The area of `triangleABC,=24sqrt(3)`, side a=6, annd side b=16. the value of `angleC` is

Options

A. `30^(@)`
B. `30^(@)` or `150^(@)`
C. `60^(@)`
D. `60^(@)` or `120^(@)`

Explanation

To find the value of angle C in triangle ABC, given the area and the lengths of two sides, we can use the formula for the area of a triangle in terms of two sides and the included angle. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Area of triangle ABC = \( 24\sqrt{3} \)
- Side \( a = 6 \) (opposite angle A)
- Side \( b = 16 \) (opposite angle B)
- We need to find angle \( C \).

2. **Use the Area Formula**:
The area \( A \) of a triangle can also be expressed as:
\[ A = \frac{1}{2} \times a \times b \times \sin(C) \]
Substituting the known values into the formula gives:
\[ 24\sqrt{3} = \frac{1}{2} \times 6 \times 16 \times \sin(C) \]

3. **Simplify the Equation**:
Calculate \( \frac{1}{2} \times 6 \times 16 \):
\[ \frac{1}{2} \times 6 \times 16 = 48 \]
Therefore, we have:
\[ 24\sqrt{3} = 48 \sin(C) \]

4. **Solve for \( \sin(C) \)**:
Rearranging the equation to isolate \( \sin(C) \):
\[ \sin(C) = \frac{24\sqrt{3}}{48} = \frac{\sqrt{3}}{2} \]

5. **Find Angle \( C \)**:
The value of \( C \) can be found by taking the inverse sine:
\[ C = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
From trigonometric values, we know:
\[ C = 60^\circ \quad \text{or} \quad C = 120^\circ \]
(since sine is positive in both the first and second quadrants).

6. **Conclusion**:
The possible values for angle \( C \) are \( 60^\circ \) and \( 120^\circ \).

### Final Answer:
The value of angle C is \( 60^\circ \) or \( 120^\circ \).

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