In any `DeltaABC.a=18,b=24` and c=30, then what is `SinC` equal?

Options

A. `1//4`
B. `1//3`
C. `1//2`
D. 1

Answer

To find the value of \( \sin C \) in triangle \( ABC \) where \( a = 18 \), \( b = 24 \), and \( c = 30 \), we can use the cosine rule and the relationship between sine and cosine.

### Step-by-Step Solution:

1. **Identify the Sides**: We have the sides of the triangle as follows:
- \( a = 18 \)
- \( b = 24 \)
- \( c = 30 \)

2. **Use the Cosine Rule**: The cosine rule states that:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
We will substitute the values of \( a \), \( b \), and \( c \) into this formula.

3. **Calculate \( a^2 \), \( b^2 \), and \( c^2 \)**:
- \( a^2 = 18^2 = 324 \)
- \( b^2 = 24^2 = 576 \)
- \( c^2 = 30^2 = 900 \)

4. **Substitute into the Cosine Rule**:
\[ \cos C = \frac{324 + 576 - 900}{2 \times 18 \times 24} \]

5. **Calculate the Numerator**:
\[ 324 + 576 - 900 = 900 - 900 = 0 \]

6. **Calculate the Denominator**:
\[ 2 \times 18 \times 24 = 36 \times 24 = 864 \]

7. **Calculate \( \cos C \)**:
\[ \cos C = \frac{0}{864} = 0 \]

8. **Find \( \sin C \)**: We know that:
\[ \sin^2 C + \cos^2 C = 1 \]
Since \( \cos C = 0 \):
\[ \sin^2 C = 1 - 0^2 = 1 \]
Therefore, taking the square root:
\[ \sin C = \sqrt{1} = 1 \]

### Final Answer:
Thus, \( \sin C = 1 \).