In `Delta`ABC, if (Sin A + SinB + SinC) (SinA + SinB - SinC) = 3SinA SinB then C =

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Write down the given equation
We start with the equation given in the problem:
\[ (\sin A + \sin B + \sin C)(\sin A + \sin B - \sin C) = 3 \sin A \sin B \]

### Step 2: Use the sine rule
From the sine rule, we know:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
This implies:
\[ \sin A = \frac{a}{2R}, \quad \sin B = \frac{b}{2R}, \quad \sin C = \frac{c}{2R} \]

### Step 3: Substitute sine values into the equation
Now, substituting these values into the original equation:
\[ \left(\frac{a}{2R} + \frac{b}{2R} + \frac{c}{2R}\right)\left(\frac{a}{2R} + \frac{b}{2R} - \frac{c}{2R}\right) = 3 \left(\frac{a}{2R}\right)\left(\frac{b}{2R}\right) \]

### Step 4: Simplify the equation
This simplifies to:
\[ \left(\frac{a + b + c}{2R}\right)\left(\frac{a + b - c}{2R}\right) = \frac{3ab}{(2R)^2} \]
Multiplying both sides by \( (2R)^2 \):
\[ (a + b + c)(a + b - c) = 3ab \]

### Step 5: Expand the left-hand side
Expanding the left-hand side:
\[ (a + b)^2 - c^2 = 3ab \]
This can be rewritten as:
\[ a^2 + b^2 + 2ab - c^2 = 3ab \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ a^2 + b^2 - c^2 = ab \]

### Step 7: Use the cosine rule
Using the cosine rule, we know:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting our result:
\[ \cos C = \frac{ab}{2ab} = \frac{1}{2} \]

### Step 8: Find angle C
The angle whose cosine is \( \frac{1}{2} \) is:
\[ C = 60^\circ \]

### Conclusion
Thus, the value of angle \( C \) is:
\[ \boxed{60^\circ} \]

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