{3}{2}\).
- The maximum value of \(\cos A + \cos B + \cos C\) is \(\frac{3}{2}\).
- This maximum value occurs precisely when the triangle is equilateral, i.e., \(A = B = C = \frac{\pi}{3}\) (or 60 degrees).
- Let's check this for an equilateral triangle: \(\cos 60^{\circ} + \cos 60^{\circ} + \cos 60^{\circ} = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}\).
- Thus, if \(\cos A + \cos B + \cos C = \frac{3}{2}\), the triangle must be equilateral. Statement 2 is also correct.
Conclusion on Statements
Since both Statement 1 and Statement 2 correctly state conditions that imply a triangle must be equilateral, both statements are correct.
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