If A, B, C are interior angles of `DeltaABC` such that `(cos A + cos B + cos C)^(2)+(sin A + sin B + sin C)^(2)=9`, then number of possible triangles is

Options

A. 0
B. 1
C. 3
D. infinite

Answer

To solve the problem, we need to analyze the given equation involving the angles of triangle ABC:

\[ (\cos A + \cos B + \cos C)^2 + (\sin A + \sin B + \sin C)^2 = 9 \]

### Step 1: Understanding the Angles of a Triangle
In any triangle, the sum of the interior angles A, B, and C is always 180 degrees:
\[ A + B + C = 180^\circ \]

### Step 2: Expressing the Sine and Cosine Sums
Using the properties of sine and cosine, we can express the sums:
\[ \cos A + \cos B + \cos C = \cos A + \cos B + \cos(180^\circ - A - B) = \cos A + \cos B - \cos(A + B) \]
\[ \sin A + \sin B + \sin C = \sin A + \sin B + \sin(180^\circ - A - B) = \sin A + \sin B + \sin(A + B) \]

### Step 3: Using the Identity
We can use the identity for the sum of squares:
\[ x^2 + y^2 = r^2 \]
where \(x = \cos A + \cos B + \cos C\) and \(y = \sin A + \sin B + \sin C\). The equation simplifies to:
\[ x^2 + y^2 = 9 \]

### Step 4: Analyzing the Maximum Values
The maximum value of \(x\) and \(y\) can be derived from the fact that the maximum value of \(\cos\) and \(\sin\) functions is 1. Thus:
\[ \cos A + \cos B + \cos C \leq 3 \quad \text{and} \quad \sin A + \sin B + \sin C \leq 3 \]

### Step 5: Finding the Condition for Equality
For the equation \(x^2 + y^2 = 9\) to hold true, both \(x\) and \(y\) must equal 3. This occurs when:
\[ \cos A + \cos B + \cos C = 3 \quad \text{and} \quad \sin A + \sin B + \sin C = 0 \]

### Step 6: Conclusion about the Angles
The only way for \(\cos A + \cos B + \cos C = 3\) is if \(A = B = C = 60^\circ\). Therefore, triangle ABC must be an equilateral triangle.

### Step 7: Number of Possible Triangles
Since the problem states that there are no restrictions on the side lengths of the triangle, we can conclude that there are infinitely many equilateral triangles that can be formed.

Thus, the final answer is:
\[ \text{Number of possible triangles} = \text{Infinite} \]

---